Estadistica Aplicada a La Ingenieria Ambiental
-
Upload
franco-follano-p -
Category
Documents
-
view
233 -
download
1
Transcript of Estadistica Aplicada a La Ingenieria Ambiental
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 1/846
QEK]A_YKN@N @Q^ÙELG@ NA IN% HQÀ_AX
KEY^K^Q^L NA KEBAEKA_Ç@ [ ^AIELCLBÇ@
NAZ@_^@GAE^L NA KEBAEKA_Ç@ IK]KC [ @GOKAE^@C
GË^LNLY AY^@NÇY^KILY Z@_@ C@
KEBAEKA_Ç@ @GOKAE^@C [ C@ IKAEIK@
N_% FËI^L_ @NLCML PQA]ANL Q_K@Y
@BLY^L NA ?::=
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 2/846
Ilz{rkbf~ « ?::=% Gë~lnly Ay~`nçy~kily z r` c` Kebaekarç @gokae~ c { c` Ikaeik`%
Fëi~lr @nlcml Pqa}anl Qrç`y
Ay zrlzkan`n nac `q~lr% Pqan` faifl ac nazùyk~l pqa g`ri` c` ca{%
@n}ar~aeik`
Zrlfkokn` c` razrlnqiikùe na ay~a ckorl# `nagày na cly aypqag`y a kna`y lrkbke`cay nac`q~lr pqa ya f`cc`e ae ay~a ~av~l# {` ya` zlr gankly acai~rùekily# gaiàekily#
ml~lilzk`nl l na iq`cpqkar l~r` mlrg`# zqay~l pqa ~lnl ay~l zar~aeaia `c nlgkekl nac` zrlzkan`n ke~acai~q`c { ay~à zrl~abknl zlr c` ca{%
Z`r` ra}kylray# irç~kily l rayað`nlray ck~ar`rkly# ` pqkaeay ya cay `ykbea c` ~`ra` na
f`iar ra}kykleay ck~ar`rk`y na ay~` lor`# cl zqanae f`iar# zra}kl `iqarnl ile ac `q~lr%
Kgzrayl ae In% Hqàrax# Ifkfq`fq`# Gëvkil
Ckor`r{ lm Ilebrayy I`~`clbkeb ke Zqocki`~kle N`~`Fëi~lr @nlcml Pqa}anl Qrç`y
Ay~a ckorl mqa zqocki`nl ae ac Ke~area~ ae Aearl na ?::= zlr c` Okockl~ai` ]kr~q`c na c`
Qek}arykn`n @q~ùelg` na In% Hqàrax%
C` nkraiikùe acai~rùeki` nac ckorl ay8
f~~z8,,ok}kr%q`ih%gv,CkorlyAcai~rlekilyCkoray,Q@IH,q`::::0%znm
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 3/846
ILE^AEKNL
Zàbke`
Ke~rlnqiikùe k
I`zç~qcl 0 Ay~`nçy~ki` Nayirkz~k}` 0'0
Namkekikùe na ay~`nçy~ki`%' Zloc`ikùe { gqay~r`%' Ay~`nçy~ki` kenqi~k}` { na kemaraeik`%'Ay~`nçy~ki` nayirkz~k}`%' ]`rk`ocay ile~keq`y { nkyira~`y%' Gankn`y na ~aenaeik` iae~r`c%'Gankn`y na nkyzarykùe%' C` }`rk`oca `ca`~lrk` ay~`en`rkx`n` x%' C`y nay}k`ikleay nac zrlgankl%' Ac r`ebl%' Yaybl { dqr~lyky%' Nky~rkoqikleay na mraiqaeik`%' Nk br`g`y na ~`ccl {flh`%
I`zç~qcl ? Zrlo`okckn`n ?'0
Zrlo`okckn`n na mraiqaeik` rac`~k}`%' Zrlo`okckn`n yqoha~k}`%' @vklg`y { zrlzkan`nay oàyki`y na c` zrlo`okckn`n%' Nk`br`g`y na ]aee { àcbaor` na ilehqe~ly%' ^ëieki`y naile~al8 _abc` na zrlnqi~l z`r` z`ray lrnae`nly# c` rabc` na gqc~kzcki`ikùe gày baear`c#rabc` m`i~lrk`c# nk`br`g`y na àrolc# zargq~`ikleay { ilgoke`ikleay%' _abc` gqc~kzcki`~k}` z`r` a}ae~ly nazaenkae~ay a kenazaenkae~ay%' _abc` `nk~k}` z`r` a}ae~ly gq~qlyavicq{ae~ay { a}ae~ly el gq~qly avicq{ae~ay%'
I`zç~qcl 9 Nky~rkoqikùe Okelgk`c a Fkzarbalgë~rki` 9'0
@zcki`ikleay baear`cay na c` nky~rkoqikùe okelgk`c%' _ac`ikùe ae~ra c` nky~rkoqikùe elrg`c
{ c` nky~rkoqikùe okelgk`c%' _ac`ikùe ae~ra c` nky~rkoqikùe okelgk`c { c` nky~rkoqikùe naZlkyyle%' C` nky~rkoqikùe fkzarbalgë~rki`%' Yqzlykikleay { zrlzkan`nay na c` nky~rkoqikùefkzarbalgë~rki`%'
I`zç~qcl > Nky~rkoqikùe na Zlkyyle >'0
@zcki`ikleay na c` nky~rkoqikùe na Zlkyyle%' Ilenkikleay pqa ya rapqkarae z`r` `zcki`r c`nky~rkoqikùe na Zlkyyle%' Mqeikleay zrlo`okcçy~ki`y na c` mqeikùe na Zlkyyle%' @zcki`ikùena c` nky~rkoqikùe na Zlkyyle nae~rl na yqy zrlzkly ~ërgkely { ilgl qe` `zrlvkg`ikùe `c` nky~rkoqikùe okelgk`c%' Zrlzkan`nay na c` nky~rkoqikùe na Zlkyyle%' Zrlocag`y na c`
nky~rkoqikùe na Zlkyyle qy`enl ac zrlbr`g` Gkek~`o%
I`zç~qcl 1 Nky~rkoqikleay na Zrlo`okckn`n Ile~keq` 1'0
Mqeikùe na naeykn`n na zrlo`okckn`n na c` }`rk`oca `ca`~lrk` ile~keq` V %' Mùrgqc`mqen`gae~`c nac iàciqcl%' Nky~rkoqikùe elrg`c { yqy i`r`i~arçy~ki`y%' _ac`ikùe ae~ra c` iqr}`elrg`c { c` okelgk`c%' Àra`y o`hl c` iqr}` elrg`c%' Nky~rkoqikùe avzleaeik`c%' Nky~rkoqikùeB`gg`%' Nky~rkoqikùe Rakoqcc%' Ke~ar}`cly na ilemk`ex` z`r` ±%' Ay~`nçy~ki` na kemaraeik`8
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 4/846
~alrç` na naikykùe ay~`nçy~ki` { zrqao`y na fkzù~ayky%' Zrqao`y na fkzù~ayky ay~`nçy~ki`y%Fkzù~ayky eqc` &Fl8+ a fkzù~ayky `c~are`~k}`y &F08# F?8# F98+%' ^kzly na arrlray K &`cm`+ { KK&oa~`+%' Zrqao`y na fkzù~ayky el ~r`nkikle`cay qy`enl ac }`clr na c` zrlo`okckn`n z%' Zrqao`yna fkzù~ayky z`r` qel { nly zrlgankly zloc`ikle`cay &±0# { ±?+%' Zrqao`y na fkzù~ayky z`r`c`y nkmaraeik`y na nly zrlgankly zloc`ikle`cay &±0 ’ ±?+# z`r` gqay~r`y br`enay &e ≡ 9:+
qy`enl c` nky~rkoqikùe elrg`c# ile }`rk`ex`y ilelikn`y a kbq`cay &ω?
0 7 ω
?
?+%' Ke~ar}`cly nailemk`ex` z`r` nly zrlgankly zloc`ikle`cay%' Zrqao`y na fkzù~ayky a ke~ar}`cly na ilemk`ex` z`r` zrlzlrikleay%'
I`zç~qcl = Nky~rkoqikleay na ~ na Ay~qnk`e~a# HK Iq`nr`n` { M ='0
Zrlzkan`nay na c` nky~rkoqikùe na ~ na Ay~qnk`e~a%' Ke~ar}`cly na ilemk`ex` z`r` ac zrlgankl zloc ikle`c ±%' Zrqao` na fkzù~ayky z`r` ±%' Zrqao` na ~ z`ra`n` z`r` na~ai~ r nkmaraeik`yae~ra nly ~r`~`gkae~ly%' Zrqao` na ~ z`r` zrlo`r c` fkzù~ayky na nly zrlgankly# iq`enl c`y}`rk`ex`y yle kbq`cay%' Zrqao` na ~ z`r` zrlo`r c` fkzù~ayky na nly zrlgankly iq`enl c`y}`rk`ex`y yle naykbq`cay%' Gai`ekygly z`r` i`ciqc`r ac }`clr na z iq`enl ya f`iae zrqao`yna fkzù~ayky el ~r`nkikle`cay%' Ke~ar}`cly na ilemk`ex` { zrqao`y na fkzù~ayky ile c` HKiq`nr`n`# &ς ?+%' @zcki`ikùe na c` HK iq`nr`n` ae iq`e~l ` c` zrqao` na olen`n na `hqy~ailgz`r`enl c`y mraiqaeik`y loyar}`n`y { c`y mraiqaeik`y ~aùrki`y%' Nky~rkoqikùe M { yq`zcki`ikùe ae c` ilgz`r`ikùe na }`rk`ex`y gqay~r`cay%'
I`zç~qcl 2 @eàckyky na ]`rk`ex` 2'0
Nkyaðly na `eàckyky na }`rk`ex` ilgzca~`gae~a `ca`~lrkx`nly { nkyaðl na oclpqay`ca`~lrkx`nly%' Gë~lnl na ilgz`r`ikleay gøc~kzcay z`r` y`oar iq`cay zloc`ikleay ylekbq`cay { iq`cay yle naykbq`cay%' @eàckyky na }`rk`ex` na nkyaðl na oclpqay
`ca`~lrkx`nly%' Yqzlykikleay nac glnacl na oclpqay `ca`~lrkly ilgzca~ly%' @eàckyky na}`rk`ex` ae nly yae~knly%' Ke~ar`iikùe ile @EL]@ na nly m`i~lray%' @eàckyky na }`rk`ex`ae ~ray yae~knly8 nkyaðl ilgzca~`gae~a `ca`~lrkl%' Ke~ar`iikùe ile @EL]@ na nkyaðlym`i~lrk`cay na ~ray ic`ykmki`ikleay%' Ahagzcly na @EL]@ qy`enl ac zrlbr`g` Gkek~`o%'
I`zç~qcl 6 _abraykùe Ckea`c Ykgzca { Gøc~kzca 6'0
Yqzlykikleay nac glnacl na rabraykùe ckea`c%' Aiq`ikleay elrg`cay z`r` i`ciqc`r acke~ariaz~l ae c` lrnae`n` ` { c` zaenkae~a o na c` iqr}` l cçea` na rabraykùe%' Ilamkikae~a nana~argke`ikùe _ ? na c` gqay~r` pqa ay~kg` ` π
? ac ilamkikae~a na na~argke`ikùe
zloc`ikle`c%' Ilamkikae~a na ilrrac`ikùe _ na c` gqay~r` pqa ay~kg` ` π# ac ilamkikae~a nailrrac`ikùe zloc`ikle`c%' Ke~ar}`cl na ilemk`ex` z`r` ac ilamkikae~a zloc`ikle`c ο ilgzleae~a na c` cçea` na rabraykùe ±[sV 7 ΰ $ ο V # ay~kg`nl zlr o# c` zaenkae~a na c`cçea`%' Ke~ar}`cl na ilemk`ex` z`r` ac z`ràga~rl zloc`ikle`c ΰ# ac ke~ariaz~l na c`lrnae`n` na c` cçea` na rabraykùe ±[sV 7 ΰ $ ο V # iq{l ay~kg`nlr ay `%' Fkzù~ayky eqc` naFl8ο 7 οl ile~r` c`y fkzù~ayky `c~are`~k}`y na F08ο 3 0 { F?8ο 5 0%' Fkzù~ayky eqc` na Fl8ΰ 7 ΰl ile~r` c`y fkzù~ayky `c~are`~k}`y na F08ΰ ≪ ΰl# F?8ΰ 5 ΰl# { na F98 ΰ 3 ΰl%' Ke~ar}`clna ilemk`ex` z`r` ±[sV na c` cçea` zloc`ikle`c ay~kg`n` zlr [ %' _abraykùe { ilrrac`ikùe
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 5/846
gøc~kzca%' Gë~lnly z`r` }`ckn`r ac glnacl na rabraykùe ckea`c ykgzca { gøc~kzca8 ` ~r`}ëy naay~`nçy~ki` na kemaraeik`y { ` ~r`}ëy nac `eàckyky bràmkil na cly rayknq`cay ay~`en`rkx`nly%Zrliankgkae~l na rabraykùe gøc~kzca qy`enl ac zrlbr`g` Gkek~`o%'
I`zç~qcl ; _abraykùe Zlckelgk`c ;'0
Glnacly zlckelgk`cay na yabqenl lrnae &d 7 ?+ ile qe` }`rk`oca kenazaenkae~a%' Glnacl na zlckelgkly na ~ariar lrnae &d 7 9+# ile qe` }`rk`oca kenazaenkae~a%' Glnacl na yabqenllrnae &iq`nrà~kil+ ile ke~ar`iikùe%' Glnacl zlckelgk`c &na yabqenl lrnae l iq`nrà~kil+#ile ~ray }`rk`ocay kenazaenkae~ay ile ke~ar`iikùe%' A}`cq`ikùe na cly glnacly narabraykùe%' Zrqao` ay~`nçy~ki` z`r` ilgz`r`r c` yqg` na cly iq`nr`nly nac arrlr &YYa+ nai`n` glnacl zrlo`nl# z`r` y`oar iq`c glnacl ay yqzarklr%' Glnacly na rabraykùe el ckea`cay{ na rabraykùe clbçy~ki`%' Glnacly na rabraykùe avzleaeik`cay z`r`gë~rkily# ile qe` ylc`}`rk`oca kenazaenkae~a%' Zrliankgkae~ly z`r` c` Knae~kmki`ikùe na }`clray `~çzkily av~ragly%Nk`beùy~kil { gk~kb`ikùe na gqc~kilckea`ckn`n%' Gankn`y z`r` ilrrabkr gqc~kilckea`ckn`nya}ar`%' Ahagzcly na zrlocag`y na rabraykùe zlckelgk`c qy`enl ac zrlbr`g` na
ilgzq~`nlr` Gkek~`o%' @q~lilrrac`ikùe ae n`~ly na yarkay na ~kagzl%' Fa~arlyian`y~kikn`n {flglyian`y~kikn`n%' Zrqao` na Rfk~a z`r` ac zrlocag` na fa~arlyian`y~kikn`n%'
I`zç~qcl 0: Ay~`nçy~ki` el Z`r`gë~rki`% Ac glnacl na Nky~rkoqikùena @EL]@ Ckora 0:'0
]ae~`h`y na cly gë~lnly el z`r`gë~rkily%' Nay}ae~`h`y na cly gë~lnly el z`r`gë~rkily%'Zrqao` na F na Drqyd`c'R`ccky z`r` `eàckyky na }`rk`ex` zlr r`ebly%' Zrqao`y na fkzù~aykyile c`y mqeikleay el z`r`gë~rki`y%' Zrliankgkae~ly na zrqao`y na Drqyd`c'R`ccky z`r`@EL]@ ykgzca%' Zrqao`y na fkzù~ayky el ~r`nkikle`cay# z`r` c` zrqao` na Drqyd`c'R`ccky#ay naikr# qy`enl ac }`clr na c` zrlo`okckn`n z%'
I`zç~qcl 00 Yarkay na ^kagzl 00'0 Ic`ykmki`ikùe na cly gl}kgkae~ly na c`y yarkay na ~kagzl%' ^aenaeik`y ` c`rbl zc`xl%'Ilgzleae~ay içickily na yarkay na ~kagzl%' ]`rk`ikleay ay~`ikle`cay%' ]`rk`ikùe krrabqc`r%'Gë~lnly z`r` aeile~r`r cçea`y na ~aenaeik`%' Cçea` na cly iq`nr`nly gçekgly { z`ràolc`yna cly iq`nr`nly gçekgly%'
I`zç~qcl 0? Yacaiikùe nac ^`g`ðl na c` Gqay~r` 0?'0 Nark}`ikùe na c` mùrgqc` z`r` ay~kg`r ac ~`g`ðl gày `zrlzk`nl na c` gqay~r` z`r` ac zrlgankl%' Yacaiikùe nac ~`g`ðl na c` gqay~r` z`r` nly zloc`ikleay%'
@zëenkiay
@zëenkia @ Cky~` na ^`oc`y Ay~`nçy~ki`y @zëenkia'@
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 6/846
@zëenkia O Okocklbr`mç` @zëenkia'O
@zëenkia I Z`zac na bràmki` @zëenkia'I
@zëenkia N Çenkia @zëenkia'N
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 7/846
k
Ke~rlnqiikùe
C` ay~`nçy~ki` { cly gë~lnly zrlo`okcçy~kily l ay~liày~kily hqab`e qe z`zac gq{
kgzlr~`e~a ae ~ln`y c`y m`yay nac ilgzlr~`gkae~l fqg`el% Ac qyl na c` zrlo`okckn`n {
na c` ay~`nçy~ki` ya f` av~aenknl# el ~`e ylcl ` c`y àra`y ~r`nkikle`cay qek}aryk~`rk`y l
ayilcày~ki`y# ykel ~`gokëe ` ~lnly cly i`gzly na c` kebaekarç`# c` `brkiqc~qr`# c`
oklclbç`# c` pqçgki`# c`y ilgqeki`ikleay# c` ailelgç # c` acai~rùeki`# c` gankike`# c`
mçyki`# c`y ikaeik`y zlcç~ki`y# c` zykilclbç`# c` yliklclbç`# c`y aeiqay~`y zlcç~ki`y# c`
gari`nl~aiek`# c` ailclbç`# c` ga~alrlclbç`# { yç yqiayk}`gae~a%
Ay~a ~av~l na zrlo`okckn`n { na ay~`nçy~ki`# ay~à nkyað`nl z`r` iqryly na zly~br`nl na c` Kebaekarç @gokae~ c { c` Ikaeik % Ay~a ckorl ay qe` ilgzkc`ikùe na
gày na ?1 ckorly na ramaraeik`y okocklbràmki`y na zrlo`okckn`n { na ay~`nçy~ki`
lrkae~`nly# el ~`e ylcl ` c` kebaekarç` `gokae~`c# ykel ~`gokëe ` c` kebaekarç` ae
baear`c# c` ailelgç`# c` pqçgki`# c` mçyki`# c` `brkiqc~qr`# c` gankike`# a~i% Ay~a ~av~l
iley~` na gày na 2:: zàbke`y pqa keicq{ae ileiaz~ly ~aùrkily# gqifly ahagzcly
zrài~kily { gqifly aharikikly% Ac `q~lr na ay~a ~av~l# yke ke~aeikleay na qm`e rya#
keicq{a qe nkyaðl na qe` mùrgqc` &pqa el `z`raia ae cly ckorly na ay~`nçy~ki`+ z`r`
ke~arzlc`r# g`eq`cgae~a# }`clray { ay~kg`r c` zrlo`okckn`n z%
Ae }arn`n# ac zrlzùyk~l na ay~a ~av~l ay ac na `{qn`r `c cai~lr ` ae~aenar cly
ileiaz~ly# kna`y { mqeikleay na c` zrlo`okckn`n { na c` ay~`nçy~ki` `zcki`nly `
zrlocag`y na c kebaekarç` `gokae~`c { ` c` ikaeik % Ay~a ~av~l naoarà yar ~`gokëe ø~kc
z`r` `pqaccly ay~qnklyly pqkaeay nayaae f`iar `zcki`ikleay na c` zrlo`okckn`n { na c`
ay~`nçy~ki` ` zrlocag`y na c` kebaekarç` ae ~ërgkely baear`cay# `yç ilgl ~`gokëe ` c`
ke}ay~kb`ikùe%
I`n` i`zç~qcl ya kekik` ile namkekikleay zar~keae~ay { ic`r`y# ~alrag`y {
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 8/846
kk
zrkeikzkly# ile g`~ark`c `oqen`e~a na bràmki`y# na g`~ark`cay nayirkz~k}ly { na
gqifly ahagzcly { aharikikly%
Zlr ahagzcl# ac I`zç~qcl 0 n` c` ke~rlnqiikùe ` c` ay~`nçy~ki` icàyki`% Ay~a
i`zç~qcl n` qe` ic`r` nky~keikùe ae~ra cl pqa ay qe` zloc`ikùe { qe` gqay~r`% Ay~a
i`zç~qcl f`oc`# `nagày# na ay~`nçy~ki` nayirkz~k}` { na nky~rkoqikleay na mraiqaeik`%
Gày `nac`e~a# ac I`zç~qcl ? f`oc` na c` ~alrç` na zrlo`okckn`n { ~lnl cl rac`ikle`nl
ile c` zrlo`okckn`n icàyki`% Nayzqëy# cly I`zç~qcly 9 { > f`oc`e na c`y nky~rkoqikleay
nkyira~`y# ilgl c` okelgk`c# c` fkzarbalgë~rki` { c` Zlkyyle% @pqç ya keicq{a ac
ileiaz~l na c` cùbki` nanqi~k}`# c` iq`c ay qe ileiaz~l na nkmçikc ae~aenkgkae~l% Ac
I`zç~qcl 1 nayirkoa c`y mqeikleay ile~keq`y na zrlo`okckn`n# ayzaik`cgae~a c`
nky~rkoqikùe elrg`c# `nagày# na c`y nky~rkoqikleay Rakoqc# avzleaeik`c# B`gg`# a~i%
Ac I`zç~qcl = f`oc` na c` ~alrç` na gqay~ral zapqaðl ilgl c` ~ na Ay~qnk`e~a# HK
iq`nr`n` { c` nky~rkoqikùe M% Ae ay~a raebcùe# ae c`y zrqao`y na fkzù~ayky# z`r` ac
ile~rlc na i`ckn`n# ya f`oc` na c` cùbki` kenqi~k}`# pqa ay qe ileiaz~l na nkmçikc
ae~aenkgkae~l { nkyiq~knl ae zlpqçykgly ckorly na ay~`nçy~ki`% @nagày# ac I`zç~qcl 2
ay~à rac`ikle`nl ile nkyaðly na `eàckyky na }`rk`ex` ilgzca~`gae~a `ca`~lrkx`nly {
nkyaðly na oclpqay `ca`~lrkx`nly% Ay~a i`zç~qcl ~`gokëe nkyiq~a glnacly m`i~lrk`cay
na nly { ~ray ic`ykmki`ikleay% Ac I`zç~qcl 6 ay~à rac`ikle`nl ile rabraykùe ckea`c
ykgzca { gøc~kzca% Ac I`zç~qcl ; ay~à rac`ikle`nl ile rabraykùe zlckelgk`c# ac iq`c
keicq{a glnacly zlckelgk`cay na yabqenl { ~ariar lrnae# ile qe` }`rk`oca
kenazaenkae~a { ile gày na nly }`rk`ocay rabrayk}`y% Ay~a i`zç~qcl f`oc` ~`gokëe na
glnacly na rabraykùe el ckea`cay na rabraykùe clbçy~ki` { na glnacly avzleaeik`cay
z`r`gë~rkily# ile qe` ylc` }`rk`oca kenazaenkae~a% Gày `nac e~a# ac I`zç~qcl 0: f`ocna zrqao`y el z`r`gë~rki`y% L~rlyç# ac I`zç~qcl 00 f`oc` na c`y yarkay na ~kagzl%
Mke`cgae~a# ac I`zç~qcl 0? f`oc` na gë~lnly z`r` yacaiikle`r ac ~`g`ðl na gqay~r`
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 9/846
kkk
gày zrlzk`nl%
Ay~a ~av~l# `nagày# keicq{a }`rkly `zëenkiay ile ~`oc`y na c`y nky~rkoqikleay
okelgk`cay# na Zlkyyle# elrg c# na ~ na Ay~qnk`e~a# na M# na HK iq`nr n`# a~i%
Kbq`cgae~a# ay~a ~av~l keicq{a qe` yarka na ramaraeik`y okocklbràmki`y% Mke`cgae~a# ay~a
ckorl na ay~`nçy~ki` keicq{a qe` yaiikùe pqa ile~kaea gày na 9>: aharikikly
rac`ikle`nly ile i`n` i`zç~qcl { ahagzcly qy`enl ac zrlbr`g` na ilgzq~`nlr`
Gkek~`o { Aviac% Ae ay~a ile~av~l# ay~a ~av~l na ay~`nçy~ki` n` gqifly ahagzcly na
zrlocag`y qy`enl ac z`pqa~a na ilgzq~`nlr Gkek~`o# ay naikr# nayirkokaenl ac qyl
nac Gkek~`o ile gkeqiklykn`n na na~`ccay4 yk~q`ikleay zrayae~`n`y zlr gq{ zlily
ckorly na ay~`nçy~ki`%
Z`r` ileicqkr# naol naikr pqa ay~a ay qe ~av~l na ay~`nçy~ki` nkyað`nl z`r` cly
ay~qnk`e~ay na kebaekarç` `gokae~`c na zlybr`nl { na c` ikaeik` ae baear`c% Ay naikr#
z`r` `pqaccly ke}ay~kb`nlray pqkaeay nayaae aeile~r`r# zrài~ki`gae~a# ~lnly cly
ileiaz~ly na c` zrlo`okckn`n { na c` ay~`nçy~ki`# pqa cay zqan` `{qn`r ae ac nay`rrlccl
na yq zrlmaykùe na kebaekarç`# ae c ke}ay~kb`ikùe l ae iq`cpqkar l~r` àra` na c
ikaeik` ae baear`c%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 10/846
Nr% Fëi~lr Pqa}anl Qrç`y
0'0
I@ZK^QCL 0
Ay~`nçy~ki` Nayirkz~k}`
Namkekikùe na ay~`nçy~ki`%' Zloc`ikùe { gqay~r`%' Ay~`nçy~ki` kenqi~k}` { na
kemaraeik`%' Ay~`nçy~ki` nayirkz~k}`%' ]`rk`ocay ile~keq`y { nkyira~`y%' Gankn`y
na ~aenaeik` iae~r`c%' Gankn`y na nkyzarykùe%' C` }`rk`oca `ca`~lrk`
ay~`en`rkx`n` x%' C`y nay}k`ikleay nac zrlgankl%' Ac r`ebl%' Yaybl { dqr~lyky%'
Nky~rkoqikleay na mraiqaeik`%' Nk`br`g`y na ~`ccl { flh`%
Ay~`nçy~ki` ay ac ay~qnkl na cly gë~lnly z`r` ilcaiikle`r# rayqgkr# lrb`ekx`r# zrayae~`r { `e`ckx`r kemlrg`ikùe na n`~ly% Ac ~ërgkel ay~ nçy~ki` ~`gokëe ya ramkara `
c` nark}`ikùe na ileicqykleay }àckn`y { ` c` mlrg`ikùe na naikykleay r`xle`ocay# ae
o`ya ` yagah e~ay `eàckyky% Ae c` ilcaiikùe na n`~ly na qe brqzl na loyar}`ikleay# `
gaeqnl ay kgzlykoca l kgzrài~kil loyar}`r ~ln` c` zloc`ikùe% Na g`ear` pqë# ae
cqb`r na av`gke`r ac brqzl ae yq ~l~`ckn`n# cc`g`nl c` zloc`ikùe l qek}aryl# ay
ile}aekae~a av`gke`r ylc`gae~a qe` z`r~a na c` zloc`ikùe cc`g`n` gqay~r`%Zloc`ikùe ya ramkara ` qe brqzl na ç~agy pqa ~kaeae qe` i`r`i~arçy~ki` ae
ilgøe% Qe` zloc`ikùe zqana yar namkekn` ilgl qe brqzl na kenk}knqly# ilgl zlr
ahagzcl# qe` zaryle`# qe `ekg`c# qe loha~l l qe` gankikùe% @nagày# qe` zloc`ikùe
zqana yar mkek~` l kemkek~ % Zlr ahagzcl# c` zloc`ikùe ileyky~ae~a na ~lnly cly ~lrekccly
zrlnqiknly ae qe` màorki # ae qe nç`# ay mkek~`% Ae ile~r`y~a# c` zloc`ikùe ileyky~ae~a
na ~lnly cly zlykocay rayqc~`nly &i`r`y l àbqkc`y+ na cly c`ex`gkae~ly yqiayk}ly na qe`
glean` ay kemkek~`% @ gaeqnl c` zloc`ikùe el avky~a zarl# yke ago`rbl# ay na
kgzlr~`eik`% Zlr ahagzcl# `c ay~qnk`r qe eqa}l ilclr`e~a z`r` ~ac`y na `cblnùe
zlnagly zrlo r ac eqa}l ilclr`e~a# ile ylc`gae~a 0: zkax`y na qe ga~rl nac g ~ark c
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 11/846
Nr% Fëi~lr Pqa}anl Qrç`y
0'?
{ f`iar gankikleay na c` rayky~aeik` nac ilclr`e~a% C` gqay~r` ileyky~a na 0: zkax`y
na `cblnùe ~r`~`n`y ile ac ilclr`e~a% C` zloc`ikùe ileyky~a na ~ln`y c`y zkax`y na
`cblnùe zlykocay na qe ikar~l ~kzl pqa zqnkar`e yar ~r`~`n`y ile ac eqa}l ilclr`e~a%Ay~` zloc`ikùe el avky~a% Yke ago`rbl# c` zloc`ikùe ~l~`c ely c` zlnagly kg`bke`r `c
ay~qnk`r c`y 0: zkax`y na `cblnùe ile ac loha~l na f`iar kemaraeik`y%
Ae ac i`yl na qe` gqay~r`# ay~l ya ramkara ` qe` ay~`nçy~ki` { ay qe ay~kg`nlr na
qe z`ràga~rl na zloc`ikùe% Zlr ahagzcl# yk V nael~` ac zrlgankl `rk~gë~kil
ay~`nçy~kil na qe` gqay~r`# ae~leiay# V ay ac ay~kg`nlr nac z`ràga~rl ± na ~lnl ac
ilehqe~l l zloc`ikùe% Yke ago`rbl# ae ile~r`y~a ilgl ya nkhl `e~ay# ay kgzrài~kil lkgzlykoca loyar}`r ~ln` c` zloc`ikùe# zlr ay~` r`xùe ya av`gke` qe` zapqað` z`r~a
nac brqzl l zloc`ikùe cc`g`n` gqay~r` ay~`nçy~ki`% @pqç# ay ile}aekae~a ke~rlnqikr
~ërgkely ~`cay ilgl gqay~r` `ca`~lrk` l `c `x`r# gqay~ral# ay~`nçy~ki` kenqi~k}` l na
kemaraeik` { ay~`nçy~ki` nayirkz~k}`% ^`gokëe ay gq{ irç~kil nky~kebqkr ae~ra cly
~ërgkely z`ràga~rly &nlena ya qy`e yçgolcly brkably+ }aryqy ay~`nçy~ki`y% Cly
z`ràga~rly ya ramkarae ` zloc ikleay kemkek~`y l mkek~`y% Yke ago`rbl# c`y ay~`nçy~ki`y
yar ramkarae ` qe` gqay~r`% Zlr ahagzcl# yk qe` gqay~r` ay razrayae~`~k}` na qe`
zloc ikùe ya zqanae y`i`r ileicqykleay kgzlr~`e~ay `iari` na ay~` zloc ikùe% Yke
ago`rbl# ay kgzlr~`e~a el~`r pqa c` gqay~r` naoa yar `ca`~lrk`# zlrpqa na l~r`
g`ear`# c` kemaraeik` `iari` na c` zloc`ikùe yarà ke}àckn`%
Ile rayzai~l ` c` ay~`nçy~ki` kenqi~k}` { ` c` ay~`nçy~ki` na kemaraeik`# ëy~`y ya
ramkarae `c zrliayl na kemarkr ileicqykleay `iari` na qe` zloc`ikùe o`yàenlya ae qe
gqay~ral `ca`~lrkl &`c `x`r+# na ~`c g`ear` pqa c` zrlo`okckn`n na ~aear qe` kemaraeik`ilrrai~` zqana yar na~argke`n` na `iqarnl ile }`rk`y fkzù~ayky ileiarekaenl c`
zloc ikùe o`hl ay~qnkl% Nkifl ae l~r`y z`c`or`y# naoknl ` pqa yagah`e~a kemaraeik` el
zqana yar `oylcq~`gae~a ikar~`# ac caebq ha na zrlo`okckn`n ay# ` gaeqnl qy nl ae c`
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 12/846
Nr% Fëi~lr Pqa}anl Qrç`y
0'9
zrayae~`ikùe na cly rayqc~`nly l ileicqykleay%
Ae ile~r`y~a# c` m`ya na ay~`nçy~ki` pqa oqyi` øeki`gae~a nayirkokr { `e`ckx`r
n`~ly na qe` nky~rkoqikùe ile~keq` &ilgl c` elrg`c+# yke y`i`r ekebqe` ileicqykùe lkemaraeik` `iari` na c` zloc`ikùe l qek}aryl# ya naelgke` ay~`nçy~ki` nayirkz~k}`%
@pqç ya keicq{ae ~ërgkely ilgl ilcaiikùe na n`~ly yke zrliay`r# mlrg`ikùe na n`~ly
ae lrnae nayiaenkae~a l `yiaenae~a &iq{` nkmaraeik` ae~ra ac g`{lr { gaely ya
naelgke` r`ebl+# nky~rkoqikleay na mraiqaeik`# pqa ay qe ~ërgkel z`r` nayirkokr ac
`rrabcl rac`~k}l na qe ilehqe~l na acagae~ly na cly }`clray na qe` }`rk`oca { na c`y
mraiqaeik` na liqrraeik` na i`n` }`clr &c` gày kgzlr~`e~a cc`g`n` iqr}` elrg`c { ~
na ay~qnk`e~a+% L~rly ~ërgkely qy`nly ae ay~`nçy~ki` nayirkz~k}` yle zrlgankly
`rk~gë~kily# zrlgankly balgë~rkily# zrlgankly `rgùekily# gank`e`y# gln`y#
zariae~kcay# nay}k`ikleay ay~àen`r# }`rk ex`y# a~i%# zarl# yke y`i r kemaraeik`y nac
brqzl pqa zrl}kaeae%
Yke ago`rbl# ile rac`ikùe ` c` ay~`nçy~ki` nayirkz~k}` { c` ay~`nçy~ki` na
kemaraeik`# ae ac i`yl na c` ay~`nçy~ki` nayirkz~k}`# ay~a ~kzl na ay~`nçy~ki` keicq{a c`
zrayae~`ikùe na ilehqe~ly na loyar}`ikleay# na ~`c g`ear` pqa zqan`e yar
ilgzraenkn`y a ke~arzra~`n`y { ykr}ae z`r` rayqgkr l nayirkokr n`~ly% Ae i`gokl# c`
ay~`nçy~ki` na kemaraeik` ya rac`ikle` ile ay~kg`ikleay na g`bek~qnay na zloc`ikleay
{ zrqao`y na `iari` na c`y i`r`i~arçy~ki`y na c` zloc`ikùe% @go`y yle ø~kcay z`r`
na~argke`r iq`c ae~ra nly ` gày iqryly na `iikùe ya ykbqae iq`enl ac iqryl ilrrai~l
ay na~argke`nl zlr qe` i`r`i~arçy~ki` z`r~kiqc`r l nayilelikn` na c` zloc`ikùe%
Ae ac i`gzl na c` kebaekarç` &ilgl ae c` kebaekarç` `gokae~`c+ { ikaeik`yavzarkgae~`cay ac qyl na c` ay~`nçy~ki` ay rapqarknl ae ac nkyaðl na zc`e~`y na `bq`y
rayknq`cay a kenqy~rk`cay# ae ac nkyaðl na ifkgaea`y kenqy~rk`cay# ae ac nkyaðl nac
apqkzl na ile~rlc na c` ile~`gke`ikùe# ae zrqao`y na rq~ke` na c`olr`~lrkl# ae
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 13/846
Nr% Fëi~lr Pqa}anl Qrç`y
0'>
~r`o`hly na ke}ay~kb`ikùe { ae c` zrlnqiikùe na i`ckn`n { iley~rqiikùe% Zlr ahagzcl#
ae ac c`olr`~lrkl yk ac gqay~ral ay zraikyl l yk c` }`rk`okckn`n na eqay~rly rayqc~`nly ay
g`{lr na cl ayzar`nl# ae~leiay f`{ pqa ilrrabkr c` }`rk`ikùe ramke`enl c`y ~ëieki`y nac`olr`~lrkl l keiragae~`enl ac ~`g`ðl na c` gqay~r`%
Ae ac i`gzl na c` ke}ay~kb`ikùe ~`c }ax ay~agly ke~aray`nly ae y`oar yk qe
i`gokl ay qe kebrankae~a pqa `mai~` c`y zrlzkan`nay nac g`~ark`c rayqc~`e~a# z`r`
ilgz`r`r c` amkikaeik` na zrliayly l na gàpqke`y zrlo`nlr`y4 z`r` na~argke`r yk cly
rayqc~`nly lo~aeknly aei`h`e ae qe` mlrg` zly~qc`n` l ylyzaif`n`% L~r` `zcki`ikùe
gq{ kgzlr~`e~a ay ac ile~rlc na c` i`ckn`n ae c` kebaekarç` kenqy~rk`c%
Ile rac`ikùe ` c`y }`rk`ocay ile~keq`y { nkyira~`y# ae ay~a i`yl ya nkia pqa qe`
}`rk`oca `ca`~lrk` ay qe` mqeikùe pqa `ykbe` qe }`clr eqgërkil ` i`n` a}ae~l ykgzca
ae qe ayz`ikl na c` gqay~r`% @yç# qe` }`rk`oca `ca`~lrk` ile~keq` zqana `yqgkr qe`
mkbqr` keeqgar`oca {# ~aùrki`gae~a# zqana `yqgkr iq`cpqkar }`clr ae~ra nly }`clray
n`nly% Zlr ahagzcl c`y `c~qr`y na qe` zaryle` zqanae yar =?%: zqcb`n`y# =9%6
Zqcb`n`y# =1%6>1= Zqcb`n`y# a~i% Ae ile~r`y~a# qe` }`rk`oca ay nkyira~` yk zqana
`yqgkr# ylc`gae~a# qe eøgarl ile~`oca na zlykocay }`clray%
Gankn`y na ~aenaeik` iae~r`c l na cli`ckx`ikùe8 ac zrlgankl# c` gank`e` { c`
gln`% Yçgolcly qy`nly ae c`y yqg`~lrk`y na ay~`nçy~ki`8
e Ac yçgolcl Ω V h ya qy` z`r` nael~`r c` yqg` na ~ln`y c`y
h70
V hy# nayna h 7 0 f`y~` h 7 E%
e Ahagzcl )0% Ω V h 7 V0 $ V? $ V9 $ %%% $ Ve
h70
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 14/846
Nr% Fëi~lr Pqa}anl Qrç`y
0'1
e
Ahagzcl )?% Ω V h[ h 7 V0[0 $ V?[? $ V9[9 $ %%%$ V E[e
h70
e Ahagzcl )9% Ω `V h 7 `V0 $ `V? $%%%$ `Ve
h70 e
7 `&V0 $ V? $#%%#$ Ve+ 7 Ω V h h70
Eù~aya c nkmaraeik` ae~ra ΩV ? { &ΩV+?
C` yqg` na cly iq`nr`nly &YY+# ay naikr# c` yqg` na c`y nay}k`ikleay `c iq`nr`nl na V
na yq zrlgankl V ya nael~` ilgl8
de
C` yqg` ~l~`c na cly iq`nr`nly 7 Ω & V k ' V +? 7 YY &0'0+k70
7 Ω V
?
' &Ω V +
?
,eAc zrlgankl `rk~gë~kil
Ac zrlgankl `rk~gë~kil ay qe }`clr ac iq`c ay ~çzkil l razrayae~`~k}l na qe ilehqe~l na
n`~ly na nky~rkoqikleay ile~keq`y% Avky~ae nkmarae~ay ~kzly na zrlgankly% Cly gày
ilgqeay yle ac zrlgankl `rk~gë~kil# c` gank`e`# c` gln`# ac zrlgankl balgë~rkil# ac
zrlgankl f`rgùekil# a~i% I`n` qel ~kaea yqy }ae~`h`y { nay}ae~ h`y nazaenkaenl na
cly n`~ly { ac zrlzùyk~l ` yabqkr% Ac zrlgankl `rk~gë~kil el ya naoa qy`r ilgl
ykeùekgl na zrlgankl l gank`# zlrpqa f`{ l~rly ~kzly na zrlgankly%
Ac zrlgankl `rk~gë~kil ay qe }`clr pqa razrayae~` qe ilehqe~l na n`~ly4 ay qe`
gankikùe na ~aenaeik` iae~r`c% Ac zrlgankl `rk~gë~kil ay ac ay~kg`nlr nac z`ràga~rl
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 15/846
Nr% Fëi~lr Pqa}anl Qrç`y
0'=
na zloc`ikùe# ± { ya namkea ilgl8
V 7 & V 0 $ V ? $ V 9 $%%%$ V e+ , e 7 Ω V h , e 7 Ω V ,e &0'?+Yk cly eøgarly V 0 # V ? # V 9 #…#V d liqrrae m 0# m ?#…#m d }aiay# ay naikr# ile n`~ly
`brqz`nly# ae~leiay8
V 7 m V k , e &0'9+
Ile c`y nky~rkoqikleay ile~keq`y# ay na el~`rya pqë# ac zrlgankl `rk~gë~kil# V
ay qe ay~kg`nlr na ±# ay naikr# nac z`ràga~rl na zloc`ikùe% Ae gq{ r`r`y li`ykleay ya
ilelia ± &~ln` c` zloc`ikùe+# ykaenl `yç# ae~leiay# ya i`ciqc` nkrai~`gae~a%
Ahagzcl )>% Ac zrlgankl na qe` gqay~r` na loyar}`ikleay na ikar~ly `eàckyky na
`bq`y# iq{ly }`clray yle 6# 9# 1# 0?# 0:# ay8
V 7 &6 $ 9 $ 1 $ 0? $ 0:+,1 7 96,1 7 2%=
Ahagzcl )1% I`ciqc`r V # na qe` gqay~r` na 1# 6# =# { ? i`yly pqa liqrrae ile qe`mraiqaeik` na na 9# ?# ># { 0%
V 7 T&9+&1+ $ &?+&6+ $ &>+&=+ $ &0+&?+W,&9$?$>$0+ 7 1%2
C` gank`e`
C` gank`e`# V u ay ac }`clr na ae gankl na qe brqzl na eøgarly q loyar}`ikleay
&zqay~`y ae mlrg` `yiaenae~a+ l ac zrlgankl `rk~gë~kil na cly nly }`clray na ae
gankl% Balgë~rki`gae~a f`oc`enl# c` gank`e` ay ac }`clr na V &`oyiky`+ilrrayzlenkae~a ` ay` cçea` }ar~ki`c pqa nk}kna ` qe fky~lbr`g` ae nly z`r~ay ~aekaenl
àra`y kbq`cay% C` gank`e` ay qe` zlykikùe na zrlgankl# gkae~r`y pqa ac zrlgankl
`rk~gë~kil ay qe zrlgankl i`ciqc`nl%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 16/846
Nr% Fëi~lr Pqa}anl Qrç`y
0'2
Ahagzcl ) =% C` gqay~r` na loyar}`ikleay 9# ># ># 1# =# 6# 6# 0: ~kaea qe` gank`e` na
V u 7 &1$=+,? 7 1%1%
Ahagzcl )2% C` gqay~r` na loyar}`ikleay 1# 1# 2# ;# 00# 0?# 09# 01# 06 ~kaea qe`
gank`e` na V u 7 00%
C` gln`
C` gln` ay qe` ay~`nçy~ki` pqa nagqay~r` ac }`clr pqa liqrra ile gày mraiqaeik` ae
qe` gqay~r` &zlekaenl cly n`~ly ae mlrg` `yiaenae~a+% Qe` nky~rkoqikùe zqana ~aear
qe` gln`# zqana yar okgln`c# a~i% Ay~a }`clr ya nael~` zlr V ˍ % Yke ago`rbl# `cbqe`y
li`ykleay c` gln` el avky~a%
Ahagzcl )6% C` gqay~r` na loyar}`ikleay ?# ?# 1# 2# ;# ;# ;# 0:# 0:# 00# 0?# 06 ~kaea
qe` gln` na V ˍ 7 ;# ay naikr# ac }`clr pqa liqrra ile gày mraiqaeik`%
Ahagzcl );% Cly }`clray 9# 1# 6# 0:# 0?# 01# 0= el ~kaeae gln`%
Ahagzcl )0:% C` gqay~r` na loyar}`ikleay ? #9# ># ># ># 1# 1# 2# 2# 2# ; ~kaea nly
gln`y# > { 2 { ay okgln`c# ay naikr# V ˍ 7 ?%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 17/846
Nr% Fëi~lr Pqa}anl Qrç`y
0'6
_ac`ikùe ae~ra ac zrlgankl rk~gë~kil# c` gank`e` { c` gln`
Yk ac zrlgankl# c` gank`e` { c` gln` ilkeiknae# ae~leiay c` nky~rkoqikùe ay ykgë~rki`4
na l~r` g`ear`# c` nky~rkoqikùe ay `ykgë~rki` ile yaybl ` c` naraif` l c` kxpqkarn`% ]ar mkbqr`y na `o`hl%
Mkbqr` 0%:% Nky~rkoqikùe lockiq` Mkbqr` 0%0% Nky~rkoqikùe lockiq`
` c` naraif` &yaybl zlyk~k}l+% ` c` kxpqkarn` &yaybl eab`~k}l+
&Ac`olr`ikùe zrlzk`+ &Ac`olr`ikùe zrlzk`+
Ahagzcl )00% Aeile~r`r ac zrlgankl `rk~gë~kil# c` gank`e` { c` gln` z`r` qe`
gqay~r` na `eàckyky na `kra na Zo iq{ly }`clray yle8 9# 1# ?# =# 1# ;# 1# ?# 6# = z`r~ay
zlr gkccùe &zzg+%
Ylcqikùe8
V 7 1%0 zzg
V u 7 &1$1+,? 7 1
V ˍ 7 eøgarl pqa liqrra ile gày mraiqaeik` 7 1
Ahagzcl )0?% Aeile~r`r ac zrlgankl# c` gank`e` { c` gln` na cly i`yly >6%2# >6%6#
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 18/846
Nr% Fëi~lr Pqa}anl Qrç`y
0';
>;%1# 1:%9# 10%=%
Ylcqikùe8
V 7 >;%6# V u 7 >;%1# V ˍ 7 el avky~aAc zrlgankl balgë~rkil
Ac zrlgankl balgë~rkil ya qy` ilgl qe nkymr`x na ~r`eymlrg`ikùe clb`rç~gki`% Ay ø~kc
z`r` zrlgank r ~`y`y na iraikgkae~l &`qgae~l l nairagae~l+ na qe` gqay~r`
ay~`nçy~ki`% C` mùrgqc` ay8
B 7 ee9?0 vvvv %%% &0'>+
Ahagzcl )09% Aeile~r`r ac zrlgankl balgë~rkil na cly }`clray 9# 1# =# =# 2# 0:# 0?Ylcqikùe8
B 7 2 0?+=+&2+&0:+&&9+&1+&=+& 7 2 >19#=::
clb B 7 0,2 clb&>19#:::+ 7 :%6:60 { `e~kclb :%6:90 7 =%>9
Avky~ae l~rly zrlgankly ilgl ac zrlgankl f`rgùekil# ac zrlgankl iq`nrà~kil#
a~i% ^`gokëe f`{ l~r`y gankn`y na cli`ckx`ikùe gày mke`y pqa nk}knae cly n`~ly ae
gày na nly z`r~ay% Zlr ahagzcl# cly iq`r~kcay nk}knae ac ilehqe~l na n`~ly ae iq`~rl z`r~ay kbq cay% Zlr ahagzcl# ac ~ariar iq`r~kc &P9+ nayirkoa c` iq r~` z`r~a yqzarklr nac
ilehqe~l na n`~ly% Ac yabqenl iq`r~kc &P?+ ay knëe~kil ` c` gank`e`% Ac zrkgar iq`r~kc
&P0+ yaz`r` c` iq`r~` z`r~a kemarklr na c`y ~ray iq`r~`y z`r~ay yqzarklray% @nagày# cly
zariae~kcay zqanae nk}knkr cly n`~ly ae 0:: z`r~ay kbq cay% Zlr ahagzcl# ac ;;`}l
zariae~kc yaz`r` ac 0! gày `c~l nac ;;! ray~`e~a# a~i%
L~r` mlrg` na }ar c` ykga~rç` na cly n`~ly ay qy`enl nk`br`g`y na i`h`%
^`gokëe f`{ cl pqa ya cc`g` nk`br`g`y na zqe~l# pqa `{qn`e# }kyq`cgae~a# ` ra}ky`r
c` ykga~rç` na cly n`~ly%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 19/846
Nr% Fëi~lr Pqa}anl Qrç`y
0'0:
C` }`rk`ex`
C` }`rk`ex`# y? ay qe` gankn` na nkyzarykùe { ely nkia# pqë ~`e~` }`rk`ikùe avky~a na
qe` loyar}`ikùe ` l~r` &l nac zrlgankl+ l na qe` gqay~r` ` l~r`% Qe` y? br`ena ~kaeagày i`yly nk}arykmki`nly# pqa qe` ile qe` }`rk`ex` zapqað`% C` }`rk`ex` y? na qe`
gqay~r` ay~`nçy~ki` &l na }`rk`y gqay~r`y+ ay ac ay~kg`nlr nac z`ràga~rl na c`
}`rk`ex`# ω? na qe` zloc`ikùe l zloc`ikleay% C` mùrgqc` na c` }`rk`ex` ay8
e y? 7 Ω & V ' V +?,&e'0+ 7 TΩ V
? ’ &Ω V +?,eW,&e ' 0+ &0'1+k70
7 YY,&e ’ 0+
Ahagzcl )0>% I`ciqc`r c` }`rk`ex` { c` nay}k`ikùe ay~àen`r na c` gqay~r` ?# ># =%
Ylcqikùe8
I`ciqc`enl V 7 > { qy`enl ac gë~lnl c`rbl ely n`8
y? 7 T&? ' >+? $ &> ' >+? $ &= ' >+?W,&9 ' 0+
7 6,? 7 >Qy`enl ac gë~lnl ilr~l8
]`rk`ex` 7 y? 7 TΩ V ? ’ &Ω V +?,eW,&e ’ 0+ ely n`rç`8
y? 7 TΩ V ? ’&Ω V +?,eW,&e ’ 0+
7 &1= ’ >6+,?
7 >
C` nay}k`ikùe ay~àen`r
C` nay}k`ikùe ay~àen`r# y ay qe` mlrg` ayzaik`c na c` nay}k`ikùe zrlgankl na c`
gank`% Ay qe` gankn` na nkyzarykùe% @ gankn` pqa `qgae~` c` nay}k`ikùe ay~àen`r l
c` }`rk`ex`# g`{lr nk}arykn`n f`orà ae~ra c`y loyar}`ikleay na qe` gqay~r`% Ay~`
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 20/846
Nr% Fëi~lr Pqa}anl Qrç`y
0'00
ay~`nçy~ki` ya n` ilgl8
y 7 ∘ TΩ V ? ’ &Ω V +?,eW , &e ’ 0+ &0'1`+
Z`r` n`~ly `brqz`nly# c` nay}k`ikùe ay~àen`r ay8
y 7 ∘ Tm hΩ V ? ’ &Ω V +?,eW , &e ’ 0+ &0'1o+
Ahagzcl )01% Z`r` ac ahagzcl na `rrko`# i`ciqc`r c` nay}k`ikùe ay~àen`r%
Ylcqikùe8
Yk c` }`rk`ex`# y? 7 ># zlr cl ~`e~l# c` nay}k`ikùe ay~àen`r# y ay8
y 7 ? y 7 > 7 ?
Ahagzcl )0=% Aeile~r`r V # y# y?# c` gank`e`# ac arrlr ay~àen`r nac zrlgankl# ac yaybl
{ c` dqr~lyky na qe` gqay~r` `c `x`r na 9= `eàckyky na mlym`~ly &ZL>'9+# ae gb,C% ¼Pqë
~`e~` mknackn`n f`{ ae cly n`~ly< C` ~`oc` na o`hl n` c` kemlrg`ikùe%
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ ]`clray na V s =0 => =2 2: 29 =; =6 2:Mraiqaeik` s 1 6 > 1 1 > 9 ?
Ylcqikùe8Qy`enl qe z`pqa~a na ilgzq~`nlr` n`8 V 7 =2%?2# y 7 9%26# y? 7 0>%90# gank`e` 7
=6# yaybl 7 ':%?? { dqr~lyky 7 ':%;1% @c hqxb`r zlr cly rayqc~`nly# f`{ qe` oqae`
`zrlvkg`ikùe ` c` nky~rkoqikùe elrg`c# zqay~l pqa V { c` gank`e` yle z`raiknly%
@nagày ac }`clr nac yaybl el nkmkara gqifl na :% Ya ca zkna `c cai~lr qy`r c` mùrgqc`
&01'o+ z`r` ilrrlolr`r cly rayqc~`nly ilgzq~`rkx`nly lo~aeknly%
Zrlzkan`nay na c` nay}k`ikùe ay~àen`rZ`r` qe` nky~rkoqikùe elrg`c ac =6%?2! na ~ln`y c`y loyar}`ikleay ay~àe keicqkn`y
ae~ra & V ' y+ { & V $ y+# ay~l ay# qe` nay}k ikùe ay~àen`r ` iq cpqkar c`nl nac
zrlgankl% Ykgkc`rgae~a# ac ;1%>1! na ~lnly cly i`yly ya keicq{ae ae~ra & V ' ?y+ {
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 21/846
Nr% Fëi~lr Pqa}anl Qrç`y
0'0?
& V $ ?y+# ay~l ay ae~ra x 7 °?% @nagày# ae ac ;;%29! na ~lnly cly i`yly ya keicq{ae ae~ra & V ' 9y+ { & V $ 9y+# ay~l ay# ae~ra x 7 °9%
Mkbqr` 0%?% Nky~rkoqikùe elrg`c gly~r`enl c`y àra`y z`r` nkmarae~ay zariae~kcay na c`
}`rk`oca ay~`en`rkx`n` x &Yzkabac# 0;=0+%
]`rk`oca ca`~lrk` ay~`en`rkx`n` x
Ay~` }`rk`oca `ca`~lrk` ay~`en`rkx`n` x gkna c`y nay}k`ikleay nac zrlgankl ae
qekn`nay na nay}k`ikùe ay~àen`r { ya n` ilgl8
x 7 &V ' V + , y% &0'=+Yq z`ràga~rl rayzai~k}l ay8
X 7 &V ' ±+,ω &0'2+
Ahagzcl )0=% I`ciqc`r c`y ykbqkae~ay zrlo`okckn`nay8
&`+ Z&x ≮ 0%?1+
&o+ Z&x 5 0%?1+
&i+ Z&x ≮ '0%?1+
&n+ Z&'%96 ≮ x ≮ 0%?1+
Ylcqikùe8
&`+ Z`r` ay~l# oqyi`gly ae c` ~`oc` na c` nky~rkoqikùe elrg`c nac raebcùe g`ri`nl ile
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 22/846
Nr% Fëi~lr Pqa}anl Qrç`y
0'09
0%? { c` ilcqge` %:1 { n` %6;>>4 zlr cl iq`c# Z&V ≮ 0%?1+ 7 %6;>>%
&o+ Z&x 5 0%?1+ 7 0 ’ Z&x ≮ 0%?1+ 7 0 ' %6;>> 7 %0:1=
i+ Z&x ≮ '0%?1+ 7 %0:1=% Zlr ykga~rç` na c` iqr}` elrg`c# ay c` gkyg` rayzqay~` pqa aeac keikyl &o+
&n+ Z&'%96 ≮ x ≮ 0%?1+ 7 &àra` na '∕ ` x 7 0%?1+ ’ &àra` na '∕ ` x 7 '%96+ 7 %6;>> '
%91?: 7 %1>?> &na c` ~`oc` na x+
L~r` g`ear` na }ar cl gkygl ay qy`enl `el~`ikùe na zrlo`okckn`nay8
Z&'%96 ≮ x ≮ 0%?1+ 7 Z&x ≮ 0%?1+ ’ Z&x ≮ '%96+
7 %6;>> ' %91?: 7 %1>?>
C`y nay}k`ikleay nac zrlgankl
C`y nay}k`ikleay nac zrlgankl yle l~r`y gankn`y na nkyzarykùe% G`~agà~ki`gae~a%%%%
e
Nay}k`ikùe nac zrlgankl 7 Ω sVh ' V s,E &0'6+ h70
Ahagzcl )02% Aeile~r`r c` nay}k`ikùe zrlgankl na cly }`clray ?# 9# =# 6# 00%
Ylcqikùe8
Ac zrlgankl `rk~gë~kil ay V 7 =
C` nay}k`ikùe zrlgankl 7 &s?'=s$s9'=s$s='=s$s6'=s$s00'=s+,1
7 ?%6
Ac r`ebl
Ac r`ebl na c`y loyar}`ikleay na qe` gqay~r` ay c` nkmaraeik` ae~ra ac eøgarl gày
br`ena { ac gày zapqaðl% @pqç# ay na el~`rya pqë# ae~ra gày br`ena ya` c` nkmaraeik`#
gày nkyzarykùe f`orà# ay naikr# c` }`rk`ex` { c` nay}k`ikùe ay~àen`r yaràe gày br`enay%
Ahagzcl )06% Aeile~r`r ac r`ebl na ?# 9# 9# 1# 1# 1# 6# 0:# 0?%
Ylcqikùe8
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 23/846
Nr% Fëi~lr Pqa}anl Qrç`y
0'0>
Ac eøgarl gày zapqaðl ay ac ? { ac gày br`ena ay ac 0?# ay~l ay# 0? ' ? 7 0:
El~ 8 Avky~ae l~r`y mqeikleay na nkyzarykùe ilgl c nkyzarykùe rac`~k}` { oylcq~` l ac
ilamkikae~a na }`rk`ikùe# a~i%Yaybl { dqr~lyky
Ac yaybl na qe` nky~rkoqikùe gkna ac br`nl na c` ykga~rç`% Yk c` iqr}` na mraiqaeik` na
qe` nky~rkoqikùe ~kaea qe av~ragl gày c`rbl ` c` naraif` nac gàvkgl iae~r`c pqa ac na
c` kxpqkarn`# c` nky~rkoqikùe ay lockiq` f`ik` c` naraif` l ile yaybl zlyk~k}l% Cl
ile~r`rkl ay ilrrai~l { ya nkia pqa ay lockiq` f`ik` c` kxpqkarn` l na yaybl eab`~k}l%
Ay~` ilenkikùe ya naelgke` ac zrkgar ilamkikae~a na yaybl na Za`ryle% Ac yaybl na c`
nky~rkoqikùe elrg`c ay kbq`c ` :%
[` ya avzckiù pqa# c` rac`ikùe ae~ra ac zrlgankl# c` gank`e` { c` gln` zqanae
n`r qe` kenki`ikùe nac br`nl na ykga~rç` na cly n`~ly na qe` nky~rkoqikùe% Zlr ahagzcl#
yk ac zrlgankl ay g`{lr pqa c` gank`e`# g`{lr pqa c` gln`# ae~leiay# c` nky~rkoqikùe
ay `ykgë~rki` ile yaybl zlyk~k}l f`ik` c` naraif`% Na l~r` g`ear`# c` nky~rkoqikùe
~kaea yaybl eab`~k}l f`ik` c` kxpqkarn`%
C` dqr~lyky na qe` nky~rkoqikùe gkna cl zqe~k`bqnl na qe` nky~rkoqikùe elrg`c%
Qe` nky~rkoqikùe pqa ~kaea qe` ikg` l zkil rac`~k}`gae~a `c~` ya cc`g` caz~ldør~ki`#
gkae~r`y pqa `pqëcc` pqa ay~à `if`~`n` ya cc`g` zc`~{dør~ki`% C` iqr}` elrg`c pqa el
ay~à zkiqn` ek `if`~`n` ya cc`g` gayldør~ki`% C` dqr~lyky na c` iqr}` elrg`c ay kbq`c
` 9%
Arrlr ay~àen`r
@nagày na razlr~`r ac }`clr na qe` ay~kg`ikùe zqe~q`c# ~`gokëe naoa kenki`rya yq zraikykùe% C` gankn` na zraikykùe qyq c ay ac arrlr ay~àen`r nac ay~kg`nlr qy`nl% Zlr
ahagzcl# cly arrlray ay~àen`ray na `cbqe`y nky~rkoqikleay na c` gqay~r` yle cly nac
zrlgankl# na zrlzlrikleay# na nay}k ikleay ay~àen`r { na gank e`y%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 24/846
Nr% Fëi~lr Pqa}anl Qrç`y
0'01
@yç# na ay~` g`ear`# cly arrlray ay~àen`ray nac zrlgankl# na c`y zrlzlrikleay l
c` gank`e` ay# rayzai~k}`gae~a8
ωV 7 ω ∘ E &0';+ω z 7 zp,E &0';`+
ωy 7 ω ?E &z`r` zloc`ikleay elrg`cay+ &0';o+
ωgan%7 ω ?E
Ϊ&z`r` e ≡ 9:+ &0';i+
^ërgkely kgzlr~`e~ay
Z`ràga~rly% Ya ramkarae ` }`clray zloc`ikle`cay% Ya qy`e cly yçgolcly brkably z`r`
nael~`rcly%
Ay~`nçy~ki`% Ya ramkara ` qe` gqay~r` ~lg`n` na qe` zloc`ikùe% Ay qe ay~kg`nlr na cly
z`ràga~rly na zloc`ikùe%
Zrlgankl `rk~gë~kil% Yk ya ilelia ~ln` c` zloc`ikùe ya qy` c` }`rk`oca ±% Yk ya ramkara
` qe` gqay~r` ay~`nçy~ki`# ya qy` c` }`rk`oca V % Na iq`cpqkar g`ear` ac zrlgankl
`rk~gë~kil ay c` yqg`~lrk` na qe brqzl na loyar}`ikleay nk}knknl ae~ra ac ~l~`c na cly
i`yly%
Zrlgankl% Ae baear`c qe zrlgankl ya ramkara ` qe` gankn` na ~aenaeik` iae~r`c%
Ahagzcly yle ac zrlgankl `rk~gë~kil# c` gank`e` { c` gln`% F`{ ~`gokëe zrlgankly
balgë~rkily# `rgùekily# a~i%
Gank`e`% Ay ac }`clr nac ç~ag iae~r`c iq`enl cly n`~ly yle `brqz`nly zlr ~`g`ðl
& V u +%
Gln`% Ay ac }`clr pqa liqrra ile gày mraiqaeik` & V ˍ +%
Nky~rkoqikùe okgln`c% Ya ramkara ` qe` nky~rkoqikùe ile nly gln`y%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 25/846
Nr% Fëi~lr Pqa}anl Qrç`y
0'0=
Gankn`y na nkyzarykùe% Ya ramkara `c br`nl na nkyzarykùe na cly n`~ly eqgërkily nac
zrlgankl% Cly gày ilgqeay yle8 ac r`ebl# c` nay}k`ikùe ay~àen`r# c` }`rk eik # c`
nay}k`ikùe zrlgankl# nay}k`ikùe na iq`r~kcay# a~i%]`rk`ex`% Ay qe` gankn` na nkyzarykùe% Ya nael~` ilgl ω? z`r` nayirkokr ~ln` c`
zloc ikùe% Yke ago`rbl# yk ya ramkara ` c` }`rk`ex` na c gqay~r # ya qy` ac yçgolcl y? {
ya nayirkoa ilgl c` yqg` na cly iq`nr`nly nk}knkn` ae~ra ac eøgarl na }`clray na c`
gqay~r` gaely qel% Ya qy` ac yçgolcl y? pqa ay ac ay~kg`nlr nac z`ràga~rl
zloc ikle`c ω?%
Nay}k`ikùe ay~àen`r% Ya lo~kaea y`i`enl c` r`çx iq`nr`n` na c` }`rk`ex` zloc`ikle`c l
na c` }`rk`ex` na c` gqay~r`%
Ilamkikae~a na }`rk`ikùe% Ay c` rac`ikùe g`~agà~ki` na c` nay}k`ikùe ay~àen`r nk}kn`
ae~ra ac zrlgankl `rk~gë~kil% Baear`cgae~a ya avzray` ilgl zlriae~`ha% Ay ø~kc z`r`
ilgz`r`r nky~rkoqikleay nlena c`y qekn`nay zqan`e yar nkmarae~ay%
]`rk`ocay nkyira~`y% ]`rk`ocay nkyira~`y ya ramkarae ` i`r`i~arçy~ki`y ~`cay ilgl ilclr#
yavl# rackbkùe# a~i%# pqa ya zqanae avzray`r ae ic`ykmki`ikleay l i`~ablrç`y iq`ck~`~k}`y%
Zlr ahagzcl# ac eøgarl e na qe` m`gkck` na ekðly `yqga }`clray na :# 0# ?# 9#%%%# zarl
pqa el zqana `yqgkr }`clray na ?%1 l na 9%61=%
]`rk`ocay ile~keq`y%' Ya ramkara ` }`rk`ocay pqa# ~aùrki`gae~a# zqanae `yqgkr
iq`cpqkar }`clr ae~ra nly }`clray n`nly% Ya zqanae avzray`r ae ic`ykmki`ikleay l
i`~ablrç`y iq`e~k~`~k}`y% Zlr ahagzcl# c` `c~qr` f na qe kenk}knql# c` iq`c zqana yar
=9%; zqcb`n`y# =1%;;>1 zqcb`n`y# ay qe` }`rk`oca ile~keq`%
Yaybl% Gkna c` ykga~rç` na qe` nky~rkoqikùe% Ac yaybl zqana yar zlyk~k}l &lockiqlf`ik` c` naraif`+ l eab`~k}l &lockiql f`ik` c` kxpqkarn`+% Yk ay yaybl ay zlyk~k}l#
ae~leiay V 5 V u 5 V ˍ % Yke ago`rbl# yk ac yaybl ay eab`~k}l# ae~leiay# ay ac ra}aryl%
C` dqr~lyky gkna cl `if`~`nl l zqe~k`bqnl na c` nky~rkoqikùe%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 26/846
Nr% Fëi~lr Pqa}anl Qrç`y
0'02
]`rk`oca ay~`en`rkx`n`% Gkna c` nay}k`ikùe nac zrlgankl ae qekn`nay na nay}k`ikùe
ay~àen`r# ykgzcagae~a# ya ramkara `c eøgarl na nay}k`ikleay ay~àen`r na qe`
loyar}`ikùe pqa ay~à `o`hl l `rrko` nac zrlgankl na c` nky~rkoqikùe%Gë~lnly bràmkily { ~`oqc`ray qy`nly ae ay~`nçy~ki` nayirkz~k}`
L~r`y ~ëieki`y }kyq`cay# pqa yle gq{ ø~kcay ae c` zrlo`okckn`n { c` ay~`nçy~ki` na
kemaraeik`# yle ac qyl na nayzcab`nly na ~`ccl { flh`y% L~rly gày yle cly nk`br`g`y na
zqe~l &avzcki`nly zly~arklrgae~a+ { cly fky~lbr`g`y% Zlr ahagzcl# z`r` iley~rqkr qe
nk`br`g` na ~`ccl { flh`# ay~` yk~q`ikùe ya avzcki` ae ac ~ùzkil na nk`br`g`y na ~`ccl {
flh`% Cly nk`br`g`y na ~`ccl { flh` yle z`raiknly ` cly fky~lbr`g`y { ykr}ae ac gkygl
zrlzùyk~l% Ay~l ay# zlrpqa cly nk br`g`y na ~ ccl { flh` ra}ac`e ac r`ebl na cly n`~ly#
gqay~r`e nlena liqrra c` ileiae~r`ikùe gày `c~` na }`clray# zrl}aae kemlrg`ikùe
`iari` na c` zrayaeik` l `qyaeik` na ykga~rç` {# zqanae kenki`r ac br`nl na ykga~rç` ae
c` iq`c cly n`~ly yle flglbëealy%
Nky~rkoqikleay na mraiqaeik`
Iq`enl ya ay~àe zrliay`enl br`enay i`e~kn`nay na n`~ly ay ile}aekae~a nky~rkoqkrcly
nae~rl na ic`yay l i`~ablrç`y# z`r` na~argke`r ac eøgarl na loyar}`ikleay pqa
zar~aeaiae ` i`n` ic`ya cc`g`n` mraiqaeik` na ic`ya% @yç# qe `rrabcl ~`oqc r na n`~ly
zlr ic`yay hqe~l ile c`y mraiqaeik y na ic`yay ilrrayzlenkae~ay ya cc`g` nky~rkoqikleay
na mraiqaeik` l ~`oc`y na mraiqaeik`y%
Namkekikùe na ~ërgkely
Ùrnaeay%' Qe lrnae ay qe `rrabcl na n`~ly eqgërkily yke zrliay`r ae lrnae na
g`bek~qn `yiaenae~a l nayiaenae~a%Ke~ar}`cl na ic`ya%' Ay qe `rrabcl pqa namkea qe` ic`ya nkb`gly na =:'=? c` iq`c ya
cc`g` ke~ar}`cl na ic`ya% Cly eøgarly ~argke`cay =: { =? ya cc`g`e cçgk~ay na ic`yay l
cçgk~ay na ic`ya kemarklr { yqzarklr% Ac ke~ar}`cl =:'=? keicq{a# ~aùrki`gae~a# c`y
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 27/846
Nr% Fëi~lr Pqa}anl Qrç`y
0'06
gankikleay 1;%1'=?%1 { ya cc`g`e cçgk~ay na ic`yay% Ay~ly ya lo~kaeae yqg`enl ac
cçgk~a yqzarklr na qe ke~ar}`cl ile ac cçgk~a kemarklr nac ykbqkae~a ke~ar}`cl na ic`ya {
nk}knkaenl ae~ra ?%Ic`yay na zqe~l ke~argankl l g`ri`y na ic`yay%' C`y ic`yay na zqe~l ke~argankl l
g`ri`y na ic`yay yle ac zqe~l gankl na qe ke~ar}`cl na ic`ya pqa ya lo~kaea yqg`enl
cly cçgk~ay yqzarklray a kemarklray { nk}knkaenl ae~ra nly% Zlr ahagzcl# ac zqe~l gankl
nac ke~ar}`cl =:'=? ay &=: $ =?+,? 7 =0 {# `yç yqiayk}`gae~a%
^`g`ðly na ke~ar}`cly na ic`ya% Ac ~`g`ðl na qe ke~ar}`cl na ic`ya ay c` nkmaraeik`
ae~ra cly cçgk~ay l ckenarly yqzarklray a kemarklray%
_abc`y z`r` f`iar nky~rkoqikleay na mraiqaeik`
0% Na~argke`r cly eøgarly gày zapqaðly { gày br`enay na cly n`~ly yke zrliay`r%
?% Nk}knkr ac r`ebl ae qe eøgarl ile}aekae~a na ke~ar}`cly na ic`yay pqa ~aeb`e ac
gkygl ~`g`ðl% Yk ay~l el ay zlykoca# qy`r ke~ar}`cly na ic`ya na nkmarae~ay ~`g`ðly%
9% Na~argke`r ac eøgarl na loyar}`ikleay pqa i`ae nae~rl na i`n` qel na ay~ly
ke~ar}`cly na ic`yay%
>% Cly cçgk~ay na ic`yay el naoae na ilkeiknkr ile cly n`~ly ra`cay% C` mùrgqc` z`r`
i`ciqc`r ac ~`g`ðl na ic`ya na qe` nky~rkoqikùe na mraiqaeik` ay8
k 7 &f ' c+ , d &0'0:+
Nlena8
k 7 ac ~`g`ðl nac ke~ar}`cl na ic`ya
f 7 ac }`clr nac ç~ag gày `c~l
c 7 ac }`clr nac ç~ag gày o`hld 7 eøgarl na ic`yay
^kzly na iqr}`y na mraiqaeik`
0% Iqr}` na mraiqaeik` ykgë~rki` l ae mlrg` na i`gz`e`% Qe ahagzcl kgzlr~`e~a ay
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 28/846
Nr% Fëi~lr Pqa}anl Qrç`y
0'0;
c` iqr}` elrg`c%
?% Iqr}` `ykgë~rki` q lockiq` iq{ly av~ragly na c` iqr}` ay~àe `c c`nl naraifl l `c
kxpqkarnl nac gàvkgl iae~r`c%9% Iqr}` na mraiqaeik` ae mlrg` na H%
>% Iqr}` na mraiqaeik` ae mlrg` na Q%
1% Iqr}` na mraiqaeik` okgln`c pqa ~kaea nly gàvkgly%
=% C` iqr}` na mraiqaeik` gqc~kgln`c pqa ~kaea gày na nly gàvkgly%
Mkbqr` 0%9 Bràmki`y gly~r`enl cly ~kzly na iqr}`y na mraiqaeik` &Yzkabac# 0;=0+%
Fky~lbr`g`y { zlcçblely na mraiqaeik`
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 29/846
Nr% Fëi~lr Pqa}anl Qrç`y
0'?:
C` mlrg` gày ilgøe na razrayae~`ikùe bràmki` na qe` nky~rkoqikùe na mraiqaeik` ay ac
fky~lbr`g`% Ay~ly fky~lbr`g`y ileyky~ae ae rai~àebqcly `n{`iae~ay# c`y `c~qr`y na cly
iq`cay razrayae~`e c`y mraiqaeik`y na ic`yay# gkae~r`y pqa yqy o`yay ya av~kaenae ae~rayqiayk}`y mrle~ar`y na ic`yay% Ay~l pqkara naikr pqa ~kaeae o`yay ylora c` `oyiky` ile
iae~rly ae c`y g`ri`y na ic`yay { ile c`y clebk~qnay kbq`c ` cly ke~ar}`cly na ic`yay%
Zlr l~rl c`nl# cly zlcçblely na mraiqaeik` yle bràmki`y na cçea`y na mraiqaeik`y
na ic`yay pqa ya br`mki`e ile~r` c`y ic`yay na g`ri`y% Ya lo~kaeae ileai~`enl cly
zqe~ly gankly na rrko` na cly rai~àebqcly ae cly fky~lbr`g`y%
Mkbqr` 0%>% Ae cly fky~lbr`g`y { zlcçblely na mraiqaeik` ya `ily~qgor` ` yqg`r c`y
av~aeykleay zp { ry z`r` c` ykbqkae~a g`ri` na ic`ya gày o`h` { gày `c~` pqa ~kaeae c`
ilrrayzlenkae~a ic`ya na mraiqaeik` na iarl% Ae ~`cay i`yly# c` yqg` na c`y àra`y na
cly rai~àebqcly ay kbq`c `c àra` ~l~`c ikriqen`n` zlr ac zlcçblel na mraiqaeik` { ac aha
na c`y apqky% &Ac`olr`ikùe zrlzk`+Nky~rkoqikleay na mraiqaeik` rac`~k}`
C` mraiqaeik` rac`~k}` na qe ke~ar}`cl na ic`ya ay c` mraiqaeik` na c` ic`ya nk}knkn`
ae~ra c` mraiqaeik` ~l~`c na ~ln`y c`y ic`yay { ya avzray` ilgl zlriae~`ha%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 30/846
Nr% Fëi~lr Pqa}anl Qrç`y
0'?0
Ahagzcl )?:% F`iar qe` ~`oc` na nky~rkoqikùe ile ke~ar}`cly na ic`ya { c` mraiqaeik`
rac`~k}` z`r` c`y `c~qr`y na 0:: ay~qnk`e~ay na qe` qek}arykn`n%
^@OC@ 0%:% @c~qr`y na cly ay~qnk`e~ay% &Yzkabac# 0;=0+% \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ Nky~rkoqikùe na c`y `c~qr`y Mraiqaeik` rac`~k}`
zlr ke~ar} cly na ic`ya na ay~qnk`e~ay &!+
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
=: ' =? zqcb`n`y 1 !
=9 ' =1 06 !
== ' =6 >? !
=; ' 20 ?2 !
2? ' 2> 6 !
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
^l~`c 0:: !
Nky~rkoqikleay na mraiqaeik`y `iqgqc`n`y { nky~rkoqikleay na mraiqaeik`y
rac`~k}`y iqgqc`n`y
@pqç ya nkyiq~kràe c`y nky~rkoqikleay na mraiqaeik`y `iqgqc`n`y { c` mraiqaeik`
rac`~k}` `iqgqc`n` pqa ya lo~kaea nk}knkaenl c` mraiqaeik` `iqgqc`n` zlr c`
mraiqaeik` ~l~`c%
Ahagzcl )?0% ^`oqc`r cly }`clray na c` ~`oc` na mraiqaeik` na 1:: loyar}`ikleay
mlrg`enl qe` ~`oc` ile cly ke~ar}`cly na ic`ya gày `zrlzk`nly# ile c` mraiqaeik`# c`
mraiqaeik` rac`~k}` &!+# c` mraiqaeik` `iqgqc`n` { c` mraiqaeik` rac`~k}` `iqgqc`n`%
Qy`r z`zac na zrlo`okckn`n { aeile~r`r ac zrlgankl `rk~gë~kil { c` nay}k`ikùe
ay~àen`r% Ilemkrg`rcly bràmki`gae~a { i`ciqc`rcly%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 31/846
Nr% Fëi~lr Pqa}anl Qrç`y
0'??
^@OC@ 0%0% Mraiqaeik`y na 1:: loyar}`ikleay na mlym`~ly &gb,C+% &Ac`olr`ikùe
zrlzk`+%
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ V m V m V m V m
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ ?: 0 ' ' ' ' ' '?0 : 9= 2 10 ?: == =?? : 92 ; 1? 0; =2 1?9 0 96 0: 19 0; =6 >?> 0 9; 00 1> 06 2: 9?1 0 >: 0? 11 06 2: 9?= 0 >0 09 1= 02 20 ??2 0 >? 0> 12 0= 2? ??6 ? >9 0= 16 0> 29 0?; ? >> 02 1; 09 2> 09: 9 >1 06 =: 0? 21 090 9 >= 06 =0 00 2= 09? > >2 0; =? 0: 22 099 1 >6 0; =9 ; 26 :9> = >; ?: => 2 2; :91 = 1: ?: =1 = 6: 0
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 32/846
Nr% Fëi~lr Pqa}anl Qrç`y
0'?9
^@OC@ 0%?% ^`oc` na mraiqaeik`y na 1:: i`yly na mlym`~ly% &Ac`olr`ikùe zrlzk`+ \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ Ke~ar}`cl na ic`ya m m% r%&!+ m% `% m% r% `% &!+
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ 3 9:%1 09 ?%= 09 ?%=9:%1'91%1 ?> >%6 92 2%>91%1'>:%1 >; ;%6 6= 02%?>:%1'>1%1 26 01%= 0=> 9?%6>1%1'1:%1 ;= 0;%? ?=: 1?%:1:%1'11%1 ;> 06%6 91> 2:%611%1'=:%1 2? 0>%> >?= 61%?=:%1'=1%1 >9 6%= >=; ;9%6=1%1'2:%1 ?0 >%? >;: ;6%:5 2:%1 0: ?%: 1:: 0::%:
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ ^l~`c 1::
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 33/846
Nr% Fëi~lr Pqa}anl Qrç`y
0'?>
Mkbqr` 0%1% Z`zac na zrlo`okckn`n gly~r`enl c`y 1:: loyar}`ikleay na mlym`~ly
rac`ikle`n`y ile c` ^@OC@ 0%?% &Ac`olr`ikùe zrlzk`+@e`ckx`enl c` Mkbqr` 0%1# ya zqana }ar pqë# z`r` i`ciqc`r ac zrlgankl ± cli`ckx`gly
%1: ae c` lrnae`n` { zlr ke~arzlc`ikùe i`ciqc`gly ac }`clr na 1:% Kbq`cgae~a# z`r`
i`ciqc`r c` nay}k`ikùe ay~àen`r ω# ely gl}agly ` %6> { zlr ke~arzlc`ikùe i`ciqc`gly
ac }`clr na 0:# pqa ay~à ae~ra 1: { =:%
Ahagzcl )??% Z`r` cly ykbqkae~ay >: n`~ly na `eàckyky na `bq` na ileiae~r`ikleay na
i`cikl# ae gb,C# ile~ay~`r c`y ykbqkae~ay zrabqe~`y8
&`+ Iley~rqkr qe` ~`oc` na mraiqaeik`y ile ke~ar}`cly na 1 { ay~kg`r ac zqe~l
ke~argankl l g`ri` na ic`ya%
&o+ Iley~rqkr l~r` ~`oc` gày ile ke~ar}`cly na ~`g`ðl ; { ay~kg`r ac zqe~l ke~argankl
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 34/846
Nr% Fëi~lr Pqa}anl Qrç`y
0'?1
l g`ri` na ic`ya%
&i+ Z`r` `goly i`yly iley~rqkr qe fky~lbr`g` { qe zlcçblel na mraiqaeik` { ~`gokëe#
ae mqeikùe na mraiqaeik` rac`~k}`%&n+ Z`r` `goly i`yly# iley~rqkr qe` bràmki` na mraiqaeik` `iqgqc`n` { mraiqaeik`
rac`~k}` `iqgqc`n`%
&a+ Qy`r z`zac na zrlo`okckn`n z`r` ay~kg`r ac zrlgankl `rk~gë~kil { c` nay}k`ikùe
ay~àen`r% Ilgz`r`r ay~ly rayqc~`nly ile ac iàciqcl nac zrlgankl { c` nay}k`ikùe
ay~àen`r qy`enl c`y mùrgqc`y ay~`nçy~ki`y%
^@OC@ 0%9% ^`oc` gly~r`enl c`y ileiae~r`ikleay na i`cikl na >: `eàckyky na `bq`%&Ac`olr`ikùe zrlzk`+
096 0=> 01: 09? 099 0?1 0>; 0120>= 016 0>: 0>2 09= 0>6 01? 0>>0=6 0?= 096 02= 0=9 00; 01> 0=10>= 029 0>? 0>2 091 019 0>: 0910=0 0>1 091 0>? 01: 01= 0>1 0?6
Ylcqikùe8
Ac r`ebl ay na 02= ' 00; 7 12 gb,C
Yk ya qy`e ke~ar}`cly na ic`ya na ~`g`ðl 1# cly ke~ar}`cly na ic`ya yle 12,1 7 0?#
`zrlvkg`n`gae~a% Yke ago`rbl# yk ya qy`e ke~ar}`cly na ic`ya na ~`g`ðl ;# cly
ke~ar}`cly na ic`ya yle 12,; 7 =# `zrlvkg`n`gae~a% C`y ~`oc`y na `o`hl gqay~r`e ay~`y
ay~kg`ikleay%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 35/846
Nr% Fëi~lr Pqa}anl Qrç`y
0'?=
^@OC@ 0%>% ^`oc` na mraiqaeik`y na c`y ileiae~r`ikleay na I`cikl &I`+ qy`enl qe
ke~ar}`cl na ~`g`ðl 1% &Ac`olr`ikùe zrlzk`+
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ Ke~ar}`cl na ic`ya G`ri` na ic`ya m m%`% m%r% m%r%`%
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ 006 ' 0?? 0?: 0 0 ?%1! ?%1!0?9 ' 0?2 0?1 ? 9 1%:! 2%1!0?6 ' 09? 09: ? 1 1%:! 0?%1!099 ' 092 091 > ; 0:%:! ??%1!096 ' 0>? 0>: = 01 01%:! 92%1!0>9 ' 0>2 0>1 6 ?9 ?:%:! 12%1!0>6 ' 01? 01: 1 ?6 0?%1! 2:%:!019 ' 012 011 > 9? 0:%:! 6:%:!016 ' 0=? 0=: ? 9> 1%:! 61%:!0=9 ' 0=2 0=1 9 92 2%1! ;?%1!0=6 ' 02? 02: 0 96 ?%1! ;1%:!029 ' 022 021 ? >: 1%:! 0::%:!
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ ^l~`c >:
^@OC@ 0%1% ^`oc` na mraiqaeik`y na c`y ileiae~r`ikleay na I` qy`enl qe ke~ar}`cl
na ~`g`ðl ;% &Ac`olr`ikùe zrlzk`+
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ Ke~ar}`cl na ic`ya Zqe~l ke~argankl m m%`% m%r% m%r%`%
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ 006 ' 0?= 0?? 9 9 2%1! 2%1!0?2 ' 091 090 1 6 0?%1! ?:%:!09= ' 0>> 0>: ; 02 ??%1! >?%1!0>1 ' 019 0>; 0? ?; 9:%:! 2?%1!01> ' 0=? 016 1 9> 0?%1! 61%:!0=9 ' 020 0=2 > 96 0:%:! ;1%:!02? ' 06: 02= ? >: 1%:! 0::%:!
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ ^l~`c >:
Cly keikyly &i+# &n+ { &a+ ya rayar}`e z`r` pqa ac ay~qnk`e~a cly f`b`%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 36/846
Nr% Fëi~lr Pqa}anl Qrç`y
0'?2
^`gokëe ya zqana i`ciqc`r ac zrlgankl `rk~gë~kil na qe` nky~rkoqikùe na
mraiqaeik`# iq`enl ya n`e cly ke~ar}`cly na ic`ya { c`y mraiqaeik`y% C` mùrgqc` z`r`
~`cay i`yly ay8 V 7 Ωm V , Ωm 7 Ωm V , eΩ &0'00+
Ahagzcl )??% Ya n`e cly ykbqkae~ay n`~ly na ~agzar`~qr`y `gokae~`cay ae br`nly
M`fraefak~ &lM+ ae c` ~`oc` na `o`hl%
^@OC@ 0%=% ^`oc` gly~r`enl cly n`~ly% &Ac`olr`ikùe zrlzk`+^agzar`~qr y &lM+ G`ri` na ic`ya & V + m m V
=: ’ =? =0 1 1 v =0 7 9:1=9 ’ =1 => 06 => v 06 7 001?== ’ =6 =2 >? =2 v >? 7 ?60>=; ’ 20 2: ?2 2: v ?2 7 06;:2? ’ 2> 29 6 29 v 6 7 16>
E 7Ωm 7 0:: Ωm V 7 =2>1
Zlr cl ~`e~l# V 7 Ωm V , Ωm 7 Ωm V , E 7 =2>1 , 0:: 7 =2%>1 lM
Nk`br`g`y na ~`ccl { flh` qy`enl ac zrlbr`g` Gkek~`o
Ahagzcl ) ?9% Z`r` kcqy~r`r c` iley~rqiikùe na qe` bràmki` na ~`ccl { flh`#ileyknëraya c` ~`oc` na `o`hl# c` iq`c gqay~r` c`y gankikleay na >: loyar}`ikleay%
^@OC@ 0%2% ^`oc` gly~r`enl c`y gankikleay na >: loha~ly% &Ac`olr`ikùe zrlzk`+%?%? >%0 9%1 >%1 9%? 9%2 9%: ?%=9%> 0%= 9%0 9%9 9%6 9%0 >%2 9%2?%1 >%9 9%> 9%= ?%; 9%9 9%; 9%09%9 9%0 9%2 >%> 9%? >%0 0%; 9%>
>%2 9%6 9%? ?%= 9%; 9%: >%? 9%1 \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
Zrliankgkae~l8
Z`r` mlrg`r ac nk`br`g` na ~`ccl { flh`# ya yaz`r` i`n` loyar}`ikùe ae nly z`r~ay
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 37/846
Nr% Fëi~lr Pqa}anl Qrç`y
0'?6
ileyky~ae~ay na qe ~`ccl { qe` flh`% Ykaenl `yç# ac ~`ccl razrayae~` ac nçbk~l pqa
zraiana `c zqe~l naikg`c {# c` flh`# ilrrayzlena `c nçbk~l ` c` naraif` nac zqe~l
naikg`c% Zlr ahagzcl# ile ac eøgarl 9%2# ac nçbk~l 9 razrayae~` ac ~`ccl { ac nçbk~l 2razrayae~` c` flh`% Na `iqarnl ` cly n`~ly na c` ^@OC@ 0%6 f`{ iq`~rl ~`ccly# ay
naikr# 0# ?# 9# >% Qe` }ax faifl ay~l# ya knae~kmki`e cly eøgarly ` c` naraif` nac
zqe~l naikg`c ilrrayzlenkae~ay ` i`n` ~`ccl% Zlr ahagzcl# z`r` ac ~`ccl 0 f`{ nly
flh`y# = { ;4 z`r` ac ~`ccl ? f`{ 1 flh`y# ay naikr# ?# 1# =# ; { 1# a~i% C` ^@OC@ 0%6
na `o`hl razrayae~` c` bràmki` na ~`ccl { flh`y z`r` ay~a zrlocag`%
El loy~`e~a# z`r` zlnar iley~rqkr c` ^@OC@ 0%6 ya zqana qy`r ac Gkek~`o na
`iqarnl ` c`y ykbqkae~ay kenki`ikleay8
Br`zf ↟ Y~ag'`en'ca`m
Ae ac raiq`nrl pqa `z`raia zlear c`y }`rk`ocay na c` ilcqge` I0 ae c` }ae~`ekcc` na
–Y~ag'`en'ca`m‗ { ae c` }ae~`ekcc` na –Keiragae~y‗ zlear 0% Ay~l zrlnqia cly n`~ly
na c` ^@OC@ 0%6 gly~r`n` `o`hl%
^@OC@ 0%6% ^`oc` gly~r`enl cly rayqc~`nly na ~`ccl { flh` ilrrayzlenkae~ay ` c`y
loyar}`ikleay na c` ^@OC@ 0%2% \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ Y~ag'`en'Ca`m Nkyzc`{8 Gankikleay na >: loha~ly
Y~ag'`en'ca`m lm Gankikleay na >: loha~ly E 7 >:Ca`m Qek~ 7 :%0:
Mraiqaeik` ^`ccly Flh`y
? 0 =;2 ? ?1==;
&?1+ 9 ::0000???999>>>11=22266;;6 > 00?9>122
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 38/846
Nr% Fëi~lr Pqa}anl Qrç`y
0'?;
Yke ago`rbl# cly rayqc~`nly na c` ^@OC@ 0%6 el n`e qe z`elr`g` `naiq`nl na c`
nky~rkoqikùe na cly n`~ly% Z`r` ragank`r ay~` yk~q`ikùe ya eaiayk~` `qgae~`r ac
eøgarl na ~`ccly ae c` bràmki`% Qe` g`ear` ykgzca na f`iarcl ay nloc`enl i`n`~`ccl% Z`r` ay~l# eqa}`gae~a ke~rlnqikr cly n`~ly ilgl ya fkxl `e~arklrgae~a { ae c`
}ae~`ekcc` na –Keiragae~y‗ zlear %1% Ay~l zrlnqia c` ~`oc` na `o`hl%
^@OC@ 0%;% ^`oc` gly~r`enl cly ~`ccly nlocay { na flh`y%
Y~ag'`en'Ca`m Nkyzc`{8 Gankikleay na >: loha~ly
Y~ag'`en'ca`m lm Gankikleay na >: loha~ly E 7 >:
Ca`m Qek~ 7 :%0:Mraiqaeik` ^`ccly Flh`y
? 0 =;9 ?( ?2 ? 1==;
&01+ 9( ::0000???999>>>06 9 11=22266;;
6 >( 00?9>9 > 122
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
C`y ~`oc`y na c`y nky~rkoqikleay na ~`ccl { flh` ya zqanae qy`r z`r` ay~kg`r cly
ke~ar}`cly na ic`ya iq`enl ya f`iae nky~rkoqikleay na mraiqaeik`% Ac zrliankgkae~l
ay ilgl ykbqa8
0% Zrkgarl ya y`i` ac r`ebl na cly n`~ly% Zlr ahagzcl# na c` ^@OC@ 0%2 ac }`clr
gàvkgl ay >%2 { ac }`clr gçekgl ay 0%=# l ya`8 r`ebl 7 >%2 ’ 0%= 7 9%0%
?% Aeyabqkn` ya ay~kg` ac `eifl nac ke~ar}`cl nk}knkaenl ac r`ebl ae~ra ac eøgarl
na ~`ccly &2 ae ay~a i`yl+# ay naikr# 9%0 , 2 7 %>%
9% @flr`# z`r` ay~kg`r ac zrkgar ke~ar}`cl na ic`ya agzax`gly ile 0%1 { ca
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 39/846
Nr% Fëi~lr Pqa}anl Qrç`y
0'9:
yqg`gly %> z`r` n`r 0%;% Ac ykbqkae~a ke~ar}`cl na ic`ya ay ?%: gày %> z`r` n`r ?%>%
Ac ykbqkae~a ke~ar}`cl na ic`ya ay ?%1 gày %> z`r` n`r ?%; { `yç yqiayk}`gae~a# ilgl
ya gqay~r` ae c` ^@OC@ 0%0: na `o`hl%^@OC@ 0%0:% ^`oc` gly~r`enl cly ke~ar}`cly na ic`ya# ac zqe~l gankl# c`
mraiqaeik`# c` mraiqaeik` rac`~k}` { c` mraiqaeik` rac`~k}` `iqgqc`n`%
Ke~ar}`cl na Zqe~l Mraiqaeik` Mraiqaeik` Mraiqaeik` rac`~k}`
ic`ya gankl &m+ rac`~k}` &m%r%+ `iqgqc`n` &m%r%`%+
0%1 ’ 0%; 0%2 ? :%:1: :%:1:
?%: ’ ?%> ?%? 0 :%:?1 :%:21?%1 ’ ?%; ?%2 > :%0:: :%021
9%: ’ 9%> 9%? 01 :%921 :%11:
9%1 ’ 9%; 9%2 0: :%?1: :%6::
>%: ’ >%> >%? 1 :%0?1 :%;?1
>%1 ’ >%; >%2 9 :%:21 0%:::
Zlr l~rl c`nl# ile cly n`~ly na c` ^@OC@ 0%0: ya zqanae f`iar fky~lbr`g`y
na mraiqaeik` rac`~k}`# ile iqr}`y elrg`cay ylorazqay~`y { iqr}`y na mraiqaeik`
rac`~k}` `iqgqc`n` z`r` i`ciqc`r gankn`y na cli`ckx`ikùe ilgl iq`r~kcay l
zariae~kcay% Zlr ahagzcl# cly iq`r~kcay nk}knae ac ilehqe~l na n`~ly ae iq`~rl z`r~ay
kbq`cay% Ykaenl `yç# ac zrkgar iq`r~kc l %?1 mr`i~kc &P0+ yaz`r` c` iq`r~` z`r~a kemarklr
na c`y ~ray iq`r~`y z`r~ay yqzarklray# ay~l ay# ac ?1! na c`y gankikleay na `o`hl% Ac
yabqenl iq`r~kc l %1: mr`i~kc &P?+ ay knëe~kil ` c` gank`e` l ya` pqa c` gk~`n na c`y
loyar}`ikleay ay~àe nao`hl na ay~a }`clr% C`y loyar}`ikleay `rrko` nac ~ariar iq`r~kc
l %21 mr`i~kc &P9+ yle c` iq`r~` z`r~a yqzarklr nac ilehqe~l na n`~ly% Mke`cgae~a# cly
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 40/846
Nr% Fëi~lr Pqa}anl Qrç`y
0'90
ke~ariq`r~kcay gknae c` nkmaraeik` ae~ra cly iq`r~kcay P0 { P?%
Na c` gkyg` g`ear`# ac ilehqe~l na n`~ly na c` gqay~r` ya zqana nk}knkr ae
0:: z`r~ay kbq`cay zlr gankl na zariae~kcay% Zlr ahagzcl# ac ;;`}l zariae~kc yaz`r`ac 0! gày `c~l nac ;;! ray~`e~a4 ac 6>`}l zariae~kc yaz`r` ac 0=! gày `c~l nac
6>! ray~`e~a% O`hl ay~`y ilenkikleay# ac 6>`}l zariae~kc ilrrayzlenkae~a `c }`clr
na c` }`rk`oca `ca`~lrk` x na c` nky~rkoqikùe elrg`c ay# `zrlvkg`n`gae~a# x 7 $0 {
zlr ykga~rç` ay x 7 '0%
Cly iq`r~kcay { zariae~kcay hqe~l ile c` ay~`nçy~ki` nayirkz~k}` ya zqanae
i`ciqc`r ile ac zrlbr`g` Gkek~`o qy`enl ac g`en`~l8
Y~`~ ↟ O`yki y~`~ky~kiy ↟ Nkyzc`{ Nayirkz~k}a Y~`~ky~kiy
Kbq`cgae~a# cly iq`r~kcay { zariae~kcay ~`gokëe ya zqanae i`ciqc`r na qe` bràmki` na
mraiqaeik` rac`~k}` `iqgqc`n` }y% }`clray na V %
Qy`enl cly n`~ly na c` ^@OC@ 0%2 }`gly ` zrlianar ` f`iar cly iàciqcly na c`
ay~`nçy~ki` nayirkz~k}`# cly iq`cay ya n`e ae c` ~`oc` na `o`hl%
^@OC@ 0%00% ^`oc` gly~r`enl c` ay~`nçy~ki` nayirkz~k}` nac ahagzcl )?9%
Nayirkz~k}a Y~`~ky~kiy8 Gankikleay na >: loha~ly
]`rk`oca E E( IqgZi~ Ga`e YA Ga`e Y~Na} ]`rk`eia Ilam]`r
Gankikleay >: : 0:: 9%>09 :%000 :%2:9 :%>;> ?:%=:
]`rk`oca Gkekgqg P0 Gank`e P9 G`vkgqg _`eba
Gankikleay 0%=:: 9%0:: 9%>:: 9%621 >%2:: 9%0::
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 41/846
Nr% Fëi~lr Pqa}anl Qrç`y
0'9?
Gankikleay na >: loha~ly
M r a p q a e i
{
>%6>%:9%??%>0%=
0?
0:
6
=
>
?
:
Ga e 9 %> 09
Y~Na} :%2:?6
E >:
Fky~lbr`g &k~f Elrg`c Iqr}a+ lm Gankikleay na >: loha~ly
Mkbqr` 0%=% Mkbqr` gly~r`enl ac fky~lbr`g` na mraiqaeik` ile iqr}` elrg`c
ylorazqay~`%
@flr`# ac zrliankgkae~l z`r` f`iar qe` bràmki` na mraiqaeik` rac`~k}` `iqgqc`n` ae
mqeikùe na cly }`clray na V ya zrliana na c` ykbqkae~a g`ear`8
0% Krya `8
I`ci ↟ Zrlo`okck~{ Nky~rkoq~kle ↟ Elrg`c
?% Ae ac raiq`nrl pqa `z`raia zqe~a`r –Iqggqc`~k}a nky~rkoq~kle‗ { `cg`iae`r cly
n`~ly na c` nky~rkoqikùe na mraiqaeik` `iqgqc`n` ae I?%
9% Z`r` f`iar c` bràmki` na mraiqaeik` rac`~k}` `iqgqc`n` }y% }`clray na V # krya `8
Br`zf ↟ Yi`~~arzcl~ ↟ Rk~f ileeai~ ckea
>% Ae c` }ae~`e` na –Yi`~~arzcl~ k~f ileeai~ ckea‗ ke~rlnqikr cly n`~ly na c`
nky~rkoqikùe na mraiqaeik` `iqgqc`n` &na c` ilcqge` I?+ }y% cly }`clray na V %
1% Ae c` }ae~`ekcc` na –Yi`~~arzcl~'Yi`ca‗# ccae`r ~lnly cly raiq`nrly%
Na ay~` g`ear`# z`r` i`ciqc`r c` nky~rkoqikùe na mraiqaeik` `iqgqc`n` zrlianar
ilgl ae ac z`yl 0 na `rrko`% ^ln`y ay~`y ùrnaeay zrlnqiae c` ~`oc` ile~aekaenl cly
}`clray na V &el ya gqay~r` `pqç+% C` bràmki` na c`y mraiqaeik`y rac`~k}`y
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 42/846
Nr% Fëi~lr Pqa}anl Qrç`y
0'99
`iqgqc`n`y { }`clray na c`y loyar}`ikleay ya f`ia ilgl ae ac z`yl 9 na `rrko`%
Na c` bràmki` na `o`hl ya zqanae caar ~lnly cly iq`r~kcay { zariae~kcay
naya`nly%
Gankikleay na >: loha~ly
N k y ~ r k o q i k l e
n a
m % r % ` %
1%:>%1>%:9%19%:?%1?%:0%1
1%:>%1>%:9%19%:?%1?%:0%1
0%:
:%6
:%=
:%>
:%?
:%:
0%:
:%6
:%=
:%>
:%?
:%:
Mkbqr` gly~r`enl c` br`mki` na m%r%`% { }`clr ay na V
Mkbqr` 0%2% Mkbqr` gly~r`enl c` bràmki` na c` mraiqaeik` rac`~k}` `iqgqc`n` }aryqy}`clray na V %
Ahagzcl )?>% Aeile~r`r cly iq`r~kcay &P0# P? { P9+ na qe` gqay~r` na 01
gankikleay na yùcknly yqyzaenknly# ae qekn`nay na gb,C# na qe` gqay~r` na `bq`
rayknq`c%
2 0; 0? 1 02 ?; 6 0; > ?2 9: 0 > 0: ?0 \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
Ylcqikùe8
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 43/846
Nr% Fëi~lr Pqa}anl Qrç`y
0'9>
Zrkgarl ya `rrabc`e cly n`~ly ae mlrg` `yiaenae~a# ay~l ay8
0# ># ># 1# 2# 6# 0:# 0?# 02# 0;# 0;# ?0# ?2# ?;# 9:
← ← ← P0 P? P9
Ac zrkgar iq`r~kc &P0+ ay 1% Ac yabqenl iq`r~kc &P?+ l c` gank`e` ay 0? { ac ~ariar iq`r~kc &P9+ ay ?0%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 44/846
Nr% Fëi~lr Pqa}anl Qrç`y
0'91
Aharikikly I`zç~qcl 0
0%0% I`ciqc`r ac zrlgankl# c` }`rk`ex` { c` nay}k`ikùe ay~àen`r na c`y loyar}`ikleay na
c` gqay~r`8 0?# =# 2# 9# 01# 0:# 06# 1% &;%1# ?2%0# 1%?+
0%?% Aeile~r`r c` nay}k`ikùe ay~àen`r { ac zrlgankl na cly }`clray8 9# =# ?# 0# 2# 1% Na
`iqarnl ` c` rac`ikùe na cly }`clray lo~aeknly nac zrlgankl { c` nay}k`ikùe ay~àen`r l
}`rk`ex`% ¼Pqë ileicqykleay ya zqanae y`i`r<
0%9% Ayirkokr cly ykbqkae~ay ~ërgkely qy`enl `el~`ikùe na yqg`~lrk`%
0:
&`+ V ?0 $ V
?? $ V
?9 $ %%%$ V
?0: &Ω Vk+
v7:
1
&o+ & V 0 $ [ 0+ $ & V ? $ [ ?+ $ %%%% $ & V 1 $ [ 1+ &Ω V k$[ k+v7:
&i+ m 0 V 0[ 0 $ m ? V ?[ ? $ m 9 V 9[ 9 $ m > V >[ >
0%>% Aeile~r`r c` nay}k`ikùe zrlgankl na8
&`+ '9# 2#';#1
&o+ ?%># 0%=# 9%6# >%0# 9%>
0%1% Ac r`ebl na cly eøgarly 1# 9# 6# ># 2# =# 0?# ># 9 ay8 &;+
0%=% Na 1: gankikleay c` gày br`ena ay 6%9> Db% Yk ac r`ebl ay %>=# aeile~r`r c`
gankikùe gày zapqað`%
0%2% Ile}ar~kr c`y ykbqkae~ay loyar}`ikleay ` qekn`nay na nay}k`ikùe ay~àen`r8 =# ?# 2#
1% &x=7:%>=# x?7'0%9;# x27:%;9# x17:+
0%6% Ayirkokr cly ykbqkae~ay ~ërgkely ae mlrg` na yqg`~lrk`%
= &`+ Ω V h
h70
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 45/846
Nr% Fëi~lr Pqa}anl Qrç`y
0'9=
>
&o+ Ω & {0 ' 9+ ? h70
1 &i+ Ω m d vd
d70 0%;% Qy`enl ac zrlbr`g` na ilgzq~`nlr` Gkek~`o# AVIAC l qe` i`ciqc`nlr` na
olcykccl# aeile~r`r8
&`+ Ac zrlgankl `rk~gë~kil &;1%6>+
&o+ C` nay}k`ikùe ay~àen`r
&i+ Ac arrlr ay~àen`r nac zrlgankl
&m+ C` }`rk`ex` &0:=%>;+
^`oc` gly~r`enl cly n`~ly nac zrlocag`% &Ac`olr`ikùe zrlzk`+Loyar}`ikùe v s 2: 2> 26 6? 6= ;: ;> ;6 0:? 0:= 00: 00> 006 0?? 0?= \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
Mraiqaeik` ms > ; 0= ?6 >1 == 61 2? 1> 96 ?2 06 001 \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
0%0:% Ae qe` nky~rkoqikùe# yk ac zrlgankl ay 1%:# c` gank`e` ay 2%: { c` gln` ay ;%:#
ile~ay~`r ` cly ykbqkae~ay aeqeik`nly8
&`+ ¼Pqë ~kzl na yaybl ~kaea ay~` nky~rkoqikùe<
&o+ ¼Nùena ya aeiqae~r` c` g`{lr ileiae~r`ikùe na }`clray<
0%00% Ae qe` nky~rkoqikùe# yk ac zrlgankl ay na 0:%:# c` gank`e` ay na 6%: { c` gln` ay
na 1%:# ile~ay~`r c`y ykbqkae~ay zrabqe~`y8
&`+ ¼Pqë ~kzl na yaybl ~kaea ay~` nky~rkoqikùe< &Yaybl zlyk~k}l+&o+ ¼Nùena ya aeiqae~r`e c` g`{lr ileiae~r`ikùe na }`clray<
0%0?% Ae qe av`gae mke`c na ay~`nçy~ki`# cly br`nly mqarle8 0::# 0::# ==# =1# =># =:#
1;# 12# 16# 1:%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 46/846
Nr% Fëi~lr Pqa}anl Qrç`y
0'92
¼Ay ay~` nky~rkoqikùe lockiq` f`ik` c` naraif` l f`ik` c` kxpqkarn`< Hqy~kmki`r ac
`rbqgae~l qy`enl c` rac`ikùe nac zrlgankl# c` gank`e` { c` gln`%
0%09% Aeile~r`r ac zrlgankl balgë~rkil na qe` gqay~r` `ca`~lrk` na na loyar}`ikleay0:# 0?# 0=% &0?%>9+
0%0>% Yk ac zrlgankl `rk~gë~kil na qe` gqay~r` na 9: i`yly ay kbq`c ` 0: { c`
nay}k`ikùe ay~àen`r ay kbq`c ` ?# i`ciqc`r c` }`rk`oca ay~`en`rkx`n` ilrrayzlenkae~a `c
}`clr na V 7 01%
0%01% C` ~`oc` na `o`hl gqay~r` cly ilamkikae~ay na ke~ackbaeik` na 11: ekðly na qe`
ayiqac` acagae~`c% Aeile~r`r8
&`+ Ac zrlgankl `rk~gë~kil% &;2%:9+
&o+ C` nay}k`ikùe ay~àen`r% &09%??+
&i+ Ac arrlr ay~àen`r nac zrlgankl &:%1=+
^`oc` gly~r`enl cly n`~ly nac zrlocag`% &Ac`olr`ikùe zrlzk`+ \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ G`ri` na s 21 26 26 6? 6= ;0 ;> ;6 0:? 0:= 00: 00> 006 0?? 0?=ic`ya & V +
Mraiqaeik` &[+ s 19 1 0: ?: >1 =: 61 2? 1> 96 ?2 06 00 1: ?
0%0=% Cly ykbqkae~ay n`~ly ay~àe rac`ikle`nly ile c`y ~agzar`~qr`y# ae lI# na 0:
rabkleay na Gëvkil% C` ~`oc` na `o`hl gqay~r` ay~` yk~q`ikùe8
^`oc` gly~r`enl cly n`~ly nac zrlocag`% &Ac`olr`ikùe zrlzk`+ \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ ^agz% Mraiqaeik` Mraiqaeik` Mraiqaeik` Mraiqaeik` rac`~k}`&lI+ `iqgqc`n` rac`~k}` &!+ `iqgqc`n`
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ ?: 9 9 9:! 9:!?0?? ? ; ?:! ;:!?9 0
^l~`c 0:
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 47/846
Nr% Fëi~lr Pqa}anl Qrç`y
0'96
&`+ Ilgzca~`r c` ~`oc` na `rrko`%
&o+ F`iar bràmki`y na mraiqaeik` }aryqy mraiqaeik` rac`~k}`%
&i+ F`iar bràmki`y na mraiqaeik` `iqgqc`n` &m%`%+ }y% mraiqaeik` rac`~k}` `iqgqc`n`&m%r%`%+%
0%02% Ya y`i` qe` gqay~r` `ca`~lrk` na `eàckyky pqçgkily na ilgzqay~ly na iclrqrly
&Ic'+ avzray`nly ae qekn`nay na gb,C zrlianae~ay na qe` gqay~r` na `bq`y
rayknq`cay% Ay~ly `eàckyky ya fkikarle qy`enl ac gë~lnl na ek~r`~l na gariqrkl nayirk~l
ae ac ~av~l Gë~lnly Ay~àen`ray% C` ~`oc` ile cly }`clray na cly iclrqrly ya n` `o`hl8
^`oc` gly~r`enl cly n`~ly nac zrlocag`% &Ac`olr`ikùe zrlzk`+ \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ 02%?# 02%0# 02%:# 02%0# 0=%;# 02%:# 02%0# 02%:# 02%9# 02%?# 0=%;#
02%:# 02%0# 02%9# 02%?# 02%># 02%0# 02%0# 02%:# 02%0
&`+ Aeile~r`r ac zrlgankl% &02%00+
&o+ Aeile~r`r c` }`rk`ex`% &:%:02+
&i+ Aeile~r`r c` nay}k`ikùe ay~àen`r% &:%09?+
&n+ F`iar qe` ~`oc` na mraiqaeik` gly~r`enl c` mraiqaeik`# c` mraiqaeik` rac`~k}` { c`mraiqaeik` rac`~k}` `iqgqc`n`% &ac cai~lr cl naoarà f`iar+
&a+ F`iar qe fky~lbr`g`% &ac cai~lr cl naoarà f`iar+
&m+ F`iar qe zlcçblel na mraiqaeik`% &ac cai~lr cl f`rà+
&b+ ¼Pqë ~`e~` ykga~rç` f`{ ae ay~` nky~rkoqikùe< &ac cai~lr rayzlenarà ` ay~l+
0%06% Ilgzca~`r c` ~`oc` na `o`hl { f`iar qe` bràmki` ae mqeikùe na cly ke~ar}`cly na
c`y ileiae~r`ikleay na NOL# na c` mraiqaeik` &m+ { na c` mraiqaeik` rac`~k}` `iqgqc`n`
&m%r%`%+%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 48/846
Nr% Fëi~lr Pqa}anl Qrç`y
0'9;
^`oc` gly~r`enl cly n`~ly% &Ac`olr`ikùe zrlzk`+ \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ Ke~ar}`cly Eøgarl na Zqe~ly Mraiqaeik`
&Ilei% NOL+ `eàckyky ke~argankly rac`~k}` &!+ \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
1:%:: ' 1;%;; 6=:%:: ' =;%;; 0:2:%:: ' 2;%;; 0=6:%:: ' 6;%;; 0>;:%:: ' ;;%;; 0:0::%:: ' 0:;%;; 10:%:: ' 00;%;; ?
0%0;% Qe` lrb`ekx`ikùe i`rk~`~k}` pqa `{qn` ` n`gekmki`nly zlr fqr`i`eay f` faifl
qe` cky~` na nle`ikleay raikokn`y nqr`e~a ac zrayae~a `ðl# ae gkcay na zayly% Ac
zrlzùyk~l na ay~a ahagzcl ay ac na f`iar qe` ~`oc na nky~rkoqikùe na mraiqaeik
aeile~r`enl cly ke~ar}`cly na ic`ya gày `zrlzk`nly qy`enl c` ~ëieki` na nk`br`g`y na
~`ccl { flh`% C` ~`oc` na `o`hl gqay~r` cly n`~ly% Z`r` ay~l f`iar cl ykbqkae~a8
&`+ I`ciqc`r ac zrlgankl { c` gank`e`% &09;# 091+&o+ F`iar qe` ~`oc` na nky~rkoqikùe na mraiqaeik` qy`enl qe nk`br`g` na ~`ccl { flh`%
Aeile~r`r cly zqe~ly ke~argankly# c` mraiqaeik`# c` m% r% { c` mraiqaeik` rac`~k}`
`iqgqc`n` { iley~rqkr qe fky~lbr`g` { qe` bràmki` na m% r% `% ile~r` }`clray na V %
^`oc` gly~r`enl cly n`~ly nac zrlocag` &Ac`olr`ikùe zrlzk`+% \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
?19%: 029%> 002%: 0;0%? 010%>06?%: 09?%: 0=?%: ?0?%; 011%;
??0%: 016%: 091%: 0?>%> =6%;6;%2 ;1%= 6>%0 091%0 0?9%?
0:0%: 0?=%1 0>?%6 ?:%? 00;%: \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 49/846
Nr% Fëi~lr Pqa}anl Qrç`y
0'>:
0%?:% C` ykbqkae~a ~`oc` n` c`y agkykleay na ùvknly na `xqmra &YL? ae ~leac`n`y
gë~rki`y+ zrl}aekae~ay na ?:: zc`e~`y yknarørbki`y cli`ckx`n`y ae ikar~` rabkùe
kenqy~rk`c%^`oc` gly~r`enl cly n`~ly% &Ac`olr`ikùe zrlzk`+
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ Agkykùe na YL? &~le+ Eøgarl na zc`e~`y
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ 0%:: ' 0%:? =0%:? ' 0%:> ?=0%:> ' 0%:= 1?0%:= ' 0%:6 16
0%:6 ' 0%0: 9;0%0: ' 0%0? 010%0? ' 0%0> 10%0> ' 0%0= 0
&`+ I`ciqc`r ac zrlgankl `rk~gë~kil na c` nky~rkoqikùe%
&o+ I`ciqc`r c` nay}k`ikùe ay~àen`r%
&i+ I`ciqc`r c` gank`e` { c` gln` na c` nky~rkoqikùe%
0%?0% Ya n`e cly ykbqkae~ay n`~ly ae c` ~`oc` na `o`hl%^`oc` gly~r`enl cly n`~ly na ay~a zrlocag`% &Ac`olr`ikùe zrlzk`+
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ @c~qr` &zqcb`n`y+ G`ri` na ic`ya & v+ Mraiqaeik` m v
=: ' =? =0 1 1 v =0 7 9:1=9 ' =1 => 06 => v 06 7 001?== ' =6 =2 >? =2 v >? 7 ?60>=; ' 20 2: ?2 2: v ?2 7 06;:
2? ' 2> 29 6 29 v 6 7 16> \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
&`+ I`ciqc`r ac zrlgankl `rk~gë~kil% Yqbaraeik`8 qy`r c` mqeikùe nac zrlgankl kbq`c `
Ωm V,Ωm
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 50/846
Nr% Fëi~lr Pqa}anl Qrç`y
0'>0
0%??% Ya n` c` ykbqkae~a ~`oc` na nky~rkoqikùe na n`~ly &ke~ar}`cly na ic`ya+ na
agkykleay na z`r~çiqc`y `~glymërki`y gaelray na 0: gkir`y zrl}aekae~ay na }`rk`y
kenqy~rk`y% &Ac`olr`ikùe zrlzk`+ \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ Gankikleay na z`r~çiqc`y Eøgarl na kenqy~rk`y
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ 1:%:: ' 1;%;; 6=:%:: ' =;%;; 0:2:%:: ' 2;%;; 0=6:%:: ' 6;%;; 0>;:%:: ' ;;%;; 0:
0::%:: ' 0:;%;; 100:%:: ' 00;%;; ?
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
&`+ I`ciqc`r c` g`ri` na ic`ya V %
&o+ I`ciqc`r ac zrlgankl `rk~gë~kil%
&i+ I`ciqc`r c` mraiqaeik` rac`~k}` &m%r%+ { c` mraiqaeik` rac`~k}` `iqgqc`n` &m%r%`%+%
&n+ F`iar qe fky~lbr`g`%
&a+ Qy`r z`zac na zrlo`okckn`n z`r` }ar pqa ~`e~` qekmlrgkn`n f`{ ae cly n`~ly%
0%?9% Ilgzca~`r cly m`c~`e~ay na c` ~`oc` na `o`hl# na qe` nky~rkoqikùe na mraiqaeik` na
c`y }kn`y na >:: ~qoly na r`nkly% @nagày# f`iar cly iàciqcly zanknly `o`hl%
&`+ Aeile~r`r ac cçgk~a yqzarklr na c` pqke~` ic`ya% &2;;+
&o+ Aeile~r`r ac cçgk~a kemarklr na c` li~`}` ic`ya% &0:::+
&i+ Aeile~r`r c` g`ri` na ic`ya na c` yëz~kg` ic`ya% &;>;%1+
&n+ Aeile~r`r cly cçgk~ay na c` øc~kg` ic`ya% &0:;;%1'00;;%1+&a+ Aeile~r`r ac ~`g`ðl nac ke~ar}`cl na ic`ya% &0::+
&m+ Aeile~r`r c` mraiqaeik` na c` iq`r~` ic`ya% &2=+
&b+ Aeile~r`r c` m%r% na c` yav~` ic`ya% &01%1!+
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 51/846
Nr% Fëi~lr Pqa}anl Qrç`y
0'>?
&f+ Aeile~r`r ac ! na cly ~qoly iq{`y }kn`y ya`e 3 =:: flr`y% &?;%1!+
&k+ Br`mki`r cly n`~ly ae z`zac na zrlo`okckn`n { caar ac zrlgankl `rk~gë~kil { c`
nay}k`ikùe ay~àen`r na c` bràmki`%&h+ F`iar qe` br`mki` na mraiqaeik` rac`~k}` `iqgqc`n` }aryqy zqe~ly gankly {
i`ciqc`r cly zariae~kcay P0# P? { P9%
^`oc` gly~r`enl cly n`~ly nac zrlocag`% &Ac`olr`ikùe zrlzk`+ \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ ]kn` na cly El% na &m+ m%r% m%`% m%r%`% Zqe~l~qoly ~qoly gankl
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
9:: ' 9;; 0>>:: ' >;; >=1:: ' 1;; 16=:: ' =;; 2=2:: ' 2;; =66:: ' 6;; =?;:: ' ;;; >60::: ' 0:;; ??00:: ' 00;; =
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
0%?>% Ya n` c` ~`oc` na nao`hl ileyky~ae~a ae qe` gqay~r` `ca`~lrk` na gankikleay na
ùvknly na ek~rùbael &EL?+# zrlianae~ay na qe` zc`e~` na ~r`~`gkae~l na `bq`y
rayknq`cay% C` ~`oc` ile cly n`~ly ya n` `o`hl%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 52/846
Nr% Fëi~lr Pqa}anl Qrç`y
0'>9
^`oc` ile cly n`~ly% &Ac`olr`ikùe zrlzk`+Gankikleay na EL? Mraiqaeik` G`ri` na m%r m%r%`%&Ke~ar}`cly+ ic`ya & V +
9%: ’ 1%: 0>=%: ’ 6%: >=;%: ’ 00%: 160?%: ’ 0>%: 2=01%: ’ 02%: =606%: ’ ?:%:?0%: ’ ?9%: >6?>%: ’ ?=%: ???2%: ’ ?;%: =
^l~`c >::
&`+ Ccae`r cly m`c~`e~ay na c` ~`oc`%
&o+ I`ciqc`r ac zrlgankl `rk~gë~kil%
&i+ Qy`enl z`zac na br`mki` na zrlo`okckn`n# br`mki`r cly n`~ly%
&n+ Na c` br`mki` na zrlo`okckn`n lo~aekn` ae ac keikyl &i+ i`ciqc`r ac zrlgankl
`rk~gë~kil { ilgz`r`rcl ile ac zrlgankl lo~aeknl ae &o+%
&a+ Na c` gkyg` br`mki` na zrlo`okckn`n ay~kg`r c` nay}k`ikùe ay~àen`r%
0%?1% Ya n` c` ~`oc` na `o`hl% \\\\\\\\\\\\\\\\\\
V Z& V + \\\\\\\\\\\\\\\\\\
: :%612>0 :%091>? :%::209 :%:::0
\\\\\\\\\\\\\\\\\
Z`r` cly zrlocag`y na `o`hl aeile~r`r c`y ykbqkae~ay yqg`~lrk`y qy`enl c` ~`oc` na
`rrko`%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 53/846
Nr% Fëi~lr Pqa}anl Qrç`y
0'>>
0
&`+ Ω z& v+ &:%;;?6+v7:
? 0
&o+ Ω z& v+ ’ Ω z& v+v7: v7:
0
&i+ Ω z& v+ &:%;;?6+v7:
9
&i+ Ω z& v+ &0%:::+v7:
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 54/846
Nr% Fëi~lr Pqa}anl Qrç`y
0'>1
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 55/846
Nr% Fëi~lr Pqa}anl Qrç`y
?'0
I@ZK^QCL ?
Zrlo`okckn`n
Zrlo`okckn`n icàyki`%' Zrlo`okckn`n na mraiqaeik` rac`~k}`%' Zrlo`okckn`n
yqoha~k}`%' @vklg`y { zrlzkan`nay oàyki`y na c` zrlo`okckn`n%' Nk`br`g`y na
]aee { cbaor` na ilehqe~ly%' ëieki`y na ile~al8 _abc` na zrlnqi~l z`r` z`ray
lrnae`nly# c` rabc` na gqc~kzcki`ikùe gày baear`c# rabc` m`i~lrk`c# nk`br`g`y na
àrolc# zargq~`ikleay { ilgoke`ikleay%' _abc` gqc~kzcki`~k}` z`r` a}ae~ly
nazaenkae~ay a kenazaenkae~ay%' _abc` nk~k}` z`r` a}ae~ly gq~qly avicq{ae~ay {
a}ae~ly el gq~qly avicq{ae~ay%'
Ac nay`rrlccl na c` ~alrç` na c` zrlo`okckn`n g`~agà~ki` liqrrkù ae ac ykbcl 02# { ay~à
rac`ikle`n` ile ac eloca mr`eiëy @e~lkea Blgo`qcn { ile ac g`~agà~kil Mr`eiëy
Oc`kya Z`yi`c% Ac ay~qnkl na c` zrlo`okckn`n ay qe` r`g` na c`y g`~agà~ki`y pqa ya
kekikù f`ia 9:: `ðly%
G`ear`y na gankr c`y zrlo`okckn`nay8
&0+ C` zrlo`okckn`n icàyki`
&?+ C` zrlo`okckn`n na mraiqaeik` rac`~k}`
&9+ C` zrlo`okckn`n yqoha~k}`
Zrlo`okckn`n icàyki`
Ac ~ërgkel zrlo`okckn`n ya ramkara `c ay~qnkl na cl `ca`~lrkl { na c` keiar~knqgora% Ac
ileiaz~l icàykil na c` zrlo`okckn`n na qe a}ae~l @ ya namkea ilgl ykbqa8 yk f`{ `
zlykocay rayqc~`nly m`}lr`ocay c` liqrraeik` nac a}ae~l @ {#o rayqc~`nly naym`}lr`ocay
` c` liqrraeik` na @# { yk ~lnly cly rayqc~`nly yle kbq`cgae~a gq~qly avicq{ae~ay &pqa
el zqanae liqrrkr c` }ax+# ae~leiay c` zrlo`okckn`n na pqa @ liqrr` ya nael~` ilgl
Z&@+# ay naikr8
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 56/846
Nr% Fëi~lr Pqa}anl Qrç`y
?'?
` Eøgarl na rayqc~`nly m`}lr`ocay `c a}ae~l @
Z&@+ 7 „„„„ 7 „„„„„„„„„„„„„„„„„„„„ &?%:+
&` $ o+ Eøgarl ~l~`c na rayqc~`nly zlykocay
L~r` g`ear` na namkekr c` zrlo`okckn`n ay8
z 7 ZrtAw 7 f , e &?':`%+
Nlena8
A 7 ac ~kzl na a}ae~l pqa ay~`gly f`ikaenl
f 7 eøgarl na g`ear`y m`}lr`ocay na pqa zqan` liqrrkr ac a}ae~l l eøgarl na zqe~ly
ae ac a}ae~l nac ayz`ikl @
e 7 eøgarl ~l~`c na zlykocay rayqc~`nly l na eøgarl na zqe~ly ae ac ayz`ikl na c`
gqay~r` &Y +
C` zrlo`okckn`n na pqa el liqrr` ac a}ae~l ay p# ay naikr8
p 7 Zrtpqa el liqrr` Aw 7 0 ' f , e 7 0 ' ZrtAw &?'0+
Zlr cl ~`e~l# z $ p 7 0
Ahagzcl )0% Yk qe` glean` ~kaea nly i`r`y nael~`n`y zlr àbqkc`y l yaccly# i`ciqc`r c`
zrlo`okckn`n na pqa y`cb` qe yaccl%
Ylcqikùe8
Qy`enl c` mqeikùe &?':+ { nah`enl pqa @ ya` ac a}ae~l yaccl { O ac a}ae~l àbqkc`#
ae~leiay# c` zrlo`okckn`n na yaccly ay8
Z&@+ 7 0 , &0 $ 0+ 7 :%1%
Ahagzcl )?% Ae ac i`yl na qe n`nl pqa ~kaea = eøgarly8 0# ?# 9# ># 1# =# yk ac n`nl ay
fleay~l# ~lnly cly eøgarly ~kaeae c` gkyg` zrlo`okckn`n na y`ckr% Ykaenl yç# i`ciqc`r
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 57/846
Nr% Fëi~lr Pqa}anl Qrç`y
?'9
c`y ykbqkae~ay zrlo`okckn`nay8
&`+ C` zrlo`okckn`n na y`i`r ac eøgarl 0
&o+ C` zrlo`okckn`n na y`i`r cly eøgarly z`ray
&i+ C` zrlo`okckn`n na y`i`r cly eøgarly 9 l >
&n+ C` zrlo`okckn`n na el y`i`r cly eøgarly 9 l >
Ylcqikùe8
&`+ Z&y`i`r ac eøgarl 0+ 7 0 , &0 $ 1+ 7 0,=%
&o+ Ae ay~a i`yl f`{ 9 eøgarly z`ray ae c`y yaky i`r`y nac n`nl# zlr cl ~`e~l# c`
zrlo`okckn`n na y`i`r cly z`ray ay8 Z&z`ray+ 7 9,&9 $ 9+ 7 0,?
&i+ @pqç# ac a}ae~l zqana liqrrkr na nly g`ear`y# ay naikr# ilgl &9 l >+% Zlr cl ~`e~l#
Z&9 l >+ 7 ?,&? $ >+ 7 0,9%
&n+ C` zrlo`okckn`n na el y`i`r ac 9 l > ay8 p 7 0 ' 0,9 7 ?,9
Ahagzcl )9% Aeile~r`r c` zrlo`okckn`n na pqa qe` z`rah` ile 9 fkhly ~aenràe8
&`+ Av`i~`gae~a ? }`rleay & V 7 ?+
&o+ 9 }`rleay { 9 fagor`y
&i+ @ cl gày nly }`rleay & V ≮ ?+
&n+ Iq`enl gaely ? }`rleay & V ≡ ?+
&a+ Gày na ? fagor`y & V 5 ?+
&m+ Gaely na ? }`rleay & V 3 ?+
Ylcqikùe8
Nahagly pqa ac a}ae~l }`rùe ya` } {# ac a}ae~l fagor`# ya` f% @pqç ac ayz`ikl
gqay~r`c Y ya zqana f`iar na qe àrolc na zrlo`okckn`n { n` 6 rayqc~`nly8
Y 7 t}}}# }}f# }f}# }ff# f}}# f}f# ff}# fffw
&`+ Z&? }`rleay ae 9 e`ikgkae~ly+ 7 Z& V 7 ?+ 7 9,6 7 :%921
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 58/846
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 59/846
Nr% Fëi~lr Pqa}anl Qrç`y
?'1
C` zrlo`okckn`n yqoha~k}` ay qe nay`rrlccl rac`~k}`gae~a raikae~a% Ay~` zrlo`okckn`n ya
namkea ilgl ac br`nl na irankokckn`n l ilemk`ex` na qe a}ae~l pqa }`rç` ile ac hqkikl
l ay~`nl na àekgl na c` zaryle`% Ay~` zrlo`okckn`n ay ø~kc ae naikykleay mke`eikar`y {
l~rly ~kzly na ~r`o`hly%
_ac`ikùe ae~ra c` zrlo`okckn`n &qy`enl nky~rkoqikleay nkyira~`y+ { c` ay~`nçy~ki`
na kemaraeik` &qy`enl nky~rkoqikleay ile~keq`y+ qy`enl cùbki` nanqi~k}` a
kenqi~k}`
C` rac`ikùe ae~ra c` zrlo`okckn`n qy`enl nky~rkoqikleay nkyira~`y ilgl c` okelgk`c#
fkzarbalgë~rki` l c` Zlkyyle { c` ay~`nçy~ki` na kemaraeik` &qy`enl nky~rkoqikleayile~keq`y ilgl c` elrg`c# c` ~ na Ay~qnk`e~a# c` nky~rkoqikùe M# b`gg`# avzleaeik`c#
a~i%+ r`nki` ae ac faifl na pqë# ae ac zrkgar i`yl# ac r`xle`gkae~l }` nac ilehqe~l l na
c` zloc`ikùe f`ik` c` z`r~a &r`xle`gkae~l nanqi~k}l l cùbki` nanqi~k}`+% Ae ile~r`y~a#
ile c` ay~`nçy~ki` na kemaraeik`# ac r`xle`gkae~l }` nayna c` gqay~r` l c` z`r~a f`ik` c`
zloc`ikùe l ~l~`c &r`xle`gkae~l kenqi~k}l l cùbki` kenqi~k}`+%
@el~`ikùe z`r` aeile~r`r zrlo`okckn`nay
C`y `el~`ikleay qy`n`y ae aeile~r`r zrlo`okckn`nay ya namkeae ilgl8 Z nael~` qe`
zrlo`okckn`n4 @# O# I nael~`e a}ae~ly ayzaiçmkily {# Z&@+# nael~` c` zrlo`okckn`n na
pqa liqrr` ac a}ae~l @%
0% Z nael~` qe` zrlo`okckn`n
?% @# O# I nael~`e a}ae~ly ayzaiçmkily
9% Z&@+ nael~` c` zrlo`okckn`n na pqa liqrr` ac a}ae~l @
>% Z &O+ nael~` c` zrlo`okckn`n na pqa liqrr` ac a}ae~l O# a~i%
@vklg`y { zrlzkan`nay oàyki`y na c` zrlo`okckn`n
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 60/846
Nr% Fëi~lr Pqa}anl Qrç`y
?'=
0% Z`r` iq`cpqkar a}ae~l @# Z&@+≡ :% @nagày# c` zrlo`okckn`n el zqana yar g`{lr pqa
0# ek ~`gzlil eab`~k}`%
?% C` zrlo`okckn`n na qe ayz`ikl gqay~r`c ay8 Z&Y + 7 0
9% Yk @0# @?#%%%%# @d ay qe` ilcaiikùe mkek~` na a}ae~ly gq~qly avicq{ae~ay &pqa el
zqana liqrrkr ` c` gkyg` }ax+# ae~leiay8
d
Z&@0 ∨ @? ∨%%%%∨ @d + 7 Ω Z&@k+ &?'9+k70
Yk @0# @?# @9#%%% ay qe` ilcaiikùe kemkek~` na a}ae~ly gq~qly avicq{ae~ay# ae~leiay8
d Z&@0 ∨ @? ∨ @9 ∨%%%+ 7 Ω Z&@k+ &?'>+
k70
Ahagzcl )=% Ay~a ay qe ahagzcl `n`z~`nl nac ckorl na _kif`rn @% Hleyle# ke~k~qc`nl
Zrlo`okckn`n { Ay~`nçy~ki` z`r` Kebaekarly na Gkccar { Mraqen &0;;>+% C`y
zrlo`okckn`nay na pqa qe ileyqgknlr pqa zrqao` ac yar}kikl na qe eqa}l nkyzlyk~k}l
`e~kile~`gke`e~a z`r` q~ly# cl ic`ykmkpqa ilgl gq{ namkikae~a# namkikae~a# yqmkikae~a#
oqael# gq{ oqael l aviacae~a yle8 :%:2# :%0?# :%02# :%?0# { :%:00% ¼Iqàcay yle c`y
zrlo`okckn`nay na pqa c`y ic`ykmki`ikleay nac nkyzlyk~k}l ya`e<8
&`+ ¼Gq{ namkikae~ay<
&o+ ¼Namkikae~ay<
&i+ Yqmkikae~ay l oqae`y<
&n+ ¼Oqaely# gq{ oqaely l aviacae~ay<
Ylcqikùe8
Zqay~l pqa c`y zlykokckn`nay yle gq~q`gae~a avicq{ae~ay &pqa el zqanae liqrrkr c`
}ax+# c` yqy~k~qikùe nkrai~` na i`n` qe` na c`y ikeil ic`ykmki`ikleay# ae c` mqeikùe &?'9+
n` ilgl rayqc~`nl8
&`+'&i+ ay8 :%:2 $ :%0? $ :%02 $ :%9? 7 :%=6
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 61/846
Nr% Fëi~lr Pqa}anl Qrç`y
?'2
&n+ :%9? $ :%?0 $ :%00 7 :%=>
^argkelclbç` qy`n` ae zrlo`okckn`n
Iq`enl ya f`oc` na zrlo`okckn`n ya keicq{ae ~ërgkely ilgl8 avzarkgae~l# rayqc~`nly#
a}ae~ly# ayz`ikl gqay~r`c# ~alrç` na ilehqe~ly &qekleay# ke~aryaiikleay# ilgzcagae~l
ilgl @"+# a}ae~ly gq~qly avicq{ae~ay# }`rk`ocay ca`~lrk`y nkyira~`y &ay~liày~ki`y na
ileha~qr` l zrlo`okckn`n+# zrlo`okckn`n na mraiqaeik` rac`~k}`# zrlo`okckn`n yqoha~k} #
~ëieki`y na ile~al &ilgoke`ikleay { zargq~`ikleay# rabc` na gqc~kzcki`ikùe { nkikùe#
a~i%+# ~alrag` na O`{ay# kenazaenaeik`# a}ae~ly gq~qly avicq{ae~ay# nk`br`g`y na
]aee# àrolcay na zrlo`okckn`n# a~i% @cbqe`y namkekikleay na ay~ly ~ërgkely ya n`e
`o`hl%
Avzarkgae~l%' Qe avzarkgae~l ay qe zrliayl pqa ely {qn` lo~aear loyar}`ikleay
na nly l gày rayqc~`nly nky~ke~ly# nlena ac rayqc~`nl pqa liqrra el zqana yar
zranaikoca ile iar~ax`# ykel ae ~ërgkely na zrlo`okckn`n%
A}ae~l%' Ay qe` ilcaiikùe na qel l gày rayqc~`nly acagae~`cay na qe avzarkgae~l% Qe
a}ae~l ay qe yqoilehqe~l na qe ayz`ikl gqay~r`c% Zlr yqoilehqe~l ya ae~kaenaiq`cpqkar z`r~a na qe ilehqe~l# keicq{aenl ac ilehqe~l ae yq ~l~`ckn`n% @pqç# ~`gokëe
zqana f`oar ilehqe~ly }`içly nael~`nly zlr ю# cly iq`cay el zlyaae ekebøe acagae~l%
A}ae~ly gq~qly avicq{ae~ay% Nly a}ae~ly @ { O yle gq~qly avicq{ae~ay l nayqeknly#
yk yq ke~aryaiikùe @ ∫ O 7 ю# ay~l ay# yk @ { O el ~kaeae acagae~ly ae ilgøe% Zlr
ahagzcl# cly a}ae~ly @ { O ya nkia pqa yle gq~qly avicq{ae~ay l nayqeknly# yk @ { O
el zqanae liqrrkr ykgqc~àea`gae~a l ae qe ylcl aey`{l na qe avzarkgae~l% Zlr aena#
yk @ { O yle a}ae~ly gq~qly avicq{ae~ay# zlr cl ~`e~l# Z&@∫ O+ 7 :% Ae ay~a raebcùe
ya zqana qy`r c` rabc` `nk~k}`# ac ~alrag` na O`{ay l a}ae~ly kenazaenkae~ay%
Ahagzcl )2% Nly a}ae~ly @ { O yle gq~qly avicq{ae~ay l nayqeknly# yk @∫ O 7 ю#
ay~l ay# yk @ { O el ~kaeae acagae~ly ae ilgøe% Ykaenl `yç# naikr yk ae qe ylcl
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 62/846
Nr% Fëi~lr Pqa}anl Qrç`y
?'6
c`ex`gkae~l na qe` glean` cly nly a}ae~ly @ { O yle gq~qly avicq{ae~ay%
Ylcqikùe8
Naoknl pqa yk i`a c` i`r`# ac àbqkc` el zqana i`ar c` gkyg` }ax { }kia}ary`# zlr cl
~`e~l# cly a}ae~ly @ { O yle gq~qly avicq{ae~ay%
Ahagzcl )6% Yk A0 ay ac a}ae~l na y`i`r qe `y na qe g`xl na 1? e`kzay { A? ay ac
a}ae~l na y`i`r qe ra{# ¼yle ay~ly a}ae~ly gq~qly avicq{ae~ay<
Ylcqikùe8
@pqç# ae ay~a i`yl# yk yle a}ae~ly gq~qly avicq{ae~ay zlrpqa el ya zqana y`i`r ac y l
ac ra{ ` c` gkyg` }ax%
Ayz`ikl gqay~r`c%' Ac ayz`ikl gqay~r`c &Y + ay ac ilehqe~l na ~lnly cly rayqc~`nly
zlykocay na qe avzarkgae~l ay~`nçy~kil% Cly ayz`ikly gqay~r`cay ya ic`ykmki`e na
`iqarnl `c eøgarl na acagae~ly &zqe~ly+ pqa ile~kaeae% Ae ay~a rayzai~l# ya zqanae
aecky~`r cly acagae~ly yaz`r`nly zlr ilg`y { aeic`qy~r`nly ae ilrifa~ay &tw+% Cly
ayz`ikly gqay~r`cay zqanae yar mkek~ly# el mkek~ly# nkyira~ly { ile~keqly% Yke ago`rbl#
cly nly ~kzly oàykily na ayz`ikly gqay~r`cay yle cly nkyira~ly { ile~keqly% Zlr
ahagzcl# qe ayz`ikl gqay~r`c nkyira~l ~kaea qe eøgarl mkek~l na a}ae~ly ykgzcay l qe
eøgarl kemkek~l ile~`oca na a}ae~ly ykgzcay% Ae ac i`yl na ayz`ikly gqay~r`cay
ile~keqly# ay~l ya ramkara iq`enl cly acagae~ly &zqe~ly+ na qe ayz`ikl gqay~r`c
iley~k~q{ae qe ile~keql# ilgl zlr ahagzcl# ~lnly cly zqe~ly na qe` cçea`4 ~lnly cly
zqe~ly na qe yabgae~l na cçea` l ~lnly cly zqe~ly na qe zc`el% Ae `cbqely
avzarkgae~ly zqana yar ø~kc aecky~`r cly acagae~ly nac ayz`ikl gqay~r`c#
yky~agà~ki`gae~a# zlr gankl na nk`br`g`y na àrolc%
Ahagzcl );% Qe ahagzcl na qe ayz`ikl gqay~r`c nkyira~l mkek~l ay ac c`ex`gkae~l na
qe` glean` nly }aiay# ac iq`c ~kaea qe ayz`ikl gqay~r`c na > a}ae~ly ykgzcay# nlena
F nael~`e i`r`y { ^ nael~`e àbqkc`y% Ay~l ay8
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 63/846
Nr% Fëi~lr Pqa}anl Qrç`y
?';
Y 7 tFF# F^# ^F# ^^w
Ahagzcl )0:% Qe ahagzcl na qe ayz`ikl el mkek~l ay~à rac`ikle`nl ile ac ykbqkae~a
avzarkgae~l% Yk qely gaiàekily aei`rb`nly na }arkmki`r c` agkykùe na ùvknly na
ek~rùbael na cly `q~ly# cay ke~aray` y`oar ac eøgarl na `q~ly pqa naoae keyzaiikle`r
`e~ay na }ar# iqàc ay ac eøgarl na accly pqa el y`~kym`iae cly rabc`gae~ly
bqoare`gae~`cay% @pqç# okae zlnrç` liqrrkr pqa mqaya ac zrkgar `q~l# ac yabqenl# ac
~ariarl# a~i%# { pqa ~q}kar`e pqa }arkmki`r gkcay na q~ly e~ay na aeile~r`r qel pqa el
iqgzc` ile cly rabc`gae~ly% N`nl ` pqa el ya y`oa pqa ~`e cahly ~aenrç`e pqa ccab`r#
zlr cl ~`e~l# ya ileyknar` qe` i`e~kn`n na `q~ly ile~`oca kemkek~`%
Ahagzcl )00% Ae ac i`yl na ayz`ikly gqay~r`cay ile qe eøgarl kemkek~l na zqe~ly
gqay~r`cay# ay~ly ya nayirkoae gahlr qy`enl gë~lnly na rabc`% Zlr ahagzcl# yk ~lnly
cly rayqc~`nly zlykocay na qe avzarkgae~l# ay ac brqzl na ikqn`nay ae ac gqenl# ile
qe` zloc`ikùe na gày na qe gkccùe# ae~leiay# ac ayz`ikl gqay~r`c Y ay8
Y 7 tvsv ay qe` ikqn`n ile qe` zloc`ikùe na gày na qe gkccùew
Ahagzcl )0?% Z`r` avzcki`r qe ayz`ikl gqay~r`c ile~keql# ileyknëraya ac avzarkgae~l
na loyar}`r ac ~kagzl z`r` ilgzca~`r qe` ~`ra` ae z`r~kiqc`r# nkb`gly# ae qe ke~ar}`cl
na : >: yabqenly% Ae ay~a i`yl# ac ayz`ikl gqay~r`c ay ile~keql naoknl pqa f`{ qe
eøgarl na }`clray kemkek~`gae~a ile~`oca# ae ac ke~ar}`cl na : ` >: yabqenly%
Y 7 t~ln`y c`y }aiay zlykocay ae~ra : { >: yabqenlyw
Qekùe%' C` qekùe na nly a}ae~ly# nkb`gly @ { O# ya nael~`e zlr ac yçgolcl @∨ O {
ya caa @ l O# { ay ac a}ae~l pqa ile~kaea ~lnly cly acagae~ly pqa zar~aeaiae @ l O l
`goly% Zlr cl ~`e~l# ac a}ae~l @∨ O liqrra# yk @ liqrra# yk O liqrra l yk goly @ { O
liqrrae%
Ahagzcl )09% Yk nah`gly pqa ac a}ae~l @ 7 t`# o# iw { O 7 to# i# n# aw# ykaenl `yç# zlr
cl ~`e~l# @ ∨ O 7 t`# o# i# n# aw
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 64/846
Nr% Fëi~lr Pqa}anl Qrç`y
?'0:
Ahagzcl )0>% Yk G 7 tvs9 3 v 3 ;w { E 7 t{s1 3 { 3 0?w# ae~leiay# aeile~r`r c` qekùe
na G ∨ E% &R`czlca 0;;9# z% 0>+
Ylcqikùe8
G ∨ E 7 txw9 3 x 3 0?w
Ke~aryaiikùe na cly a}ae~ly% C` ke~aryaiikùe na nly a}ae~ly @ { O# ya nael~` zlr ac
yçgolcl @∫ O# pqa ya caa /@ { O/% C` ke~aryaiikùe @∫O ay ac brqzl na zqe~ly ae ac
a}ae~l nac ayz`ikl @ { ae ac a}ae~l nac ayz`ikl O% Zlr cl ~`e~l# ac a}ae~l @∫ O
liqrra# ylc`gae~a# yk `goly a}ae~ly @ { O liqrrae% @pqç# c` z`c`or` ic`}a –{‗ ya
ramkara `c a}ae~l ile~aekaenl ~lnly cly acagae~ly pqa yle ilgqeay l pqa ay~àe ae`goly# @ { O%
Ahagzcl )01% Yk Y 7 t:# 0# ?# 9# ># 1# =# 2# 6# ;w# @ 7 t:# ?# ># =# 6w# O 7 t0# 9# 1# 2# ;w#
I 7 t?# 9# ># 1w { N 7 t0# =# 2w# aeile~r`r8
&`+ @ ∫ O%
&o+ @ ∫ I
Ylcqikùe8
&`+ Naoknl pqa ae @∫O el f`{ ekebøe acagae~l ae ilgøe# zlr cl ~`e~l# @∫O 7ю
{ el zqanae liqrrkr ` c` gkyg` }ax%
&o+ Naoknl ` pqa# ylc`gae~a ac ? { ac > yle ilgqeay ae `goly a}ae~ly @ { I# zlr cl
~`e~l# @ ∫ I 7 t?# >w
Ahagzcl )0=% Yk nah`gly pqa G 7 t`# a# k# l# qw { E 7 tr# y# ~w# zlr cl ~`e~l# G∫ E 7
έ# cl iq`c nkia pqa G { E el ~kaeae acagae~ly ae ilgøe { pqa el zqanae liqrrkr c`
gkyg` }ax%
Ilgzcagae~l%' Ac ilgzcagae~l na qe a}ae~l @# nael~`nl zlr @"# ay ac ilehqe~l na
~lnly cly rayqc~`nly ae ac ayz`ikl gqay~r`c Y # pqa el ay~àe ile~aeknly ae @%
Ahagzcl )02% Yk @ 7 t:# 0# ?# 9# >w# O 7 t9# ># 1# =w { I 7 t0# 9# 1w# ae~leiay#
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 65/846
Nr% Fëi~lr Pqa}anl Qrç`y
?'00
aeile~r`r8
&`+ @ ∨ O
&o+ @ ∨ I
&i+ @ ∫ O
&n+ @ ∫ I
&a+ @"
&m+ t@ ∨ Iw"
Ylcqikùe8
&`+ @ ∨ O 7 t:# 0# ?# 9# ># 1# =w 7 Y
&o+ @ ∨ I 7 t:# 0# ?# 9# ># 1w
&i+ @ ∫ O 7 t9#>w
&n+ @ ∫ I 7 t0#9w
&a+ @" 7 t1#=w
&m+ &@ ∨ I+" 7 t=w
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 66/846
Nr% Fëi~lr Pqa}anl Qrç`y
?'0?
Mkbqr` ?%:% Nk`br`g` gly~r`enl cly ayz`ikly gqay~r`cay { cly a}ae~ly% &Hlfeyle#
0;;2+%
Ahagzcl )06% _amkrkëenlya c zrlocag` e~arklr razrayae~`r ile yçgolcly na ]aee c`y
ykbqkae~ay rabkleay8
&`+ ># =# 2
&o+ 0#>
&i+ 0# ?# 1# 2
&n+ 0# ?
&a+ 0# 9# >
Ylcqikùe8
&`+ &@ Q I+
&o+ &@ ∫ I+
&i+ &@ Q O+
&n+ &@ ∫ O+&a+ &@ Q O+ ∫ I+
Ahagzcl )0;% Yk Y 7 tckorl# i`~`ckx`nlr# ikb`rrkccl# pqçgkil# kebaekarl# rag`ifaw {# yk
nah`gly pqa @ 7 ti`~`ckx`nlr# rag`ifa# ckorl# ikb`rrkcclw# ae~leiay @" 7 tpqçgkil#
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 67/846
Nr% Fëi~lr Pqa}anl Qrç`y
?'09
kebaekarlw
Ahagzcl )?:% Ac ayz`ikl gqay~r`c na qe avzarkgae~l `ca`~lrkl ya n` ilglY 7 t@@#
@E# E@# EEw% Yk A0 7 t@@# @E# E@w { A? 7 t@E# E@# EEw# ae~leiay# aeile~r`r8
&`+ A0 ∨ A?
&o+ A0 ∫ A?
&i+ A0"
&n+ A?"
Ylcqikùe8
&`+ A0 ∨ A? 7 t@@# @E# E@# EEw&o+ A0 ∫ A? 7 t@E# E@w
&i+ A0" 7 tEEw
&n+ A?" 7 t@@w
A}ae~ly gq~qly avicq{ae~ay%' Nly l gày a}ae~ly ya nkia pqa yle gq~qly avicq{ae~ay
l nayqeknly# iq`enl el f`{ acagae~ly ilgqeay ae~ra yk% Z`r` ay~l ya qy` c`
ykgolclbç` na ke~aryaiikleay# ay naikr# @ ∫ O 7 έ# ay~l nkia pqa @ { O el ~kaeae
acagae~ly ae ilgøe% Ay~l ely nkia pqë# iq`enl qel na cly rayqc~`nly liqrra# cly l~rly
el zqanae liqrrkr `c gkygl ~kagzl% Zlr ahagzcl# iq`enl ya c`ex` qe n`nl# c` y`i`n`
na qe 0 { qe ? yle a}ae~ly gq~qly avicq{ae~ay# naoknl ` pqa# yk ac y`ca ac 0# el zqana
y`ckr ac ?# c` gkyg` }ax% Kbq`cgae~a# ile cly e`kzay yk y`ca qe ra{ el zqana y`ckr qe y
l iq`cpqkar l~r` i`r~` nac g`xl na i`r~`y%
Yk A0 { A? yle a}ae~ly gq~qly avicq{ae~ay# ae~leiay8
ZrtA0A?w 7 :%
Yk A0 $ A? nael~`e cly a}ae~ly na pqa# {` ya` pqa A0 l A? l `goly liqrr`e# ae~leiay8
ZrtA0 $ A?w 7 ZrtA0w $ ZrtA?w ' ZrtA0A?w
Ae baear`c z`r` a}ae~ly gq~qly avicq{ae~ay8
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 68/846
Nr% Fëi~lr Pqa}anl Qrç`y
?'0>
ZrtA0 $ A?w 7 ZrtA0w $ ZrtA?w &?'1+
Ahagzcl )?0% Na cly ykbqkae~ay a}ae~ly# na~argke`r# iqàcay a}ae~ly yle gq~qly
avicq{ae~ay { iqàcay el cl yle%
&`+ G`eqm`i~qr`enl qe ilgzleae~a acai~rùekil namai~qlyl%
G`eqm`i~qr`enl qe ilgzleae~a acai~rùekil oqael%
&o+ Zrlo`enl qe yqha~l ile qe ilamkikae~a na ke~acai~l 5 0::%
Zrlo`enl qe yqha~l ile qe ilamkikae~a na ke~acai~l 3 ;1
&i+ Yacaiikle`enl qe gënkil# pqkae ay ikrqh`el
Yacaiikle`enl qe gënkil pqkae ay gqhar
&n+ Yacaiikle`enl qe ~kzl ile zaryle`ckn`n nlgke`e~a
Yacaiikle`enl qe ~kzl na zaryle`ckn`n yqgky`%
Ylcqikùe8
Ae ay~a i`yl# cly keikyly &`+# &o+# &n+ yle a}ae~ly gq~qly avicq{ae~ay% Yke ago`rbl# ac
keikyl &i+ ay a}ae~l el gq~ql avicq{ae~a%
Ahagzcl )??% Yqzùeb`ya pqa f`{ 9 nky~rkoqknlray na `q~ly8 ac nky~rkoqknlr na BG
}aena Ifa}rlca~# Zle~k`i { Oqkid4 ac nky~rkoqknlr na c` Mlrn }aena Gariqr{ { Mlrn {#
ac nky~rkoqknlr na c` Ifr{ycar }aena Zc{glq~f { Ifr{ycar% Yk qe avzarkgae~l ileyky~a
ae loyar}`r c` g`ri` nac ykbqkae~a q~l }aenknl# ae~leiay# cly a}ae~ly @ 7 tIfa}rlca~#
Zle~k`i# Oqkidw { O 7 tMlrn# Gariq{w yle gq~qly avicq{ae~ay zlrpqa ac ykbqkae~a
`q~l }aenknl el zqana yar zrlnqi~l na BG l na Mlrn%
Ahagzcl )?9% Nly a}ae~ly @ { O yle gq~qly avicq{ae~ay l nayqeknly# yk @∫ O 7έ#
ay~l ay# yk @ { O el ~kaeae acagae~ly ae ilgøe% Ykaenl `yç# naikr yk ae qe ylcl
c`ex`gkae~l na qe` glean` cly a}ae~ly @ { O yle gq~qly avicq{ae~ay%
Ylcqikùe8
Naoknl ` pqa# yk i`a c` i`r` na c` glean`# c` i`r` lzqay~` el zqana i`ar ` c` gkyg`
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 69/846
Nr% Fëi~lr Pqa}anl Qrç`y
?'01
}ax { }kia}ary`% Zlr cl ~`e~l# cly nly a}ae~ly @ { O yle gq~qly avicq{ae~ay%
Zrlo`okckn`n ilenkikle`c%' Ya namkea ilgl c` zrlo`okckn`n na pqa qe a}ae~l @ liqrr`#
iq`enl ya y`oa pqa ac a}ae~l O f` liqrrknl { ya nael~` ilgl Z &@sO+% ^`gokëe c`
zrlo`okckn`n na pqa qe a}ae~l O liqrr`# iq`enl ya y`oa pqa ac a}ae~l @ f` liqrrknl#
ya nael~` zlr Z &Os@+% C`y mqeikleay qy`n`y z`r` ~`cay mkeay yle8
Z &@ ∫ O+ Z&@ ∫ O+
Z &Os@+ 7 „„„„„„4 Z &@sO+ 7 „„„„„ &?'=+
Z&@+ Z&O+
Ahagzcl )?>% Yk Z&N+ 7 :%69# Z&@+ 7 :%6? { Z&N∫@+ 7 :%26# aeile~r`r cly ykbqkae~ay
aeqeik`nly8
&`+ Z&@sN+
&o+ Z&Ns@+
Ylcqikùe8
&`+ Z&@sN+ 7 Z&N ∫ @+,Z&N+
7 :%26,:%69
7 :%;>
&o+ Z&Ns@+ 7 Z&N ∫ @+,Z&@+
7 :%26,:%6?
7 :%;1
Ahagzcl )?1% Cly rayqc~`nly lo~aeknly na ?== gqay~r`y na `kra ya ic`ykmki`e na
`iqarnl c` zrayaeik` na nly glcëiqc`y r`r`y% Ya`e @8 ac a}ae~l mlrg`nl zlr ~ln`y c`y
gqay~r`y na kra ae c` pqa ya aeiqae~r` c` glcëiqc` r`r` 0# { O8 ac a}ae~l mlrg`nl zlr
~ln`y c`y gqay~r`y na `kra nlena ay~à zrayae~a c` glcëiqc` r`r` ?% Yk ya i`ciqcù pqa c`
zrlo`okckn`n Z&@∫O+ 7 0?,== { Z&@+ 7 9=,?==# ae~leiay# i`ciqc`r c` zrlo`okckn`n nac
a}ae~l mlrg`nl zlr ~ln`y c`y gqay~r`y na `kra ile c` glcëiqc` ?# n`nl ac a}ae~l
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 70/846
Nr% Fëi~lr Pqa}anl Qrç`y
?'0=
mlrg`nl zlr ~ln`y c`y gqay~r`y na `kra ile c` glcëiqc` 0% &Gle~blgar{ a~ `c% 0;;=+
Ylcqikùe8
Z&Os@+ 7 Z&@ ∫ O+ , Z&@+
7 &0?,?==+ , &9=,?==+
7 0?,9=
Ahagzcl )?=% _amkrkëenlya `c zrlocag` `e~arklr# aeile~r`r Z&@sO+# yk Z&O+ ay kbq`c `
9:,?==%
Ylcqikùe8
Z&@sO+ 7 Z&@ ∫ O+ , Z&O+ 7 0?,?==,&9:,?==+ 7 0?,9:
A}ae~ly kenazaenkae~ay { nazaenkae~ay%' Ae ay~a i`yl# yke ago`rbl# iq`enl f`oc`gly
na zrlo`okckn`n ilenkikle`c ya keicq{ae cl pqa ya cc`g`e a}ae~ly kenazaenkae~ay {
a}ae~ly nazaenkae~ay% Zlr ahagzcl# yk c` liqrraeik` na qe a}ae~l# el i`gok` c`
zrlo`okckn`n na c` liqrraeik` nac l~rl a}ae~l# ae~leiay# ya nkia pqa cly nly a}ae~ly
yle kenazaenkae~ay% Yke ago`rbl# yk iq`cpqkar` na ay~`y ilenkikleay el ya y`~kym`iae#
cly nly a}ae~ly ya nkiae pqa yle nazaenkae~ay# ay naikr# Z&@sO+≪ Z&@+%
Ae ac i`yl ayzaik`c na pqa @ { O ya`e kenazaenkae~ay# ay naikr# na g`ear` pqa#
Z&@sO+ 7 Z&@+# ay~l ilenqia ` c` rabc` ayzaik`c na gqc~kzcki`ikùe8
Z&@ ∫ O+ 7 Z&@+ ² Z&O+ &?'2+
Ahagzcl )?2% Aeile~r`r c` zrlo`okckn`n na y`i`r nly i`r`y ae nly c`ex`gkae~ly na qe`
glean` fleay~`%
Ylcqikùe8
Zqay~l pqa c` zrlo`okckn`n na c`y i`r`y ay na :%1 zlr i`n` c`ex`gkae~l { cly nly
c`ex`gkae~ly yle kenazaenkae~ay# c` zrlo`okckn`n ay &0,?+&0,?+ 7 ³
Ahagzcl )?6% Ya y`i`e nly i`r~`y# `ca`~lrk`gae~a# na qe g`xl na 1? e`kzay% ¼Pqë
zrlo`okckn`n f`{ na lo~aear nly `yay yk<
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 71/846
Nr% Fëi~lr Pqa}anl Qrç`y
?'02
&`+ C` zrkgar` i`r~` ya raagzc`x` `e~ay na pqa ya y`pqa c` yabqen`%
&o+ C` zrkgar` i`r~` el ya raagzc`x` `e~ay na pqa ya y`pqa c` yabqen` i`r~`%
Ylcqikùe8
&`+ N`nl pqa ae~ra cly 1? e`kzay f`{ iq`~rl `yay# c` zrlo`okckn`n na y`i`r nly `yay ay
na8 &0,09+&0,09+ 7 0,0=;%
&o+ N`nl pqa ae~ra cly 10 e`kzay ray~`e~ay# `c y`i`r qe `y nac m`hl na i`r~`y# pqan`e
ylcl 9 `yay# ae~leiay# c` zrlo`okckn`n ay8 &>,1?+&9,10+ 7 0,??0% @pqç ya }a pqa ay~a ay
qe a}ae~l nazaenkae~a# zlrpqa 0,??0≪ 0,0=;# {` pqa cly a}ae~ly yle nazaenkae~ay#
iq`enl f`{ gqay~ral yke raagzc`xl%
Ahagzcl )?;% Z`r` nly a}ae~ly H { D ya y`oa pqa Z&H+ 7 :%=:# Z&D+ 7 :%> { Z&H∫D+ 7
:%0:% Naikr yk ay~ly nly a}ae~ly yle kenazaenkae~ay%
Ylcqikùe8
Naoknl ` pqa Z&H∫ D+ 7 :%0:# Z&H,D+ 7 Z&H ∫D+,Z&D+ 7 :%0:,:%>: 7 :%?1# ae~leiay#
ykaenl pqa Z&H,D+ 7 :%?1 ≪ Z&H+ 7 :%= { cly nly a}ae~ly yle nazaenkae~ay%
Ahagzcl )9:% Aeile~r`r Z&@sO+# yk Z&O+ 7 ?:,?= { Z&@∫ O+ 7 9:,?=
Ylcqikùe8
Qy`enl c` mqeikùe Z&@sO+ 7 Z&@ ∫ O+,Z&O+ { yqy~k~q{aenl n`8
Z&@sO+ 7 &9:,?=+,&?:,?=+
7 =::,=2=
7 :%666
]`rk`oca `ca`~lrk` &}`+%' Mqen`gae~`cgae~a# f`{ nly ~kzly na }`rk`ocay `ca`~lrk`y8
}`rk`ocay ca`~lrk`y nkyira~`y { }`rk`ocay ca`~lrk`y ile~keq`y% C` }`rk`oca ca`~lrk` ay
qe` mqeikùe pqa `ykbe` qe eøgarl ra`c ` i`n` rayqc~`nl ae qe ayz`ikl gqay~r`cY % Ay
qe }`clr na qe` mqeikùe eqgërki`gae~a namkeknl ylora Y # ay naikr# qe` rabc` pqa
`ylik` qe eøgarl ` i`n` rayqc~`nl ae ac ayz`ikl gqay~r`c Y % @cbqely ay~`nçy~kily
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 72/846
Nr% Fëi~lr Pqa}anl Qrç`y
?'06
rac`ikle`e ac ~ërgkel /}`rk`oca `ca`~lrk`/ ile ac ~ërgkel /ay~liày~kil/# pqa ya
rac`ikle` ile ileha~qr` l zrlo`okckn`n% F`{ }`rk`ocay `ca`~lrk`y okelgk`cay# na
Zlkyyle# fkzarbalgë~rki`y# }`rk`ocay na c` nky~rkoqikùe elrg`c# na c` nky~rkoqikùe na ~
na ay~qnk`e~a# na HK iq`nr`n`# na Mkyfar# a~i%
Ay~liày~kil%' Ay qe ~ërgkel pqa ke}lcqir` qe` }`rk`oca `ca`~lrk` l pqa rac`ikle`
i`yq`ckn`n l zrlo`okckn`n%
]`rk`oca `ca`~lrk` nkyira~` &}`n+%' C` }`n ay qe ilehqe~l l r`ebl na }`clray mkek~ly l
kemkek~`gae~a ile~`ocay ae eøgarly% C` }`n ya `ylik` ile nky~rkoqikleay na Oarelqcck#
na Zlkyyle# balgë~rki`# fkzarbalgë~rki`# eab`~k}` okelgk`c# a~i% Qe ahagzcl na }`n
mkek~` ay ac eøgarl na q~ly g`eah`nly ile qe` mcl~` na = }afçiqcly# ay naikr# nlena v
7 :# 0# ?# 9# ># 1% Yke ago`rbl# qe ahagzcl na }`n kemkek~`gae~a ile~`oca ay ac eøgarl
na zaryle`y pqa ae~r`e ` qe` ~kaen` na ilgzr`y i`n` gay%
]`rk`oca `ca`~lrk` ile~keq` &}`i+%' C` }`i ya namkea ilgl ac r`ebl na qe` }`rk`oca
`ca`~lrk V pqa ile~kaea qe ke~ar}`cl kemkek~l l mkek~l na eøgarly ra`cay% Zlr ahagzcl#
yk V ay ac }`clr nac zayl na qe` zaryle`# ac r`ebl na V ay V ≡ :% C`y nky~rkoqikleay
ile~keq`y ylik`n`y ile }`i yle c` nky~rkoqikùe elrg`c# c` m`gkck` na c`y nky~rkoqikleay
b`gg`# oa~`# c` nky~rkoqikùe avzleaeik`c# c` HK iq`nr`n`# c` ~ na ay~qnk`e~a# a~i%
Nk`br`g`y na ]aee { àcbaor` na ilehqe~ly
Nk`br`g` na ]aee%' Ay qe nkyzlyk~k}l bràmkil z`r` razrayae~`r ac ayz`ikl gqay~r`c {
c`y lzar`ikleay pqa kgzcki`e a}ae~ly% Ac kebcëy H% ]aee nay`rrlccù ay~a ~kzl na
nk`br`g` z`r` razrayae~`r# bràmki`gae~a# cly rayqc~`nly na qe avzarkgae~l% Ac
ileiaz~l na c`y rabc`y na a}ae~ly gq~qly avicq{ae~ay { }`rk`y l~r`y rabc`y na
zrlo`okckn`n ya zqanae razrayae~`r ile nk`br`g`y na ]aee% Z`r` iley~rqkr qe
nk`br`g` na ]aee qe ayz`ikl ya aeic`qy~r` razrayae~`enl ac ~l~`c na ~lnly cly
rayqc~`nly zlykocay%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 73/846
Nr% Fëi~lr Pqa}anl Qrç`y
?'0;
C`y rabc`y na c`y ~ray lzar`ikleay oàyki`y nac àcbaor` na ilehqe~ly z`r` mlrg`r
qekleay# ke~aryaiikleay { ilgzcagae~ly na a}ae~ly ya nayirkoae ae c` ^@OC@ ?%0%
^@OC@ ?%0% ^`oc` gly~r`enl c`y ca{ay nac àcbaor` na ilehqe~ly% &Ac`olr`ikùe zrlzk`+
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
Ca{ `ylik`~k}`8 &@ ∨ O+ ∨ I 7 @ ∨ &O ∨ I+
&@ ∫ O+ ∫ I 7 @ ∫ &O ∫ &O ∫ I+
Ca{ ilegq~`~k}`8 @ ∨ O 7 O ∨ @
@ ∫ O 7 O ∫ @
Ca{ nky~rkoq~k}`8 @ ∨ &O ∫ I+ 7 &@ ∨ O+ ∫ &@ ∨ I+
@ ∫ &O ∨ I+ 7 &@ ∫ O+ ∨ &@ ∫ I+
Ca{ay na Glrb`e8 &@ ∨ O+" 7 @" ∫ O"&@ ∫ O+" 7 @" ∨ O"
Ca{ay ilgzcagae~`rk`y8 @ ∨ @" 7 Y
@ ∫ @" 7 έ
&@"+" 7 @
Y" 7 έ# έ" 7 Y
Ca{ay knëe~ki`y8 @ ∨ έ 7 @
@ ∫ Y 7 @
@ ∨ Y 7 Y
@ ∫ έ 7 έ Ca{ay ile c` gkyg` zl~aeik`8 @ ∨ @ 7 @
@ ∫ @ 7 @
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 74/846
Nr% Fëi~lr Pqa}anl Qrç`y
?'?:
Mkbqr` ?%?% Cly aypqag`y na o`hl gqay~r`e cbqely nk`br`g`y na ]aee% &Ac`olr`ikùe
zrlzk`+
^ëieki`y na ile~al
Eqgarly`y rabc`y na ile~al f`e yknl qy`n`y z`r` ile~`r ac eøgarl na zqe~ly ae
gqay~raly% Iq`enl cly nk}aryly rayqc~`nly na qe avzarkgae~l yle kbq`cgae~a
zrlo`ocay# c` ~`ra` na i`ciqc`r zrlo`okckn`nay ya ranqia ` ile~`r% Ay~`y ~ëieki`y na
ile~al yle ø~kcay z`r` ile~`r ac eøgarl na a}ae~ly pqa ilgzleae ac eqgar`nlr {,l ac
naelgke`nlr na qe` zrlo`okckn`n%
Ahagzcly na ~ëieki`y na ile~al yle8
0% C` rabc` nac zrlnqi~l z`r` z`ray lrnae`nly
?% C` rabc` nac zrlnqi~l gày baear`c
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 75/846
Nr% Fëi~lr Pqa}anl Qrç`y
?'?0
9% M`i~lrk`cay
>% Qyl na nk`br`g`y na àrolc
1% Zargq~`ikleay
=% Ilgoke`ikleay
C` rabc` nac zrlnqi~l z`r` z`ray lrnae`nly
C` mlrg` gày oàyki` na ile~al ay c` rabc` nac zrlnqi~l ge% Zlr ahagzcl# yk ac zrkgar
acagae~l q loha~l na qe z`r lrnae`nl ya zqana yacaiikle`r ae e0 mlrg`y# { zlr i`n`
qe` na ay~`y e0 mlrg`y ya zqana yacaiikle`r qe yabqenl acagae~l nac z`r ae e? mlrg`y#
ae~leiay# ykaenl `yç# ay~l ay qe` rabc` nac zrlnqi~l%
Ahagzcl )90% ¼Iqàe~ly zqe~ly gqay~r`cay f`{ ae qe ayz`ikl gqay~r`c Y # iq`enl qe
z`r na n`nly ya c`ex`e qe` }ax<
Ylcqikùe8
Ac zrkgar n`nl zqana i`ar ae e0 7 = g`ear`y% Z`r` i`n` qe` na ay~`y = g`ear`y# ac
yabqenl n`nl zqana ~`gokëe i`ar ae e? g`ear`y% Zlr cl ~`e~l# ac z`r na n`nly zqanae
i`ar ae e0e? 7 &=+ &=+ 7 9=% Ac ayz`ikl gqay~r`c ay8
Y 7 t0'0# 0'?# 0'9# 0'># 0'1# 0'=# ?'0# ?'?# ?'9# ?'># ?'1# ?'=# 9'0# 9'?# 9'9# 9'># 9'1# 9'
=# >'0# >'?# >'9# >'># >'1# >'=# 1'0# 1'?# 1'9# 1'># 1'1# 1'=# ='0# ='?# ='9# ='># ='1# ='=w
Ahagzcl )9?% Ae qe ay~qnkl gënkil cly z`ikae~ay ya ic`ykmki`e ae lifl g`ear`y na
`iqarnl ` pqa ~aeb`e ~kzl na y`ebra# ay naikr# @O$# @O
'# @
$# @
'# O
$% O
'l L
$# L
'{
~`gokëe na `iqarnl `# `pqëccly pqa ~aeb`e zraykùe `c~`# o`h` l elrg`c% Aeile~r`r ac
eøgarl na g`ear`y ae c`y iq`cay qe z`ikae~a ya zqan` ic`ykmki`r%
Ylcqikùe8
e0 7 6 ~kzly na y`ebra { e? 7 9 zraykleay `r~ark`cay% Zlr cl ~`e~l# e0 e? 7 &6+ &9+ 7 ?>
g`ear`y%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 76/846
Nr% Fëi~lr Pqa}anl Qrç`y
?'??
_abc` na gqc~kzcki`ikùe gày baear`c
C` rabc` nac zrlnqi~l z`r` d'`rrabcly ya namkea ilgl ykbqa8 Yk qe` lzar`ikùe zqana
yar faif` ae e0 g`ear`y {# yk z`r` i`n` qe` na ay~`y g`ear`y# qe` yabqen` lzar`ikùe
zqana yar faif` ae e? g`ear`y# {# yk zlr i`n` qe` na ay~`y nly zrkgar`y lzar`ikleay#
qe` ~ariar` lzar`ikùe zqana yar faif` ae e9 g`ear`y {# yç yqiayk}`gae~a# ae~leiay# c`
yaiqaeik` na d lzar`ikleay l `rrabcly zqana yar faif` ae e0# e?# e9#%%%# ed `rrabcly# ay
naikr8
e0e?e9#…#ed &?'6+
Ahagzcl )99% Yqzùeb`ya pqa qe ickae~a naya` key~`c`r qe ~acëmlel rkgckea { ya zqana
yacaiikle`r na e0 7 0: ilclray nailr`~k}ly pqa ya yqzlea ay~àe nkyzlekocay ae e? 7 9
clebk~qnay na i`ocay ile e9 7 ? ~kzly na ~lely rl~`~k}ly% Ae~leiay# ¼iqàe~ly rrabcly ya
zqanae f`iar<
Ylcqikùe8
e0e?e9 7 &0:+&9+&?+
7 =: `rrabcly
Ahagzcl )9>% Yk i`n` icçeki` ae qe iae~rl gënkil# ~kaea > ayzaik`cky~`y nac ilr`xùe# 9
ayzaik`cky~`y ae gankike` ke~are` { nly ikrqh`ely baear`cay# ¼iqàe~`y g`ear`y avky~ae
na yacaiikle`r qe gënkil na i`n` ~kzl< &El~`8 ae ay~a raebcùe# nac zqe~l na }ky~` nac
`q~lr na ay~a ckorl# el zqana f`oar ayzaik`cky~`y gënki`y na i`n` qe` na c`y z`r~ay#
ùrb`ely l yky~ag`y nac iqarzl# ilgl ilgøegae~a ya iraa% Yk `yç mqar`# ay~l
apqk}`cnrç` naikr pqa i`n` ùrb`el l yky~ag` nac iqarzl mqeikle` kenazaenkae~agae~a
nac ray~l nac lrb`ekygl4 cl iqàc el ay ilrrai~l% Ay~l ya naoa ` pqë# ac iqarzl ay~à
ilgzqay~l zlr ùrb`ely l yky~ag`y ile~kebae~ay l nazaenkae~ay# iq{l mqeikle`gkae~l
nazaena# ae ~qrel# na c` nkraiikùe pqa ya ca në ` ~lnl ac lrb`ekygl ilgl qekn`n
kenazaenkae~a% Ac faifl na pqa qe ùrb`el l yky~ag` nac iqarzl ay~ë `z`rae~agae~a
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 77/846
Nr% Fëi~lr Pqa}anl Qrç`y
?'?9
aemargl# ay~l el pqkara naikr pqë# ylc`gae~a# aya ùrb`el ae z`r~kiqc`r ay~à aemargl#
ykel pqa ~ln` c` pqçgki` nac iqarzl ay~à c~ar`n`# ilgl rayqc~`nl na }kn` e~ke`~qr`c%
Ay~a r`xle`gkae~l ay~à rac`ikle`nl ile c` ~ayky na Fkzùir`~ay ileyzkr`~kl qe`+%
Ylcqikùe8
e0e?e9 7 &>+&9+&?+ 7 ?>
_abc` m`i~lrk`c
N`nl qe çe~abrl zlyk~k}l e# ac zrlnqi~l na ~lnly cly eøgarly ae~arly nayna e f`y~` 0
ya cc`g` m`i~lrk`c e { ya ayirkoa e!% Ae baear`c# e! 7 e&e ’ 0+&e ’ ?+&e ’ 9+…%0% Zlr
namkekikùe :! 7 0% @pqç eù~aya pqa 0:! 7 0:²;!4 1! 7 >²>!# { e! 7 e&e ’ 0+!
Gày `nac`e~a# iq`enl ya nkyiq~` ac ~ag` na zargq~`ikleay ya }arà pqa# c`
nkmaraeik` ae~ra c` rabc` m`i~lrk`c { c` rabc` na zargq~`ikleay# ay c` ykbqkae~a8 c` rabc`
m`i~lrk`c nkia iqàe~ly `rrabcly yle zlykocay# iq`enl ya qy`e ~lnly cly nkmarae~ay
loha~ly na e% Yke ago`rbl# iq`enl ya f`oc` na zargq~`ikleay# ya yacaiikle`e
ylc`gae~a `cbqely na cly loha~ly e# el ~lnly# ilgl ae ac i`yl na c` rabc` m`i~lrk`c%
Ahagzcl )91% I`ciqc`r cly ykbqkae~ay m`i~lrk`cay8
&`+ 0:!
&o+ 1!
&i+ ;!,:!
Ylcqikùe8
&`+ 0:! 7 9#=?6#6::
&o+ 1! 7 0?:
&i+ ;!,:! 7 9=?#66:,0 7 9=?#66:
Ahagzcl )9=% Qe i`enkn`~l zrayknaeik`c zc`ea` }kyk~`r i`n` qel na ?6 ay~`nly na qe
z`çy% ¼Iqàe~`y rq~`y nkmarae~ay yle zlykocay<
Ylcqikùe8
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 78/846
Nr% Fëi~lr Pqa}anl Qrç`y
?'?>
C`y i`zk~`cay na cly nkmarae~ay ?6 ay~`nly ya zqanae `rrabc`r ae ?6! g`ear`y# na ~`c
mlrg` ac eøgarl na nkmarae~ay rq~`y ay ?6! 7 9%:>;v0:?;
%
Ahagzcl )92% Ae c` m`iqc~`n na kebaekarç`# ae ikar~` lmkike`# cly ayirk~lrkly na >
oai`rk`y ya zleae ae cçea` ile~r` qe` z`ran% I`n` oai`rk` ya zqana yae~`r ae iq`cpqkar
ayirk~lrkl% ¼Iqàe~ly `rrabcly z`r` yae~`r ` c`y oai`rk`y yle zlykocay<
Ylcqikùe8
Qy`enl e! 7 >! 7 &>+&9+&?+&0+ 7 ?>
Nk`br`g`y na àrolc
Ae c`y rabc`y na zrlnqi~l l rabc` na gqc~kzcki`ikùe ya zqana qy`r qe` ilemkbqr`ikùe
cc`g`n` nk`br`g` na àrolc# z`r` razrayae~`r aypqagà~ki`gae~a# ~ln`y c`y zlykokckn`nay
{ i`ciqc`r iq`cpqkar zrlo`okckn`n ae cly rayqc~`nly lo~aeknly nac nk`br`g` na àrolc%
Na ay~` g`ear`# cly ayz`ikly gqay~r`cay zqanae nayirkokrya bràmki`gae~a ae ~ërgkely
na qe nk`br`g` na àrolc%
Ahagzcl )96% Yqzùeb`ya pqa qe` ilgzq~`nlr` zqan` yacaiikle`r# ca`~lrk`gae~a# qel
na nly m`i~lray# _f &zlyk~k}l { eab`~k}l+ { qel na ~ray ~kzly na y`ebra% I`ciqc`r c`
zrlo`okckn`n na y`i`r qe m`i~lr _f zlyk~k}l ile ~kzl na y`ebra @%
Ylcqikùe8
Qy`enl c` rabc` na gqc~kzcki`ikùe e0 e? 7 &?+ &9+ 7 = ya f`ia ay~a iàciqcl% Yke ago`rbl#
`pqç ay nkmçikc }kyq`ckx`r c`y ilgoke`ikleay i`ciqc`n`y ae c` zrlo`okckn`n% El loy~`e~a#
ac qyl na qe nk`br`g` na àrolc ykgzckmki` ay~` ~`ra`%
Ahagzcl )9;% Ile rac`ikùe `c zrlocag` `e~arklr f`iar qe nk`br`g` na àrolc z`r`
rac`ikle`r ac m`i~lr _f { ac ~kzl na y`ebra%
Ylcqikùe8
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 79/846
Nr% Fëi~lr Pqa}anl Qrç`y
?'?1
M`i~lr _f ^kzl na y`ebra _ayqc~`nl
@ $@
$ L $L
O $O
@ '@
' L 'L
O 'O
Mkbqr` ?%9% Nk`br`g` gly~r`enl ac m`i~lr _f# ac ~kzl na y`ebra { ac rayqc~`nl%
&Ac`olr`ikùe zrlzk`+
Nac nk`br`g` na àrolc na `rrko` zlnagly }ar pqa ac ayz`ikl gqay~r`c ay8
Y 7 t$@# $L# $O# '@# 'L# 'O+
Av`gke`enl ay~` yk~q`ikùe }agly pqë# qe` ylc` r`g` ilrrayzlena 8 $@% Zlr cl ~`e~l#
c` zrlo`okckn`n na y`i`r ay~a `rrabcl ay na 0,=%
Ahagzcl )>:% Yqzùeb`ya pqa ya pqkar` aeile~r`r c` zrlo`okckn`n na qe kem`e~a# pqa ya`
qe` fagor` ile lhly `xqcay% @yqgkr pqa c` zrlo`okckn`n na }`rleay { fagor`y aykbq`c { pqa zqan`e y`ckr ile ilclray na lhly i`mëy# }arnay# `xqcay l i`y~`ðly%
Ylcqikùe8
Qy`enl c` rabc` na zrlnqi~ly n`8 e0 e? 7 &?+ &>+ 7 6% C` zrlo`okckn`n na qe` fagor`
ile lhly `xqcay ay 0,6%
Zarl# f`ikaenl qe nk`br`g` na àrolc ykgzckmki`gly ac iàciqcl na c` zrlo`okckn`n na
y`i`r qe` fagor` ile lhly `xqcay%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 80/846
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 81/846
Nr% Fëi~lr Pqa}anl Qrç`y
?'?2
A~`z` 0 A~`z` ? A~`z` 9
Zrkgar` glean` Yabqen` glean` ^ariar` glean`
Mkbqr` ?%1% Nk`br`g` na àrolc nac avzarkgae~l na c`ex`r c`y ~ray glean`y
ykgqc~àea`gae~a# nlena Y 7 ylcay { @ 7 àbqkc`y% &Ac`olr`ikùe zrlzk`+
Ile ay~a nk`br`g` na àrolc }agly pqa f`{ 6 rayqc~`nly `c c`ex`r qe` glean` ~ray
}aiay ileyaiq~k}`y l ~ray glean`y ykgqc~àea`gae~a% Ac ayz`ikl gqay~r`c ay8
Y 7 t&YYY+# &YY@+# &Y@Y+# &Y@@+# &@YY+# &@Y@+# &@@Y+# &@@@+w
&o+ C` zrlo`okckn`n na i`kb`e av`i~`gae~a 9 ylcay ay8
Z&ylcay 7 9+ 7 0,6
&i+ C` zrlo`okckn`n na pqa i`kb`e iq`enl gaely ? ylcay ay8
Z&ylcay ≡ ?+ 7 >,6 7 0,?
&n+ C` zrlo`okckn`n na i`kb`e ` cl gày nly àbqkc`y ay8
Z&àbqkc`y ≮ ?+ 7 raylc}arya zlr ac cai~lr &a+ C` zrlo`okckn`n na ~lnl ac ilehqe~l gqay~r`c ay8 Z&Y + 7 0 l ya`8
7 Z&YYY+$Z&YY@+$Z&Y@Y+$Z&Y@@+$Z&@YY+$Z&@Y@+$Z&@YY+$Z&@@@+
7 0,6 $ 0,6 $ 0,6 $ 0,6 $ 0,6 $ 0,6 $ 0,6 $ 0,6 7 0
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 82/846
Nr% Fëi~lr Pqa}anl Qrç`y
?'?6
Ahagzcl )>?% Qe` z`rah` na raikëe i`y`nly naya` ~aear > fkhly%
&`+ Aecky~`r ac ayz`ikl gqay~r`c%
&o+ ¼Iq`c ay c` zrlo`okckn`n na ~aear 9 }`rleay< ¼> }`rleay<
&i+ ¼Iqàc ay c` zrlo`okckn`n na ~aear zqr`y fagor`y< ¼Gày na ? fagor`y<
Ylcqikùe8
&`+ Y 7 t}}}}# }}}f# }}f}# }}ff# }f}}# }f}f# }ff}# }fff# f}}}# f}}f# f}f}# f}ff#
ff}}# ff}f# fff}# ffffw
&o+ Z&9 }`rleay+ 7 ?,6 7 ³4 Z&> }`rleay+ 7 0,0=
&i+ Z&zqr`y fagor`y+ 7 0,0=4 Z&gày na ? fagor`y+ 7 1,0=
Zargq~`ikleay
Qe` zargq~`ikùe ay qe `rrabcl lrnae`nl na loha~ly l i`yly% Na ay~` g`ear`# f`y~`
`flr` ya f` nkyiq~knl# øeki`gae~a# c`y rabc`y nac zrlnqi~l z`r` z`ray lrnae`nly { c`
rabc` na gqc~kzcki`ikùe gày baear`ckx`n`% Ilgl ya nkhl# ay~`y rabc`y nkiae pqa# cly
acagae~ly yqiayk}ly na qe d'`rrabcl ya yacaiikle`rle na ilehqe~ly nkmarae~ay { ile
lzikleay ile raagzc`xl z`r` ac gkygl acagae~l pqa zqan` z`raiar gày na qe` }ax%
Yke ago`rbl# ae ac i`yl na c`y zargq~`ikleay# }`gly ileyknar`r qe mlenl mkhl
mlrg`nl zlr e nky~ke~ly acagae~ly { yqzlekaenl pqa ya mlrg` qe d'`rrabcl# `c
yacaiikle`r yqiayk}`gae~a na ay~a ilehqe~l# yke raagzc`xl# z`r` pqa qe acagae~l
zqan` `z`raiar ` cly yqgl ae qe` na c`y d zlykikleay%
Namkekikùe8 Qe` zargq~`ikùe ay qe rrabcl na ~lnly l z`r~a na qe ilehqe~l na loha~ly#
nlena ac lrnae ay na kgzlr~`eik` &ae ile~r`y~a ile c` ilgoke`ikùe ae c` iq`c }aragly
pqa ac lrnae el ay na kgzlr~`eik`+%
^alrag` 08 Ac eøgarl na zargq~`ikleay na loha~ly ~lg`nly ~lnly ` qe ~kagzl ay e!
Ay~a ~alrag` ely n` ac eøgarl ~l~`c na ~lnly cly loha~ly ~lg`nly ~lnly ` qe ~kagzl
&ac iq`c ay ac ayz`ikl gqay~r`c+%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 83/846
Nr% Fëi~lr Pqa}anl Qrç`y
?'?;
Ahagzcl )>9% Qy`enl qe` i`ciqc`nlr` na olcykccl# a}`cq`r c`y ykbqkae~ay
zargq~`ikleay8 &`+ 6Z9# &o+ =Z># &i+ 01Z0# &n+ 9Z9
Ylcqikùe8
&`+ 6Z9 7 e! , &e ' r+! 7 6!,&6 ’ 9+! 7 99=
&o+ =Z> 7 &=+&1+&>+&9+ 7 9=:
&i+ 01Z0 7 01
&n+ 9Z9 7 &9+&?+&0+ 7 =
Ahagzcl )>>% Ac eøgarl na zargq~`ikleay na c`y iq`~rl ca~r`y# `# o# i# n &~lg`n`y
~ln`y ` qe ~kagzl+ ay8
e! 7 >! 7 ?>
Ay~` zargq~`ikùe ay# ra`cgae~a# qe` rabc` m`i~lrk`c# zlrpqa ya ~lg`rle ~ln`y c`y ca~r`y
` qe ~kagzl%
^alrag` ?8 Ac eøgarl na zargq~`ikleay na e loha~ly nky~ke~ly ~lg`nly qe ~kagzl r
&qe` z`r~a elgày+ ya n` ilgl8
eZr 7 e! , &e ' r+! &?';+
Ahagzcl )>1% Nly olca~ly na c` cl~arç` ya y`i`e na ?: z`r` ac zrkgarl { yabqenl cqb`r%
Aeile~r`r ac eøgarl na zqe~ly gqay~r`cay ae ac ayz`ikl% Aeile~r`r ~`gokëe ~lnl ac
ayz`ikl gqay~r`c Y %
Ylcqikùe8
@pqç cly loha~ly yle ~lg`nly na ? ae ? ay naikr# e 7 ?: { r 7 ? { qy`gly c` mùrgqc`8
eZr 7 e! , &e ' r+! 7 ?:Z? 7 ?:!,&?: ' ?+!
7 96:
@flr`# yk pqaragly ~lnl ac ayz`ikl gqay~r`c pqkara naikr pqa cly }`gly ~lg`r ~lnly
` qe ~kagzl r% Ay~l nkia pqa c` mùrgqc`8
eZr 7 e! , &e ' r+! ya ranqia ` e! l ya` ?:! 7 ?%>9v0:06
%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 84/846
Nr% Fëi~lr Pqa}anl Qrç`y
?'9:
Ahagzcl )>=% ¼Na iqàe~`y g`ear`y zqana c` Ylikan`n Pqçgki` Gavki`e` yacaiikle`r
` 9 ilemaraeiky~`y z`r` 9 ilemaraeik`y nkmarae~ay# yk f`{ øeki`gae~a 1 maif`y
nkyzlekocay<
Ylcqikùe8
@pqç e 7 1 { r 7 9 qy`enl eZr 7 e! , &e ' r+! { yqy~k~q{aenl cly }`clray n`8
eZr 7 e! , &e ' r+! 7 1Z9 7 1! , ?! 7 =:% Ae rayqgae# `pqç }agly pqë# yk pqaragly ~ln`y
c`y zargq~`ikleay zlykocay l ~lnl ac ayz`ikl gqay~r`c# ae~leiay# qy`gly e! Zarl# yk
pqaragly# øeki`gae~a# qe` z`r~a# qy`gly eZr 7 e!,&e ' r+!
Ahagzcl )>2% ¼Iqàc ay ac eøgarl na zargq~`ikleay na c`y ca~r`y # o# i ~lg`n`y ~ln`y
` qe ~kagzl<
Ylcqikùe8
Yaky# }%b %# `o# o`# `i# i`# oi# io
Ahagzcl )>6% Ileyknëraya qe` i`rrar` na 0: i`o`ccly { qe zragkl na av`i~` z`r`
iq`cpqkar` pqa zqan` ayilbar ac lrnae av`i~l nac zrkgarl f`y~` ac nëikgl cqb`r%
@yqgkaenl pqa ~lnly cly i`o`ccly ~kaeae c` gkyg` lzlr~qekn`n na b`e`r# ¼Iqàe~ly
`rrabcly f`{<
Ylcqikùe8
0:Z0: 7 9#=?6#6:: zargq~`ikleay
Ahagzcl )>;% O`hl c`y ilenkikleay nac zrlocag` )2# ¼Iqàc ay c` zrlo`okckn`n na b`e`r
yk ya ilgzr` qe ylcl olca~l<
Ylcqikùe8
Z&Ile qe ylcl olca~l+ 7 0 , 0:Z0: 7 0,9#=?6#6::
7 ?%2=v0:'2
Ahagzcl )1:% Yqzùeb`ya pqa f`{ = z`r~ay nkmarae~ay z`r` yar `cg`iae`n`y# zarl
ylc`gae~a# f`{ > i`h`y nkyzlekocay% ¼Iqàe~`y zargq~`ikleay yle zlykocay<
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 85/846
Nr% Fëi~lr Pqa}anl Qrç`y
?'90
Ylcqikùe8
@pqç# e 7 = { r 7 ># ay naikr8 =Z> 7 9=:
^alrag` 9% Ac eøgarl na nkmarae~ay zargq~`ikleay na e loha~ly# na cly iq`cay e0 yle
na qe` ic`ya# e? yle na qe` yabqen` ic`ya#%%%ed yle na qe` d'ëykg` ic`ya ya n` ilgl8
e! , &e0! e?!%%ed !+ &?'0:+
Nlena8 e! ay ac ~l~`c na cly loha~ly
Ahagzcl )10% ¼Na iqàe~`y g`ear`y zqanae `rrabc`rya ae qe ilrnùe acëi~rkil 9 mlily
rlhly# > `g`rkccly { ? `xqcay ae ; zlr~`càgz`r`y<
Ylcqikùe8
Qy`enl c` rabc` na z`r~kikùe e!,&e0!e?!%%ed !+
Nlena# e! 7 ;# e0 7 9# e? 7 > { e9 7 ?# n`8
;! , &9! >! ?!+ 7 0?=:
Ahagzcl )1?% Qe ilcabkl hqab` 0? hqably nqr`e~a c` ~agzlr`n`% Na iq`e~`y g`ear`y
zqana ac apqkzl ~argke`r c` ~agzlr`n` ile 2 hqably b`e`nly# 9 zarnknly { ? agz`~ay<
Ylcqikùe8
Qy`enl c` mqeikùe &?';+ ile e! 7 0?# e0 7 2# e? 7 9 { e9 7 ? { yqy~k~q{aenl n`8
0?!,T&2!+&9!+&?!+ 7 2;?:
L~r` mlrg` na }ar c`y zargq~`ikleay ay iq`enl ay~`gly ke~aray`nly ae ac
eøgarl na g`ear`y na z`r~kr qe ilehqe~l na e loha~ly ae r yqoilehqe~ly cc`g`n`y
iacn`y%
^alrag` >% Ac eøgarl na g`ear`y na z`r~kr qe ilehqe~l na e loha~ly ae r iacn`y ile e0
acagae~ly ae c` zrkgar` iacn`# e? acagae~ly ae c` yabqen` {# `yç yqiayk}`gae~a# ay8
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 86/846
Nr% Fëi~lr Pqa}anl Qrç`y
?'9?
e
eee r %%#
?0
7 eIe0#e?%%er 7 e! , e0! e?!%%%er ! &?'00+
Nlena8
e0 $ e? $ er 7 e
Ahagzcl )19% Ae iqàe~`y g`ear`y zqanae 2 ikae~çmkily yar ykbe`nly qe iq`r~l ~rkzca
{ ` ? iq`r~ly nlocay ae qe fl~ac%
Ylcqikùe8
2
?#?#9
7 2! , &9!?!?!+ 7 ?0:
Ahagzcl )1>% Na iqàe~`y g`ear`y ya zqanae `ilgln`r ` 0: }k`harly ae qe fl~ac
`ykbeàenlcly ae ? iq`r~ly ~rkzcay { 9 iq`r~ly nlocay<
Ylcqikùe8
Qy`enl c` mqeikùe &?'00+ { yqy~k~q{aenl n`8
0:! , &9! 9! ?! ?! ?!+ 7 0?#=::
Ilgoke`ikleay
Qe` ilgoke`ikùe ay qe `rrabcl na loha~ly# yke kgzlr~`r ac lrnae% Ac eøgarl na
ilgoke`ikleay na e loha~ly ~lg`nly ` qe ~kagzl r zqana ayirkokrya ilgl eIr %
^alrag`8 Ac eøgarl na ilgoke`ikleay na e loha~ly nky~ke~ly ~lg`nly qe ~kagzlr ay
qe` ilgoke`ikùe4 ay~l ay# ac eøgarl na yqoilehqe~ly na ~`g`ðl r pqa zqanae
yacaiikle`rya na qe ilehqe~l na e loha~ly nky~ke~ly nlena ac lrnae el ay kgzlr~`e~a
&ilgl ae ac i`yl na c` zargq~`ikùe# ae c` iq`c ac lrnae yk ay kgzlr~`e~a+% C`
ilgoke`ikùe ya nael~` zlr c` mqeikùe8
eIr 7 e! , r! &e ' r+! &?'0?+
Nlena8
eIr ay c` ilgoke`ikùe# pqa ~`gokëe ya zqana nael~`r ilgl Ier
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 87/846
Nr% Fëi~lr Pqa}anl Qrç`y
?'99
Ahagzcl )11% A}`cq`r 2I>%
Ylcqikùe8
Qy`enl c` mùrgqc` &?'0?+ eIr 7 e! , r!&e ' r+! { yqy~k~q{aenl cly }`clray n`8
7 2I9 7 2! , >! 9!
7 91
Ahagzcl )1=% Qe m`orki`e~a na cc`e~`y f`ia 0: ~kzly na eaqgà~kily z`r` nkmarae~ay
~`g`ðly { pqkara zraz`r`r qe` z`r~kn` pqa ile~aeb` = ~kzly na cc`e~`y% ¼Iqàe~`y
ilgoke`ikleay na cc`e~`y ay~àe nkyzlekocay<
Ylcqikùe8
Qy`gly c` mqeikùe na ilgoke`ikùe# c` iq`c ay qe `rrabcl na loha~ly# yke kgzlr~`r ac
lrnae% @pqç ya qy` eqa}`gae~a# c` mqeikùe &?'0?+ namkekn` ilgl8
eIr 7 e!, r!&e ' r+! 7 eZr , r!
@pqç# e 7 0:# r 7 =% Yqoy~k~q{aenl ay~ly }`clray ae c` mqeikùe na `rrko` n`8
0:I= 7 0:! , =! >! 7 ?0:
Ahagzcl )12% Qe brqzl na ~ray keyzai~lray }` ` keyzaiikle`r c`y `i~k}kn`nay na qe`
kenqy~rk` ile~`gke`e~a% Ac brqzl ya }` ` mlrg`r yacaiikle`enl cly ~ray `bae~ay na qe
brqzl na 1% ¼Iqàe~ly brqzly nkmarae~ay ya zqanae mlrg`r ykbqkaenl qe lrnae
namkeknl< ¼Ykbqkaenl qe lrnae kenamkeknl<
Ylcqikùe8
Z`r` ac zrkgar i`yl# yarç` qe` zargq~`ikùe# zlrpqa ya pqkara qe lrnae namkeknl%
Qy`enl c` mùrgqc` eZr 7 e!,&e ' r+! ile e 7 1 { r 7 9 { yqy~k~q{aenl cly }`clray n`8
1Z9 7 1! , &1 ' 9+! 7 1!,9! 7 ?:
Z`r` ac yabqenl i`yl# l ya` qe lrnae kenamkeknl# yarç` qe` ilgoke`ikùe# zlrpqa ac
lrnae el ay na kgzlr~`eik`# ay naikr# qy`enl c` mùrgqc` &?'0?+8
1I9 7 0:
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 88/846
Nr% Fëi~lr Pqa}anl Qrç`y
?'9>
L~r` }`rk`ikùe na ilgoke`ikùe ya namkea ilgl ac eøgarl na ilgoke`ikleay na
e loha~ly ~lg`nly 0# ?# 9#%%% e qe ~kagzl% Na ay~` g`ear`# ae baear`c# z`r` iq`cpqkar
çe~abrl zlyk~k}l e ya n` zlr c` mqeikùe na `o`hl8
eI0 $ eI? $ eI9 $ %%% $ eIe 7 ?e
’ 0 &?'09+
Ahagzcl )16% Qe` zaryle` ~kaea ikeil glean`y na nkmarae~ay naelgke`ikleay%
¼Iqàe~`y yqg`y nkmarae~ay na nkearl ya zqanae mlrg`r<
Ylcqikùe8
C` glean` ya zqana yacaiikle`r {` ya` qe` na 1 glean`y# nly na 1 glean`y#…%#
ikeil na 1 glean`y% Qy`enl c` mqeikùe na `rrko` &?'09+ { yqy~k~q{aenl cly }`clray
`zrlzk`nly n`8
1I0 $ 1I? $ 1I9 $ 1I> $ 1I1 7 1 $ 0: $ 0: $ 1 $ 0
7 90
L~r` mlrg` na f`iar ay~a zrlocag` yarç` r`xle`enl na c` ykbqkae~a g`ear`% I`n`
glean` ya zqana g`eah`r na nly g`ear`y# ` gankn` pqa ya yacaiikle` l el ya
yacaiikle`% Naoknl ` pqa i`n` qe` na c`y nly g`ear`y na ~r`~`r ile qe` na c`y
glean`y ay `ylik`n` ile nly g`ear`y na qy`r# ile i`n` qe` na c`y l~r`y glean`y# ac
eøgarl na g`ear`y na ~r`~`r ile c`y ikeil glean`y ay qy`enl c` rac`ikùe ?1
g`ear`y%
Zarl c` i`e~kn`n ?1 g`ear`y keicq{a ac i`yl ae ac iq`c ekebqe` glean` ya yacaiikle`%
Zlr cl ~`e~l# ac eøgarl rapqarknl na yqg`y na nkearl ay na ?1
’ 0 7 90%
Nae~rl nac ~ùzkil na ilgoke`ikleay# ~`gokëe ya zqana keicqkr ac qyl na c` rabc`
fkzarbalgë~rki` &Zm`mmaeoarbar a~ c% 0;62+% Ykaenl yç# yqzùeb`ya pqa f`{ e loha~ly
ae qe brqzl {# pqa e0 yle na qe ~kzl { e? yle na l~rl ~kzl% Ac eøgarl na brqzly na r
loha~ly# nlena r 0 yle nac zrkgar ~kzl { r ? yle nac yabqenl ~kzl# pqa zqanae yar
mlrg`nly zlr gankl na y`i`r r loha~ly na e# ya n` zlr8 e0
e0Ir 0 ² e?Ir ? nlena e0 $ e? 7 e4 r 0 $ r ? 7 r &?'0>+
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 89/846
Nr% Fëi~lr Pqa}anl Qrç`y
?'91
Ahagzcl )1;%Qe raicq~`nlr na qe` mkrg` na agzcaly f` faifl ae~ra}ky~`y ile 0:
kebaekarly# na cly iq`cay = yle kebaekarly ik}kcay { > el% Ac raicq~`nlr pqkara agzca`r
1 na cly 0: kebaekarly ae~ra}ky~`nly% ¼Iqàe~ly brqzly zlykocay na cly ikeil kebaekarly
agzca`nly ile~aenràe av`i~`gae~a ~ray kebaekarly ik}kcay<
Ylcqikùe8
Nah`r pqa e0 7 = { e? 7 > q qy`r c` rabc` fkzarbalgë~rki` &?'0>+% Ae ac yqobrqzl na
~`g`ðl r 7 1# pqaragly r 0 7 9 kebaekarly ik}kcay { r ? kbq`c ` cly pqa el yle kebaekarly
ik}kcay% Ae~leiay# ac eøgarl na brqzly na ~`g`ðl 1 na ay~` i`~ablrç` ay8
=! >!
=I9²>I? 7 „„„„„ ² „„„„„ 7 &?:+&=+ 7 0?:
9!&= ’ 9+! ?!&> ’ ?+!
@pqç eù~aya pqa c` rabc` fkzarbalgë~rki` ay –zlekaenl hqe~ly‗ ac zrlnqi~l { c` rabc`
na ilgoke`ikleay z`r` lo~aear ac rayqc~`nl%
A}ae~ly kenazaenkae~ay { nazaenkae~ay%' Nly a}ae~ly @ { O ya nkia pqa yle
kenazaenkae~ay yk c` liqrraeik` na @ el mai~` c` zrlo`okckn`n na c` liqrraeik` na O# ay
naikr8
Z&@sO+ 7 Z&@+ &?'01+
l okae Z&Os@+ 7 Z&O+ &?'0=+
A}ae~ly nazaenkae~ay%' Yk c` liqrraeik` l el liqrraeik` na @# `mai~` c` zrlo`okckn`n
na liqrraeik` na O# ae~leiay# cly a}ae~ly yle nazaenkae~ay% @nagày# z`r` ~ray a}ae~ly
kenazaenkae~ay# nkb`gly# A0# A?# A9 c` zrlo`okckn`n ay8
ZrtA0A?A9w 7 ZrtA0wZrtA?sA0wZrtA9sA0A?w%
Ahagzcl )=:% Ya yacaiikle` `ca`~lrk`gae~a qe` i`r~` na qe` o`r`h` ilgøe na 1?
i`r~`y% Yk @ ay ac a}ae~l na pqa c` i`r~` acabkn` ya` qe `y { O ya` ac a}ae~l na pqa ya`
qe ilr`xùe# ae~leiay# @ { O yle a}ae~ly kenazaenkae~ay# {` pqa Z&@O+ 7 0,1?# Z&@+ 7
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 90/846
Nr% Fëi~lr Pqa}anl Qrç`y
?'9=
>,1? { Z&O+ 7 09,1?% Ay~l ya naoa ` pqa f`{ > `yay { 09 i`r~`y na ilr`xleay%
Ahagzcl )?% Ileyknar`r ac ayz`ikl gqay~r`cY 7 t@# O# I# N+# nlena Z&@+ 7 Z&N+ 7 %9
{ Z&O+ 7 Z&I+ 7 %?% &Daccar a~ `c% 0;;:+
&`+ Ykaenl `yç# namkekr cly a}ae~ly8
0 7 t@# Ow
? 7 tO# Iw
9 7 tI# Nw
&o+ ¼Iqàc na cly ykbqkae~ay z`ray na a}ae~ly yle kenazaenkae~ay l nazaenkae~ay<
&o+ 0 { ?
&i+ ? { 9
&n+ 0 { 9
Ylcqikùe8
&`+ 0 7 t@# Ow 7 %9# %?
? 7 tO# Iw 7 %?# %?
9 7 tI# Nw 7 %?# %9
&o+ Cly a}ae~ly 0 { ? yle kenazaenkae~ay
&i+ Cly a}ae~ly ? { 9 yle kenazaenkae~ay
&n+ Cly a}ae~ly 0 { 9 yle nazaenkae~ay
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 91/846
Nr% Fëi~lr Pqa}anl Qrç`y
?'92
_abc` gqc~kzcki`~k}` z`r` a}ae~ly nazaenkae~ay a kenazaenkae~ay
Ae `cbqe`y li`ykleay ya zqanae raylc}ar zrlocag`y na zrlo`okckn`n# zlr gankl na
ile~`r ac eøgarl na zqe~ly ae qe ayz`ikl gqay~r`c# ac iq`c ya ramkara ilgl c` rabc`
gqc~kzcki`~k}`% C` rabc` gqc~kzcki`~k}` l na ile~al na eøgarl na zqe~ly ae qe ayz`ikl
gqay~r`c ya qy` ae ay~a i`yl% Yke ago`rbl# zlnagly }ar pqa ay~` rabc` zcki` z`r` nly
a}ae~ly nazaenkae~ay { z`r` nly a}ae~ly kenazaenkae~ay% Ay~` rabc` na gqc~kzcki`ikùe
ay yqbarkn` zlr c` namkekikùe na zrlo`okckn`n ilenkikle`c rrko` nayirk~`% Ay~` rabc` na
zrlo`okckn`n ilenkikle`c ya n` ilgl8
Z&@sO+ 7 Z&@sO+,Z&O+# Z&O+≪
: &?'02+Zlnagly raayirkokr ay~` aiq`ikùe z`r` lo~aear8
Z&@sO+ 7 Z&O+ Z&@sO+ &?'06+
C` rabc` gqc~kzcki`~k}` z`r` nly a}ae~ly nazaenkae~ay ay8
Z&@ { O+ 7 Z&@+ Z&Os@+ { Z&@ { O+ 7 Z&O+ Z&@sO+ &?'0;+
Pqa mke`cgae~a# ~`gokëe ya ayirkoa ilgl8
Z&@ ∫ O+ 7 Z&O+ Z&@sO+ &?'0;`+
7 Z&@+ Z&Os@+ &?'?:o+
Nlena8
Z&@sO+ ya ramkara ` c` zrlo`okckn`n ilenkikle`c na pqa ac a}ae~l @ liqrr` n`nl pqa O
{` liqrrkù { Z&Os@+ ya ramkara c` zrlo`okckn`n ilenkikle`c na pqa ac a}ae~l O liqrr`
n`nl pqa @ {` liqrrkù%
Ae }arn`n# c` rabc` gqc~kzcki`~k}` z`r` a}ae~ly nazaenkae~ay ay c` zrlo`okckn`n
na c` ke~aryaiikùe &@sO+ na nly a}ae~ly @ { O% Ay~l nkia pqa# c` zrlo`okckn`n na
liqrraeik` ilehqe~` na a}ae~l @ { a}ae~l O ay kbq`c ` c` zrlo`okckn`n ilenkikle`c na
@ n`nl O zlr c` zrlo`okckn`n g`rbke`c na O%
C` rabc` gqc~kzcki`~k}` z`r` nly a}ae~ly kenazaenkae~ay ay8
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 92/846
Nr% Fëi~lr Pqa}anl Qrç`y
?'96
Z&@ { O+ 7 Z&@+ Z&O+ &?'?0+
L okae Z&@ ∫ O+ 7 Z&@+ Z&O+ &?'?0`+
Ahagzcl )=0% Ae~ra 9 nkyily na ilgzq~`nlr` qel ay~à namai~qlyl% Nly na accly ya
yacaiikle`e ca`~lrk`gae~a# zarl ac zrkgarl ay raagzc`x`nl# e~ay na y`i`r ac yabqenl
nkyil% ¼Iqàc ay c` zrlo`okckn`n na pqa `goly nkyily ay~ëe oqaely<
Ylcqikùe8
Nahagly pqa @ ya` ac a}ae~l na y`i`r qe nkyil oqael {# O# ac a}ae~l na y`i`r qe
yabqenl nkyil oqael% Ae~leiay# c` zrlo`okckn`n na @ ay Z&@+ 7 ?,9 { c` zrlo`okckn`n
na O ay Z&O+ 7 ?,9% Naoknl ` pqa f`{ raagzc`xl# ay~l ely cca}` ` c` rabc`
gqc~kzcki`~k}` na a}ae~ly kenazaenkae~ay% Zlr cl ~`e~l8
Z&@ { O+ 7 Z&@ ∫ O+ 7 Z&@+ Z&O+
7 &?,9+&?,9+ 7 >,;
Ahagzcl )=?% ]ake~a qekn`nay na qe zrlnqi~l g`eqm`i~qr`nl ya yk~ø`e ae qe nazùyk~l%
Nly na ay~`y qekn`nay ay~àe namai~qly`y% Yk ya keyzaiikle`e ~ln`y c`y ?: qekn`nay#
¼iqàc ay c` zrlo`okckn`n na yacaiikle`r &`ca`~lrk`gae~a+# c`y ? qekn`nay namai~qly`y<
Ylcqikùe8
Nah`r pqa @ ya` c` zrkgar` qekn`n namai~qly` { O c` yabqen` qekn`n namai~qly`%
Ae~leiay# pqaragly aeile~r`r c` zrlo`okckn`n na ke~aryaiikùe na cly nly a}ae~ly# ay
naikr# &@ ∫ O+% Cly a}ae~ly yle ic`r`gae~a nazaenkae~ay# zlrpqa c` zrlo`okckn`n na
pqa c` yabqen` qekn`n ya` namai~qly` nazaena na pqa yk c` zrkgar` qekn`n ya` l el
namai~qly`% @pqç8 Z&@+ 7 ?,?: { Z&O,@+ 7 0,0; Yqy~k~q{aenl ay~ly }`clray ae c`
mqeikùe na c` rabc` na gqc~kzcki`ikùe z`r` a}ae~ly nazaenkae~ay n`8
Z&@ ∨ O+ 7 Z&@+ Z&O,@+
7 &?,0:+&0,0;+
7 :%::1?=
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 93/846
Nr% Fëi~lr Pqa}anl Qrç`y
?'9;
El~`8 C` zrlo`okckn`n ilenkikle`c Z&O,@+ ay kbq`c ` 0;# zlrpqa yk @ liqrra &qe`
qekn`n namai~qly` yacaiikle`n` ae c` zrkgar` y`i`n`+# ae~leiay# pqan`ràe ylc`gae~a
0; qekn`nay z`r` yar yacaiikle`n`y ae c` yabqen` y`i`n`% Zlr cl ~`e~l# c` zrlo`okckn`n
na yacaiikle`r c`y ? qekn`nay namai~qly`y ay na :%::1?=# c` iq`c ay gq{ kgzrlo`oca# ay
naikr# 1 lzlr~qekn`nay ae 0#:::!
Zlr l~r` z`r~a# c` rabc` gqc~kzcki`~k}` z`r` a}ae~ly kenazaenkae~ay ya namkea
ilgl8
Z&@ ∫ O+ 7 Z&@+ Z&O+ &?'??+
Ahagzcl )=9% Yqzùeb`ya pqa ae qe` i`h` f`{ ?: mqykocay# na cly iqàcay 1 ay~àe
namai~qlyly% Yk ya yacaiikle`e ? mqykocay `ca`~lrk`gae~a# ae yqiaykùe# ¼iqàc ay c`
zrlo`okckn`n na pqa cly ? mqykocay yacaiikle`nly ay~ëe namai~qlyly<
Ylcqikùe8
Nahagly pqa @ ya` ac a}ae~l na y`i`r ac zrkgar mqykoca namai~qlyl {# O# ya` ac a}ae~l
na y`i`r ac yabqenl mqykoca namai~qlyl% Yk ke~arzra~`gly @∫ O ilgl ac a}ae~l na
pqa @ liqrra { O ac a}ae~l nayzqëy na pqa @ liqrrkù# ae~leiay# c` zrlo`okckn`n na @
ay Z&@+ 7 1,?: {# c` zrlo`okckn`n na O ay Z&O+ 7 >,0;% Zlr cl ~`e~l#
Z&@ ∫ O+ 7 Z&@+ Z&Os@+
7 &1,?:+&>,0;+
7 0,0;
Ahagzcl )=>% Ae cly hqably na cly n`nly &fleay~ly+ c` yqg` na qe ~l~`c na 2 zqe~ly na
cly nly n`nly b`e`% ¼Iqàc ay c` zrlo`okckn`n na pqa qe hqb`nlr c`eia nly 2
ileyaiq~k}ly<
Ylcqikùe8
Cly nly a}ae~ly yle kenazaenkae~ay# zlrpqa ac rayqc~`nl nac yabqenl c`ex`gkae~l el
`mai~` `c rayqc~`nl nac zrkgarl% @pqç ac ayz`ikl gqay~r`c ay na &9=+&9=+ 7 0#?;= { c`
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 94/846
Nr% Fëi~lr Pqa}anl Qrç`y
?'>:
zrlo`okckn`n na y`i`r c` yqg` na 2 ay na =# ay naikr# &?$1# 1$?# 9$># >$9# 0$=# =$0+%
Qy`enl c` rabc` gqc~kzcki`~k}` z`r` a}ae~ly kenazaenkae~ay n`8
Z&@ ∫ O+ 7 Z&@+ ² Z&O+
7 &=,9=+&=,9=+ 7 0,9=
Ahagzcl )=1% ¼Iqàe~ly zqe~ly gqay~r`cay f`{ ae qe ayz`ikl gqay~r`c# iq`enl qe z`r
na n`nly ya c`ex`e qe` }ax<
Ylcqikùe8
Ac zrkgar n`nl zqana i`ar ae e0 7 = g`ear`y% Z`r` i`n` qe` na ay~`y = g`ear`y# ac
yabqenl n`nl zqana i`ar ae e?
g`ear`y% Zlr cl ~`e~l8
e0 e? 7 &=+&=+ 7 9= g`ear`y zlykocay
Ac ayz`ikl gqay~r`c ay8
Y 7 t0'0# 0'?# 0'9# 0'># 0'1# 0'=# ?'0# ?'?# ?'9# ?'># ?'1# ?'=# 9'0# 9'?# 9'9# 9'># 9'1# 9'=#
>'0# >'?# >'9# >'># >'1# >'=# 1'0# 1'?# 1'9# 1'># 1'1# 1'=# ='0# ='?# ='9# ='># ='1# ='=w
_abc` `nk~k}` z`r` a}ae~ly gq~qly avicq{ae~ay { a}ae~ly el gq~qly avicq{ae~ay
Ae gqif`y `zcki`ikleay na c` ~alrç` na zrlo`okckn`n# ay~`gly ke~aray`nly ae ilgoke`r
zrlo`okckn`nay na a}ae~ly pqa ay~àe rac`ikle`nly na cbqe` g`ear`% Ae ay~a i`yl ya qy`
c` rabc` nk~k}`% @yç# c` rabc` nk~k}` ya qy` z`r` ilgzq~`r c` zrlo`okckn`n na c` qekùe na
nly a}ae~ly% Ay~` rabc` `zcki` z`r` a}ae~ly el gq~qly avicq{ae~ay {# ~`gokëe# z`r`
a}ae~ly gq~qly avicq{ae~ay%
Zlr ahagzcl yk @ { O yle a}ae~ly gq~qly avicq{ae~ay ac glnacl `nk~k}l ay8
Z&@ ∫ O+ 7 Z&@+ $ Z&O+ &?'?9+
Pqa ~`gokëe ya zqana ayirkokr ilgl8
Z&@ l O+ 7 Z&@+ $ Z&O+ &?'?9`+
El~`% Ac yçgolcl Z&@ l O+ ya ramkara c zrlo`okckn`n na iq`cpqkar` na cly a}ae~ly @ l
O liqrr`e l# okae# pqa `goly liqrr`e%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 95/846
Nr% Fëi~lr Pqa}anl Qrç`y
?'>0
Yke ago`rbl# yk cly i`yly @ { O el yle a}ae~ly gq~qly avicq{ae~ay# ac glnacl
`nk~k}l ay8
Z&@ ∨ O+ 7 Z&@+ $ Z&O+ ' Z&@ ∫ O+ &?'?>+
Ay~` mqeikùe ~`gokëe ya zqana avzray`r ilgl8
Z&@ l O+ 7 Z&@+ $ Z&O+ ' Z&@ { O+ &?'?1+
El~`8 Ac yçgolcl Z&@ { O+ ya qy` z`r` nael~`r c` zrlo`okckn`n na pqa goly a}ae~ly @
{ O liqrrkràe%
Ahagzcl )==% ¼Iqàc ay c` zrlo`okckn`n na lo~aear qe = ae ac zrkgar l yabqenl
c`ex`gkae~l na qe n`nl l# ae `goly c`ex`gkae~ly<Ylcqikùe8
@pqç# qy`gly c` rabc` `nk~k}` z`r` a}ae~ly el gq~qly avicq{ae~ay8
Z&@0 l @?+ 7 Z&@0+ $ Z&@?+ ' Z&@0 { @?+
L okae Z&@ l O+ 7 Z&@+ $ Z&O+ ' Z&@ { O+
Z`r` ay~l# nahagly pqa @0 nael~a ac a}ae~l na qe = ae ac zrkgar c`ex`gkae~l { @? ac
a}ae~l na qe = ae ac yabqenl c`ex`gkae~l% Pqaragly aeile~r`r c` zrlo`okckn`n na Z&@0
l @?+# cl iq`c pqkara naikr pqa ay~`gly oqyi`enl pqa ac = `z`raxi`# {` ya` ae ac zrkgar
c`ex`gkae~l l ae ac yabqenl c`ex`gkae~l l ae `goly c`ex`gkae~ly% Na g`ear` pqa8
Z&@0+ 7 0,=# Z&@?+ 7 0,= { Z&@0 { @?+ 7 0,9=
Yqoy~k~q{aenl ~lnly ay~ly }`clray ae c` mùrgqc` n`8
Z&@0 l @?+ 7 0,= $ 0,= ' 0,9= 7 00,9=
Ahagzcl )=2% C` zrlo`okckn`n na pqa G`rke` z`ya g`~agà~ki`y ay na ?,9# { c`
zrlo`okckn`n na pqa z`ya ac iqryl na kebcëy ay >,;% Yk c` zrlo`okckn`n na z`y`r `goly
iqryly ay na 0,># ¼iqàc ay c` zrlo`okckn`n na pqa G`rke` z`ya# iq`enl gaely qel na
ay~ly iqryly<
Ylcqikùe8
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 96/846
Nr% Fëi~lr Pqa}anl Qrç`y
?'>?
Nah`r pqa @ 7 ?,9 ya` ac a}ae~l na z`y`r g`~agà~ki`y { O 7 >,; ac a}ae~l na z`y`r
kebcëy { Z&@ { O+ 7 0,> ac a}ae~l na z`y`r g`~agà~ki`y a kebcëy# ae~leiay zlr c` rabc`
`nk~k}`8
Z&@ l O+ 7 Z&@+ $ Z&O+ ' Z&@ { O+
Z&@ ∨ O+ 7 Z&@+ $ Z&O+ ' Z&@ ∫ O+
Z&?,9 l 0,>+ 7 Z&?,9+ $ Z&0,>+ ' Z&?,9 { 0,>+
7 ?,9 $ >,; ' 0,> 7 90,9=
Ahagzcl )=6% ¼Iqàc ay c` zrlo`okckn`n na pqa qe` i`r~` yacaiikle`n`# ca`~lrk`gae~a# na
qe g`xl na 1? i`r~`y ya` qe ra{ l qe ilr`xùe<Ylcqikùe8
Naoknl ` pqa f`{ qe ~r`yc`z`nl# ya qy` c` rabc` `nk~k}` z`r` a}ae~ly el gq~qly
avicq{ae~ay Z&@ l O+ 7 Z&@+ $ Z&O+ ' Z&@ { O+% Ykaenl `yç# nahagly pqa @ 7 qe ra{
iq`cpqkar` { O 7 zraiky`gae~a qe ilr`xùe iq`cpqkar`% Zlr cl ~`e~l# Z&@+ 7 >,1?# Z&O+ 7
09,1?# Z&@ { O+ ra{ l ilr`xleay 7 0,1?% @pqç# ay cùbkil pqa c` zrlo`okckn`n ilehqe~`
&Qe` zrlo`okckn`n pqa gkna c` }arkykgkck~qn na pqa zqan`e liqrrkr nly gày a}ae~ly
c` gkyg` }ax+# na qe ra{ { qe ilr`xùe nao` na ray~`rya qe` }ax% Na el yar `yç ya
keicqkrç` nly }aiay ae aeile~r`r c` zrlo`okckn`n na pqa qe` i`r~` yacaiikle`n`
`ca`~lrk`gae~a mqar`# {` ya` qe ra{ l qe ilr`xùe% Avky~a qe ~r`yc`z`nl na rayqc~`nly# cl
iq`c pqkara naikr pqa avky~a c` zrlo`okckn`n na pqa ac ra{ &@+ { qe ilr`xùe &O+ liqrr`e
`c gkygl ~kagzl% Zlr cl ~`e~l8
Z&@ l O+ 7 Z&@+ $ Z&O+ ' Z&@ { O+
7 >,1? $ 09,1? ’ 0,1? 7 0=,1?
Ahagzcl )=;% Ay~a ay qe zrlocag` y`i`nl nac ckorl Y~`~ky~ki`c @e`c{yky mlr Naikykùe
G`dkeb na Glrrky F`goqrb &0;6;+# ac iq`c ay~à rac`ikle`nl ile c` zrlo`okckn`n na
lo~aear qe = ae ac zrkgarl l yabqenl c`ex`gkae~l na qe n`nl l ae goly c`ex`gkae~ly%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 97/846
Nr% Fëi~lr Pqa}anl Qrç`y
?'>9
Ay~l ay cl gkygl pqa i`ciqc`r c` zrlo`okckn`n na lo~aear qe =# iq`enl gaely qe` }ax
ae nly c`ex`gkae~ly na qe n`nl%
Ylcqikùe8
Nah`r pqa @0 nael~a c` y`ckn` na qe = ae ac zrkgar c`ex`gkae~l nac n`nl { @? razrayae~a
c` y`ckn` na qe = ae ac yabqenl c`ex`gkae~l% Pqaragly aeile~r`r ac } clr na Z&@0 l @?+%
Z`r` ay~l `e`ckiagly cly rayqc~`nly zlykocay nac zrkgarl { yabqenl c`ex`gkae~l%
0#0 ?#0 9#0 >#0 1#0 =#0
0#? ?#? 9#? >#? 1#? =#?
0#9 ?#9 9#9 >#9 1#9 =#9
0#> ?#> 9#> >#> 1#> =#>
0#1 ?#1 9#1 >#1 1#1 =#1
0#= ?#= 9#= >#= 1#= =#=
C` zrlo`okckn`n na pqa qe = y`cb` ae `goly c`ex`gkae~ly ay Z&@0 { @?+ 7 0,9=% C`
zrlo`okckn`n na pqa qe = y`cb` ae ac zrkgar c`ex`gkae~l ay Z&@0+ 7 0,= { ae ac yabqenl
c`ex`gkae~l ay Z&@?+ 7 0,=% Ae~leiay# `zcki`enl c` rabc` `nk~k}` n`8
Z&@0 l @?+ 7 Z&@0+ $ Z&@?+ ’ Z&@0 { @?+
7 0,= $ 0,= ’ 0,9=
7 00,9=
@pqç eù~aya pqa ay eaiay`rkl ray~`rca 0,9= z`r` a}k~`r qe ~r`yc`z`nl%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 98/846
Nr% Fëi~lr Pqa}anl Qrç`y
?'>>
Mkbqr` ?%=% C`y Mkbqr`y &`+# &o+ { &i+ gqay~r`e ac qyl na nk`br`g`y na ]aee z`r` c` rabc`
`nk~k}`# z`r` a}ae~ly gq~qly avicq{ae~ay { el gq~qly avicq{ae~ay% &Ac`olr`ikùe zrlzk`+
Ahagzcl )2:% Qe` ilgzq~`nlr` baear` `ca`~lrk`gae~a ac øc~kgl nçbk~l na qe eøgarl
~acamùekil% Aeile~r`r c` zrlo`okckn`n na pqa ac rayqc~`nl ya` qe 6 l ; &^rklc`# 0;6=+%
Ylcqikùe8
Cly rayqc~`nly na cly eøgarly 6 { ; yle a}ae~ly gq~qly avicq{ae~ay# zlr cl ~`e~l# ya
qy` c` mqeikùe &?'?9+% Ae~leiay# nahagly pqa Z&@+ 7 6 { Z&O+ 7 ;# { zcki`enl c` rabc`
`nk~k}` Z&@ ∫ O+ 7 Z&@+ $ Z&O+ { yqy~k~q{aenl n`8
Z&6 l ;+ 7 Z&6+ $ Z&;+ ’ Z&6 { ;+
7 0,0: $ 0,0: ' :
7 0,1
Ahagzcl )20% Yk A0 ay ac a}ae~l na y`i`r qe y na qe g`xl na i`r~`y { A? ay ac a}ae~l na
y`i`r qe ra{# ae~leiay# ZrtA0w 7 >,1? { ZrtA?w 7 >,1? 7 0,09 { c` zrlo`okckn`n na y`i r#
{` ya` qe `y l qe ra{ ay na8 ZrtA0 $ A?w 7 ZrtA0w $ ZrtA?w 7 0,09 $ 0,09 7 ?,09%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 99/846
Nr% Fëi~lr Pqa}anl Qrç`y
?'>1
Ae~leiay# zlr cl ~`e~l# naoknl ` pqa `goly ac `y { ac ra{ el zqanae yar y`i`nly na qe
ylcl ~krl# zlr cl ~`e~l# yle a}ae~ly gq~qly avicq{ae~ay { ya qy` c` mqeikùe &?'?9+%
Ahagzcl )2?% Yk A0 ay ac a}ae~l na y`i`r qe `y { A? ay ac a}ae~l na y`i`r qe` ayz`n`#
ae~leiay# A0 { A? el yle a}ae~ly gq~qly avicq{ae~ay naoknl ` pqa ac `y na ayz`n`y
zqana yar y`i`nl% Ykaenl `yç# ya qy` c` mqeikùe &?'?1+ z`r` a}ae~ly el gq~qly
avicq{ae~ay% Zlr cl ~`e~l# c` zrlo`okckn`n na y`i`r {` ya` qe y l qe` ayz`n` l goly ay8
ZrtA0 $ A?w 7 ZrtA0w $ ZrtA?w ' ZrtA0A?w 7 >,1? $ 09,1? ' 0,1? 7 >,09
Ahagzcl )29% ¼Iq`c ay c` zrlo`okckn`n na lo~aear qe yaky ae ac zrkgarl l yabqenl
c`ex`gkae~l na qe n`nl l# ae `goly c`ex`gkae~ly na qe n`nl fleay~l<
Ylcqikùe8
@pqç# qy`gly c` rabc` `nk~k}` z`r` a}ae~ly el gq~qly avicq{ae~ay# ay naikr# c` mqeikùe
&?'?1+% Z`r` ay~l nahagly pqa @0 nael~a ac a}ae~l na qe yaky ae ac zrkgar c`ex`gkae~l {
@? nael~a ac a}ae~l na qe yaky ae ac yabqenl c`ex`gkae~l% Pqaragly aeile~r`r c`
zrlo`okckn`n na Z&@0 l @?+# cl iq`c nkia pqa ay~`gly oqyi`enl pqa ac eøgarl yaky
`z`raxi`# {` ya` ae ac zrkgar c`ex`gkae~l l ae ac yabqenl c`ex`gkae~l l ae `goly
c`ex`gkae~ly% Ae~leiay8
Z&@0+ 7 0,=# Z&@?+ 7 0,= { Z&@0 { Z@?+ 7 0,9=
Yqy~k~q{aenl ~lnly ay~ly }`clray ae c` mqeikùe &?'?2+ n`8
Z&@ ∨ O+ 7 Z&@+ $ Z&O+ ’ Z&@ ∫ O+
Z&@0 ∨ @?+ 7 0,= $ 0,= ’ 0,9= 7 00,9=
Ahagzcl )2>% ¼Iqàc ay c` zrlo`okckn`n na pqa qe` i`r~` yacaiikle`n`# ca`~lrk`gae~a# na
qe g`xl na 1? i`r~`y ya` qe `y l qe ilr`xùe<
Ylcqikùe8
@pqç# eqa}`gae~a# ya qy` c` rabc` nk~k}` z`r` a}ae~ly el gq~qly avicq{ae~ay% Z`r` ay~l
nahagly pqa @ 7 qe y iq`cpqkar` { O 7 zraiky`gae~a qe ilr`xùe iq`cpqkar`% Qy`enl ac
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 100/846
Nr% Fëi~lr Pqa}anl Qrç`y
?'>=
g`xl na 1? i`r~`y &pqa ile~kaea iq`~rl ?¾y# iq`~rl 9¾y# iq`~rl >¾y# ………# iq`~rl 0:¾y#
iq`~rl yl~`y# iq`~rl rake`y# iq`~rl ra{ay { iq`~rl yay# ile yqy ilrrayzlenkae~ay mkbqr`y
na ~rëolcay# ilr`xleay# ayz`n`y { nk`g`e~ay+# zlr cl ~`e~l# z r` qe y iq`cpqkar`# Z&@+ 7
>,1? # z`r` qe ilr`xùe iq`cpqkar`# Z&O+ 7 09,1? {# z`r` `yay l ilr`xleay# Z&@ { O+
kbq`c 0,1?% @pqç# eqa}`gae~a# ilgl ae ac i`yl nac ahagzcl )=6 ay cùbkil yqgkr pqë#
c` zrlo`okckn`n ilehqe~` &qe` zrlo`okckn`n pqa gkna c` zrlo`okckn`n na pqa zqan`e
liqrrkr nly l gày a}ae~ly ` c` gkyg` }ax+# na qe y { qe ilr`xùe nao` ray~`rya qe` }ax%
Na el yar `yç# ya keicqkrç` nly }aiay ae aeile~r`r c` zrlo`okckn`n na pqa qe` i`r~`
yacaiikle`n` `c `x`r mqar`# {` ya` qe `y l qe ilr`xùe% Avky~a qe ylorazqay~l na
rayqc~`nly# cl iq`c nkia pqa avky~a c` zrlo`okckn`n na pqa ac `y &@+ { qe ilr`xùe &O+
y`cb`e ` c` gkyg` }ax% Zlr cl ~`e~l8
Z&`y l ilr`xùe+ 7 Z&`y+ $ Z&ilr`xùe+ ’ Z&`y { ilr`xùe+
7 >,1? $ 09,1? ’ 0,1?
7 0=,1?
Ahagzcl )21% Ae ay~a ahagzcl# z`r` kcqy~r`r c` ca{ `nk~k}` na zrlo`okckn`n# ae c` iq`c
avky~ae ~r`yc`z`nly# ya zqana f`iar qy`enl nk`br`g`y na ]aee% Z`r` ay~l# ya f`ia ac
ykbqkae~a avzarkgae~l na c`ex`r nly glean`y% Ykaenl `yç# ay~kg`r c` zrlo`okckn`n na
y`i`r# iq`enl gaely qe` i`r`# {` ya` ae ac zrkgar c`ex`gkae~l l ae ac yabqenl
c`ex`gkae~l &Ygk~f# 0;61+%
Ylcqikùe8
Zrkgar`gae~a# aecky~`r cly iq`~rl zlykocay rayqc~`nly zlekaenl F 7 i`r`y { 7 ylcay#
ay naikr# F^# FF# F { ^% @pqç# z`r` a}k~`r qe ~r`yc`z`nl# ya qy` c` rabc` nk~k}` z`r`
a}ae~ly el gq~qly avicq{ae~ay% Ac nk`br`g` na ]aee na `o`hl kcqy~r` ic`r`gae~a# ac
~r`yc`z`nl pqa zqnkar` liqrrkr# yk ya yqg`r` c` zrlo`okckn`n na qe` i`r` ae ac zrkgar
c`ex`gkae~l# gày c` zrlo`okckn`n na qe` i`r` ae ac yabqenl c`ex`gkae~l pqa n`rç`· $
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 101/846
Nr% Fëi~lr Pqa}anl Qrç`y
?'>2
· 7 0# cl iq`c yarç` keilrrai~l% Ae ay~a i`yl# c` zrlo`okckn`n na qe` i`r` ae ac zrkgar
c`ex`gkae~l ay na :%14 c` zrlo`okckn`n na qe` i`r` ae ac yabqenl c`ex`gkae~l ay :%1 {#
c` zrlo`okckn`n na i`r`y ae `goly c`ex`gkae~ly ay na :%?1% Zlr cl ~`e~l# c` zrlo`okckn`n
na qe` i`r`# {` ya` ae ac zrkgarl l yabqenl c`ex`gkae~l ay8
Z&F l ^+ 7 Z&F+ $ Z&^+ ' Z&F { ^+
7 · $ · ' ³ 7 µ
Ac ~r`yc`z`nl l c` razrayae~`ikùe nac zl~aeik`c na qe nloca ile~al &FF+ ya n` `o`hl%
Mkbqr` ?%2% Mkbqr` aypqagà~ki` gly~r`enl qe nk`br`g` na ]aee kenki`enl ac ~r`yc`z`nl
na i`r`y &FF+# pqa liqrra ae c` ke~aryaiikùe na @ { O &Ygk~f# 0;61+%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 102/846
Nr% Fëi~lr Pqa}anl Qrç`y
?'>6
_abc` gqc~kzcki`~k}` z`r` gày na nly a}ae~ly
L~r` rabc` ø~kc z`r` i`ciqc`r c` zrlo`okckn`n na qe a}ae~l ay ac glnacl na c` rabc`
gqc~kzcki`~k}`% Ay~` rabc` ya namkea ilgl c` zrlo`okckn`n na c` liqrraeik` ilehqe~` pqa
ac a}ae~l @ { ac a}ae~l O ya` kbq`c ` c` zrlo`okckn`n ilenkikle`c nac a}ae~l @ n`nl ac
a}ae~l O gqc~kzcki`nl zlr c` zrlo`okckn`n g`rbke`c na O%
^alrag` 08 Yk ae qe avzarkgae~l# cly a}ae~ly nazaenkae~ay @0# @?# @9#%%%@d zqanae
liqrrkr# ae~leiay8
Z&@0 ∫ @? ∫ @9 ∫ %%% @d + 7 Z&@0+Z&@?s@0+Z&@9s@0 ∫ @?+ %%%
%%%Z&@d s@
0 ∫ @
? ∫ %%% ∫ @
d'0+ &?'?=+
^alrag` ?8 Yk cly a}ae~ly @0# @?# @9#%%% @d yle kenazaenkae~ay# ae~leiay8
Z&@0 ∫ @? ∫ @9%%% ∫ @d + 7 Z&@0+Z&@?+Z&@9+…%Z&@d + &?'?2+
Ahagzcl )2=% ^ray e`kzay ya y`i`e ae yqiaykùe# yke raagzc`xl% Aeile~r`r c`
zrlo`okckn`n na pqa liqrr` ac a}ae~l @0 ∫ @? ∫ @9# iq`enl @0 ay ac a}ae~l na pqa c`
zrkgar` i`r~` ya` qe `y rlhl# @? ya` pqa c` yabqen` i`r~` ya` qe 0: l qe` yl~` {# @9 ya`
ac a}ae~l na pqa c` ~ariar` i`r~` ya` g`{lr pqa qe 9# zarl gaelr pqa qe 2%
Ylcqikùe8
Zrkgarl }`gly ` namkekr cly a}ae~ly8
@08 c` zrkgar` i`r~` ay qe `y rlhl &`pqç# eù~aya pqa f`{ elgày ? `yay rlhly+
@?8 c` yabqen` i`r~` ya` qe 0: l qe` yl~` &f`{ iq`~rl 0:¾y { iq`~rl yl~`y+
@98 c` ~ariar` i`r~` ya` g`{lr pqa 9 zarl gaelr pqa 2 &f`{ nlia i`r~`y ae~ra ac 9 { ac 2+%
Cly }`clray yle8 Z&@0+ 7 ?,1?4 Z&@?s@0+ 7 6,104 Z&@9s@0 ∫ @?+ 7 0?,1:% &@pqç eù~aya
pqa# ae c` zrkgar` y`i`n` yle 1? i`r~`y# zarl ae c` yabqen` y`i`n` ac eøgarl na i`r~`y
o`h` ` 10 { ae c` ~ariar` y`i`n` o h` ` 1: i`r~`y+%
Zlr cl ~`e~l8
Z&@0 ∫ @? ∫ @9+ 7 Z&@0+Z&@?s@0+Z&@9s@0 ∫ @?+
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 103/846
Nr% Fëi~lr Pqa}anl Qrç`y
?'>;
7 &?,1?+&6,10+&0?,1:+
7 6,1#1?1
Ahagzcl )22% F`iar ac gkygl ahagzcl )0 na rrko` zarl# ae ay~` li`ykùe# ile raagzc`xl
na i`r~`y%
Ylcqikùe8
@c f`oar raagzc`xl na i`r~`y# ac zrlocag` ya ranqia ` c` rabc` gqc~kzcki`~k}` z`r`
a}ae~ly kenazaenkae~ay% Cly }`clray na c`y }`rk`ocay yle8
Z&@0+ 7 ?,1?4 Z&@?+ 7 6,1?4 { Z&@9+ 7 0?,1?
Aeyabqkn`# yqoy~k~q{aenl cly }`clray ae c` avzraykùe na `o`hl n`8
Z&@0 ∫ @? ∫ @9+ 7 Z&@0+Z&@?+Z&@9+
7 &?,1?+&0?,1?+&0?,1?+
7 :%::?
Ahagzcl )26% Iq`~rl i`r~`y ya y`i`e ae yqiaykùe% Aeile~r`r c` zrlo`okckn`n na pqa c`
zrkgar` i`r~` ya` qe ra{4 c` yabqen` ya` qe ; na nk`g`e~ay4 c` ~ariar` ya`# iq`enl
gaely# qe` yl~` &`yqgkaenl pqa ac `y ya` c` øc~kg` i`r~`+ {# c` iq`r~` i`r~` ya` qe 2
eabrl%
Ylcqikùe8
Nahagly pqa @ ya` iq`cpqkar` na cly > ra{ay4 O ya` zraiky`gae~a qe ; na nk`g`e~ay4 I
ya` kbq`c ` nlia i`r~`y# ay naikr# nayna c` yl~` f`y~` ac `y4 { N ya` iq`cpqkar` na cly nly
yka~ay eabrly%
Ykaenl `yç# Z&@+ 7 >,1?# Z&O+ 7 0,10# Z&I+ 7 0=,1:# Z&N+ 7 ?,>;
Zlr cl ~`e~l8
Z&@ ∫ O ∫ I ∫ N+ 7 &>,1?+&0,10+&0=,1:+&?,>;+
7 0?6,=#>;2#>:: 7 %::::?
Ahagzcl )2;% Nahagly pqa qe z`r na n`nly ya`e c`ex`nly qe` ylc` }ax% C`y ~`oc`y na
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 104/846
Nr% Fëi~lr Pqa}anl Qrç`y
?'1:
`o`hl gqay~r`e cly rayqc~`nly zlykocay# c`y zrlo`okckn`nay { yq razrayae~`ikùe% F`iar
qe` bràmki` pqa }`{` ae mqeikùe na Z& V +# ay naikr# 0,9=# ?,9=# a~i% &Ac ay~qnk`e~a cl
f`rà+% Ylcqikùe8
^@OC@ ?%0% Nk`br`g` gly~r`enl c` nky~rkoqikùe na zrlo`okckn`nay iq`enl ya c`ex`e
nly n`nly qe` ylc` }ax% &Ac`olr`ikùe zrlzk`+
El% ëvk~ly s 0 ? 9 > 1 = 1 > 9 ? 0
Yqg` & V + s ? 9 > 1 = 2 6 ; 0: 00 0?
Zrlo`okckn`n s0,9= ?,9= 9,9= >,9= 1,9= =,9= 1,9= >,9= 9,9= ?,9= 0,9=
^@OC@ ?%?% Zrlo`okckn`nay iq`enl ya c`ex`e nly n`nly% &Ac`olr`ikùe zrlzk`+
Yqg` na cly n`nly Eøgarl na ëvk~ly Zrlo`okckn`n
? 0 0,9=
9 ? ?,9=
> 9 9,9=
1 > >,9=
= 1 1,9=
2 = =,9=
6 1 1,9=
; > >,9=
0: 9 9,9=
00 ? ?,9=
0? 0 0,9=
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 105/846
Nr% Fëi~lr Pqa}anl Qrç`y
?'10
^@OC@ ?%9% _ayqc~`nly iq`enl ya c`ex`e nly n`nly qe` ylc` }ax% &Ac`olr`ikùe zrlzk`+
Zrkgar n`nl Yabqenl n`nl _ayqc~`nl Yqg` na cly eøgarly
0 0'0 ?? 0'? 9
9 0'9 >
0 > 0'> 1
1 0'1 =
= 0'= 2
0 ?'0 9
? ?'? >
9 ?'9 1
? > ?'> =
1 ?'1 2
= ?'= 6
0 9'0 >
? 9'? 1
9 9'9 =
9 > 9'> 2
1 9'1 6
= 9'= ;
0 >'0 1
? >'? =
9 >'9 2
> > >'> 61 >'1 ;
= >'= 0:
0 1'0 =
? 1'? 2
9 1'9 6
1 > 1'> ;
1 1'1 0:
= 1'= 00
0 ='0 2
? ='? 69 ='9 ;
= > ='> 0:
1 ='1 00
= ='= 0?
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 106/846
Nr% Fëi~lr Pqa}anl Qrç`y
?'1?
Aharikikly I`zç~qcl ?
?%0% Yk qe` glean` ~kaea nly i`r`y nael~`n`y zlr àbqkc`y l ylcay# ¼iqàc ay c`
zrlo`okckn`n na pqa y`cb` qe ylc< &:%1+
?%?% Ae ac i`yl na qe n`nl pqa ~kaea = eøgarly l i`r`y# ae~leiay# yk ac n`nl ay fleay~l#
~ln`y cly eøgarly nac 0 `c = ~kaeae c` gkyg` zrlo`okckn`n na i`ar% Ae~leiay# ¼iqàc ay c
zrlo`okckn`n na y`i`r qe 0<
?%9% Ae ac c`ex`gkae~l na qe n`nl# ¼iqàc ay c` zrlo`okckn`n na pqa ya gqay~rae cly
eøgarly 9 l >< ¼Iqàc ay c` zrlo`okckn`n na el y`i`r qe 9 l qe >< &?,9+
?%>% Yk qe` zaryle` ay yacaiikle`n` `c `x`r na qe brqzl na ?: zykiùclbly { 9:ylikùclbly# ¼iqàc ay c` zrlo`okckn`n na yacaiikle`r qe ylikùclbl<
?%1% ¼Iqàc na cly ykbqkae~ay el ay qe` zrlo`okckn`n< 9,2# ?# '0,?# 9,># ;;,0:0# :# 0# 1#
0%00# 0%:::0# :%:::0# :%::0# :%;;;;% &?# 1# 0%00# 0%:::0
?%=% C` zrlo`okckn`n na pqa Hq`e ay~ë }k}l ae ?: ðly ay na :%2 { c` zrlo`okckn`n na pqa
Zanrl ay~ë }k}l ae ?: ðly ay :%1% ¼Iqàc ay c` zrlo`okckn`n na pqa goly ay~ëe }k}ly ae
?: `ðly<
?%2% Yk A0 { A? ya`e cly a}ae~ly na /i`r`y nac pqke~l c`ex`gkae~l/ { /i`r`y ae ac yav~l
c`ex`gkae~l/ na qe` glean`# ae~leiay# cly a}ae~ly A0 { A? yle a}ae~ly kenazaenkae~ay%
¼Iqàc ay c` zrlo`okckn`n na pqa y`cb`e i`r`y ae `goly c`ex`gkae~ly< &0,>+
?%6% ¼Iqàc ay c` zrlo`okckn`n na y`i`r iq`enl gaely qe = ae nly c`ex`gkae~ly na qe
n`nl fleay~l< Yqbaraeik`8 Qy`r c` rabc` na `nkikùe%
?%;% @yqgkaenl pqa cly }`rleay { c`y fagor`y liqrr`e kbq`cgae~a { pqa ac yavl na
iq`cpqkar fkhl ya` kenazaenkae~a na iq`cpqkar` na cly farg`ely l farg`e`y# aeile~r`r ac
ayz`ikl gqay~r`c { aeile~r`r c` zrlo`okckn`n na pqa qe` z`rah` ile 9 fkhly ~aenràe8
&`+ av`i~`gae~a ? }`rleay% &9,6+
&o+ Av`i~`gae~a ? fagor`y% &9,6+
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 107/846
Nr% Fëi~lr Pqa}anl Qrç`y
?'19
&i+ Iq`enl gaely ? }`rleay &Z& V ≡ ?++
?%0:% C`ex`r qe` glean` ? }aiay% Aeile~r`r cly ykbqkae~ay a}ae~ly8
&`+ Aeile~r`r ac ayz`ikl gqay~r`c%
&o+ Aeile~r`r c` zrlo`okckn`n na pqa y`cb`e av`i~`gae~a qe` i`r` { qe àbqkc`%
?%00% Aeile~r`r ac eøgarl na zargq~`ikleay na c`y ca~r`y `# o# i ~lg`n`y nly ` qe
~kagzl% &=+
?%0?% Aeile~r`r ac eøgarl na ilgoke`ikleay na c`y ca~r`y `# o# i ~lg`n`y nly ` qe
~kagzl%
?%09% Z`r` nly a}ae~ly @ { O# Z&@+ 7 :%0:# Z&O+ 7 :%>: { Z&@∫O+ 7 :%:1% Na~argke`r8&`+ Z&@sO+ &:%0?1+
&o+ Z&Os@+% &:%1:+
?%0>% Yk Z&O+ 7 ?21:,0:#::: { Z&@∫ O+ 7 :%0># aeile~r`r Z&@sO+%
?%01% Nahagly pqa A ya` ac a}ae~l na pqa# cly eøgarly z`ray na qe n`nl# ya`e ?# ># =%
Aeile~r`r c` zrlo`okckn`n na pqa y`cb`e ay~ly a}ae~ly% &0,?+
?%0=% Qe brqzl na ileyqgknlray ileyky~a na 6: ay~qnk`e~ay# 9: na cly iq`cay yle
gqharay% Yk qe ay~qnk`e~a ay yacaiikle`nl `ca`~lrk`gae~a na ay~a brqzl# aeile~r`r c`
zrlo`okckn`n na el ayilbar ` qe` gqhar%
?%02% Na cly ykbqkae~ay a}ae~ly naikr iq`cay a}ae~ly yle gq~qly avicq{ae~ay8
&`+ Yacaiikle`enl qe ay~qnk`e~a pqkae `~kaena c`y ic`yay na ay~`nçy~ki` rabqc`rgae~a%
Yacaiikle`enl qe ay~qnk`e~a pqkae zlyaa qe` ilgzq~`nlr`%
&o+ Yacaiikle`enl ` qe` zaryle` ile zacl rqokl%
Yacaiikle`enl ` qe` zaryle` ile lhly i`mëy%
&i+ Yacaiikle`enl qe iqryl `i`nëgkil rapqarknl% &a}ae~l gq~ql avicq{ae~a+
Yacaiikle`enl qe iqryl acai~k}l
?%06% C` zrlo`okckn`n na pqa qe }qacl na `}kùe y`cb` ` ~kagzl ay na Z&N+ 7 :%694 c`
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 108/846
Nr% Fëi~lr Pqa}anl Qrç`y
?'1>
zrlo`okckn`n na pqa ac }qacl ccabqa ` ~kagzl ay na Z&@+ 7 :%6?4 {# c` zrlo`okckn`n na
pqa y`cb` ~kagzl { ccabqa ~kagzl ay na Z&N∫@+ 7 :%26% @yøg`ya qe` zrlo`okckn`n
ilenkikle`c% Aeile~r`r c` zrlo`okckn`n na pqa ac `}kùe8
&`+ Ccabqa ` ~kagzl n`nl pqa z`r~kù ` ~kagzl%
&o+ Y`cb` ` ~kagzl n`nl pqa `rrkoù ` ~kagzl%
?%0;% Yqzùeb`ya pqa qe` i`h` ile~kaea 9 olc`y oc`ei`y { ? olc`y eabr`y% @yøg`ya pqa
el f`{ raagzc`xl {# zlr cl ~`e~l# yle a}ae~ly nazaenkae~ay% Ykaenl `yç# i`ciqc`r cly
ykbqkae~ay aeqeik`nly8
&`+ ¼Iqàc ay c` zrlo`okckn`n na pqa c` zrkgar` olc` y`i`n` ya` eabr`< &?,1+
&o+ ¼Iqàc ay c` zrlo`okckn`n na c` yabqen` olc` y`i`n` ya` eabr` n`nl pqa c` zrkgar`
olc` y`i`n` mqa eabr`< &0,>+
&i+ ¼Iqàc ay c` zrlo`okckn`n na pqa `go`y olc`y y`i`n`y ya`e eabr`y< &0,0:+
?%?:% Qy`enl c` mkbqr` na `o`hl { c` ykgolclbç` na nk`br`g`y na ]aee namkekr c`y
ykbqkae~ay rabkleay8
&`+ _abkleay 0 { ?
&o+ _abkleay 0 { 9
&i+ _abkleay 0# ?# 9# ># 1# { 2
&n+ _abkleay > { 2
&a+ _abkùe 0
&m+ _abkleay ?# =# 2
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 109/846
Nr% Fëi~lr Pqa}anl Qrç`y
?'11
Mkbqr` gly~r`enl cly nk`br`g`y na ]aee% &Mqae~a8 Gle~blgar{ a~ `c%0;;=+
?%?0% Yqzùeb`ya pqa ya ay~qnk`e 0:#::: zaryle`y na ?: `ðly { ya aeiqae~r` pqa ;;=0
}k}karle ?0 ðly% Aeile~r`r c` zrlo`okckn`n na pqa qe` zaryle` na ?: ðly }`{` }k}kr
?0 `ðly% &%;;=0+
?%??% Qe ay~qnkl aeiqay~ù ` qe brqzl na 0:: zrlmaykleky~`y pqa ileyky~ç` na >:
kebaekarly &na cly iq`cay c` gk~`n ar`e gqharay+ { ` =: `rpqk~ai~ly &na cly iq`cay c`
gk~`n ar`e gqharay+% Aeile~r`r c` zrlo`okckn`n na pqa qe zrlmaykleky~` yacaiikle`nl
`ca`~lrk`gae~a ya` kebaekarl l gqhar% @yqgkr qe` rabc` `nk~k}`%
?%?9% ¼Iqàc ay c` zrlo`okckn`n na pqa qe` i`r~` yacaiikle`n` `c `x`r na qe g`xl na 1?
e`kzay ya` qe` rake` l qe ilr`xùe< @yqgkr qe` rabc` `nk~k}` z`r` a}ae~ly el gq~qly
avicq{ae~ay &>,09+
?%?>% ¼Iqàc ay c` zrlo`okckn`n na y`i`r qe = ae ac zrkgarl l yabqenl c`ex`gkae~l na qe
n`nl fleay~l l# ae `goly c`ex`gkae~ly<
?%?1% Qe kebaekarl m`orki`e~a na gl~lray ca zraliqz`e ~ray ~kzly na zrkeikz`cay namai~ly%
Zlr ahagzcl# @ ay ac a}ae~l ae ac pqa ac aha nac gl~lr ay nag`yk`nl br`ena# O ac a}ae~l
ae ac pqa c`y oloke`y yle ke`naiq`n`y { I ac a}ae~l ae ac pqa c`y ileavkleay acëi~rki`y
yle key`~kym`i~lrk`y% Na yar `yç# avzray`r }aro`cgae~a pqë a}ae~ly ay~àe razrayae~`nly
zlr c`y ykbqkae~ay rabkleay nac nk`br`g` na ]aee% &Hlfeyle# 0;;2+
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 110/846
Nr% Fëi~lr Pqa}anl Qrç`y
?'1=
&`+ _abkùe ?% &N`nl pqa c` rabkùe ? ay~à ae @ { O# zarl el ae I# ay~l nkia pqa# ac aha ay
nag`yk`nl br`ena { c`y oloke`y yle ke`naiq`n`y+
&o+ _abkùe 0 { 9 hqe~`y
&i+ _abkleay 9# 1# = { 6 hqe~`y &Naoknl ` pqa ~ln`y ay~`y rabkleay ay~àe mqar` na c`
rabkùe @# ay~l razrayae~` ac a}ae~l ae pqa ac aha ay nag`yk`nl c`rbl l namai~qlyl+
Mkbqr` gly~r`enl cly ayz`ikly gqay~r`cay { a}ae~ly% &Mqae~a8 Hlfeyle 0;;2+
?%?=% _amkrkëenlya c zrlocag` e~arklr razrayae~`r ile yçgolcly na ]aee c`y ykbqkae~ay
rabkleay8
&`+ ># =# 2
&o+ 0#>
&i+ 0# ?# 1# 2
&n+ 0# ?
&a+ 0# 9# >%
?%?2% Ae ay~qnkly na fkbkaea kenqy~rk`c { yabqrkn`n na lorarly na qe` kenqy~rk` ya
nayiqorkù pqa ac 6 eaiayk~`rle ol~`y na fqca z`r` zrl~aiikùe ile~r` nayi`rb`y
acëi~rki`y# 01 eaiayk~`rle i`yily zrl~ai~lray z`r` c` i`oax` {# 9 eaiayk~`rle# goly#
ol~`y na fqca zrl~ai~lr`y { i`yily zrl~ai~lray z`r` c` i`oax`% ¼Iqàc ay c` zrlo`okckn`n
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 111/846
Nr% Fëi~lr Pqa}anl Qrç`y
?'12
na pqa qe ~r`o`h`nlr yacaiikle`nl# c x`r# eaiayk~`rà# {` ya`# ol~`y zrl~ai~lr`y na fqca l
i`yily zrl~ai~lray z`r` c` i`oax`< Yqbaraeik`8 qy`r ac glnacl `nk~k}l% &:%?:+
?%?6% Ya c`ex` qe` glean` nly }aiay% Aeile~r`r c` zrlo`okckn`n na y`i`r qe` i`r`# {`
ya` ae ac zrkgar c`ex`gkae~l l yabqenl c`ex`gkae~l l ae goly c`ex`gkae~ly% @yqgkr
pqa F 7 i`r`y# ^ 7 àbqkc`y%
?%?;% Qe` ilgzq~`nlr` baear`# `ca`~lrk`gae~a# ac øc~kgl nçbk~l na qe eøgarl
~acamùekil% I`ciqc`r8
&`+ C` zrlo`okckn`n na pqa ac rayqc~`nl ya` qe 6 l ;% &0,1+
&o+ C` zrlo`okckn`n na pqa ac rayqc~`nl ya` qe eøgarl ele l gaelr pqa >% &:%2+
?%9:% Aeile~r`r c` zrlo`okckn`n na y`i`r qe ~l~`c na 2 l 00 iq`enl qe z`r na n`nly ya
c`ex`e%
?%90% ¼C` zrlo`okckn`n na y`i`r qe `y l qe ra{ na qe g`xl na 1? i`r~`y< &?,09+
?%9?% Iqàc ay c` zrlo`okckn`n na y`i`r# {` ya` qe y l qe` ayz`n` l goly ae qe` y`i`n`
na i`r~`y na qe g`xl na 1? e`kzay%
?%99 ¼Iqàe~`y ilgkn`y ileyky~ae~ay na qe` ylz`# qe agz`ran`nl# qe zly~ra { qe
ramrayil yle zlykocay# yk zlnagly yacaiikle`r > ylz`y# 9 ~kzly na agz`ran`nly# 1 zly~ray
{ > ramrayily< &?>:+
?%9>% Nly glean`y ya c`ex`e% ¼Iqàc ay c` zrlo`okckn`n na pqa go`y glean`y i`kb`e ae
àbqkc`y< Qy`r rabc` gqc~kzcki`~k}`%
?%91% Qe` z`rah` na raikëe i`y`nly zc`ea` ~aear 9 fkhly% Aeile~r`r cly ykbqkae~ay
aeqeik`nly8
&`+ C` zrlo`okckn`n pqa ~lnly cly fkhly ya`e flgoray% &0,6+
&o+ C` zrlo`okckn`n na 9 fagor`y% &0,0=+
&i+ C` zrlo`okckn`n na av`i~`gae~a ? }`rleay% &9,6+
&n+ C` zrlo`okckn`n na 9 }`rleay { 9 fagor`y% &0,=>+
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 112/846
Nr% Fëi~lr Pqa}anl Qrç`y
?'16
&a+ C` zrlo`okckn`n na ~aear ` cl gày ? }`rleay% &9,6+
&m+ C` zrlo`okckn`n na ~aear iq`enl gaely ? }`rleay% &>,6+
@yqgkr pqa cly }`rleay { c`y fagor`y ~kaeae c` gkyg` lzlr~qekn`n { pqa ac yavl na
i`n` fkhl ya` kenazaenkae~a nac yavl nac l~rl% F`iar qe nk`br`g` na àrolc z`r` m`ikck~`r
ac iùgzq~l%
?%9=% Ile ramaraeik` `c zrlocag` `e~arklr# yk c` m`gkck` mqar` na > fkhly# ¼iqàc yarç` c`
zrlo`okckn`n na mqar`e > }`rleay {,l > fagor`y<
?%92% Ya y`i`e nly i`r~`y `c `x`r na qe g`xl na 1? e`kzay% ¼Pqë zrlo`okckn`n f`{ na
lo~aear nly `yay yk<
&`+ C` zrkgar` i`r~` ay razqay~` `e~ay na y`i`r c` yabqen` i`r~`% &0,=;+
&o+ C` zrkgar` i`r~` el ay razqay~` `e~ay na y`i`r c` yabqen` i`r~`% @yqgkr qe` rabc`
gqc~kzcki`~k}`% &0?,?=1?+
?%96% F`{ 0: rlccly na zacçiqc` ae qe` i`h` { 9 ay~àe namai~qlyly% Ya y`i`e ? rlccly qel
na~rày nac l~rl% ¼Iqàc ay c` zrlo`okckn`n na yacaiikle`r qe rlccl namai~qlyl yabqknl zlr
l~rl rlccl namai~qlyl# yke el f`{ raagzc`xl< Qy`r rabc` gqc~kzcki`~k}`%
?%9;% _ayzlenar ` c`y ykbqkae~ay zrabqe~`y4
&`+ ¼Iqàe~ly rayqc~`nly f`{ ae qe ayz`ikl gqay~r`c# iq`enl ya c`ex`e qe z`r na n`nly
qe` ylc` }ax< &9=+
¼Iqàc ay ëy~a<
&o+ ¼Iqàe~ly zqe~ly gqay~r`cay f`{ ae qe ayz`ikl gqay~r`c iq`enl ya c`ex`e 9 n`nly
ykgqc~àea`gae~a<
?%>:% Qe nkyað`nlr na qe` eqa}` yqonk}kykùe lmraia ` cly ilgzr`nlray na i`y`y# qe`
yacaiikùe na ay~kcly av~arklray na kebcëy# røy~kil# ilclek`c# { av~arklr ~r`nkikle`c
ilgoke`nly ile ~kzly na r`eifl# na nly zkyly { qe nayek}ac% ¼Na iqàe~`y g`ear`y ya
zqana lrnae`r qe` na ay~`y i`y`y ile ayly ay~kcly na iley~rqiikùe< F`iar qe nk`br`g`
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 113/846
Nr% Fëi~lr Pqa}anl Qrç`y
?'1;
na àrolc% Yqbaraeik`8 qy`r c` rabc` nac zrlnqi~l e0e?% &0?+
?%>0% Qe ay~qnkl na ~ràmkil }afkiqc`r kenki` pqa na 9#21= q~ly pqa ya iari`e c` zc`x`#
612 ae~r`e ae ac `z`ri`gkae~l% ¼Iqàc ay c` zrlo`okckn`n na pqa qe `q~l el ae~ra ae ac
`z`ri`gkae~l< &Z&612+ 7 :%?9# p 7 <+
?%>?% Ae qe` zrqao` c` zrkgar` zrabqe~` ay na m`cyl { }arn`narl {# c` yabqen` zrabqe~`
ay na yacaiikùe gøc~kzca ile zlykocay rayzqay~`y na `# o# i# n# a% &`+ ¼Iqàe~`y yaiqaeik`y
na zlykocay rayzqay~`y f`{ ae ay~`y nly zrabqe~`y< &o+ Qy`r qe nk`br`g` na àrolc {
aecky~`r ac ayz`ikl gqay~r`c% &0:+
?%>9% Ae ac nkyaðl na qe yky~ag` na ilgzq~`nlr`# yk qe o{~a ya namkea ilgl qe`
yaiqaeik` na 6 ok~y {# i`n` ok~ naoa yar : l 0# ¼iqàe~ly o{~ay nkmarae~ay yle zlykocay<
?%>>% Avzckpqa ae yqy zrlzk`y z`c`or`y cl pqa ykbekmki`e cly ykbqkae~ay ~ërgkely8
&`+ Avzarkgae~l `ca`~lrkl
&o+ Ayz`ikl gqay~r`c
&i+ A}ae~l
?%>1% F`oc`enl na m`i~lrk`cay# a}`cq`r 1:! Yqbaraeik`8 qy`r c` `zrlvkg`ikùe na Y~arckeb8
e! u ∘?ϊe ee
a'e
&9%:>v0:=>
+
?%>=% Ya c`ex` qe` glean` 9 }aiay ileyaiq~k}`y% F`iar qe nk`br`g` na àrolc ile cly
rayqc~`nly na ylcay { àbqkc`y { ac ayz`ikl gqay~r`c% I`ciqc`r cl ykbqkae~a8
&`+ Eøgarl na ylcay ay iq`enl gaely ?%
&o+ Yabqenl c`ex`gkae~l yle ylcay%
&i+ Ac eøgarl na ylcay ay av`i~`gae~a ?%
&n+ Yabqenl c`ex`gkae~l yle àbqkc`y%
&a+ ^lnly cly c`ex`gkae~ly gqay~r`e c` gkyg` kg`bae%
&m+ Ac eøgarl na ylcay ay gaelr pqa ?%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 114/846
Nr% Fëi~lr Pqa}anl Qrç`y
?'=:
&b+ Ac yabqenl c`ex`gkae~l el yle ylcay%
&f+ Ac eøgarl na ylcay ay na iq`enl gaely ?%
&k+ Ac eøgarl na ylcay ay el gày na 9%
&h+ Ac eøgarl na àbqkc`y ay ` cl gày 9%
&d+ Ac eøgarl na ylcay pqa avian`e ac eøgarl na àbqkc`y%
?%>2% ¼Na iqàe~`y g`ear`y nkmarae~ay qe` yaiikùe ykenki`c ile ?1 gkagorly zqana
acabkr qe zrayknae~a { qe }kiazrayknae~a< &=::+
?%>6% Yk qe n`nl ya c`ex` 9 }aiay ileyaiq~k}`y# ¼Iqàc ay c` zrlo`okckn`n na pqa y`cb` qe
9<
?%>;% Ya yacaiikle`e 9 i`r~`y# yqiayk}`gae~a# na qe g`xl na 1?# ae~leiay# aeile~r`r ac
eøgarl na rayqc~`nly yk8
&`+ F`{ raagzc`xl &0>:#=:6+
&o+ Yk el f`{ raagzc`xl &09?#=::+
?%1:% ¼Na iqàe~`y g`ear`y zqanae `ilgln`rya 1 i`eki`y na nkmarae~ay ilclray ae qe`
mkc`<
?%10% ¼I`ciqc`r na iqàe~`y g`ear`y zqanae mlrg`rya yaky zaryle`y z`r` yqokr ` qe
`q~loøy< &2?:+
?%1?% Qe i`enkn`~l zrayknaeik`c zc`ea` f`iar i`gz`ð` zlcç~ki`% Aeile~r`r ac eøgarl na
zargq~`ikleay yk8
&`+ Zc`ea` }kyk~`r ~lnly cly ay~`nly na c` _azøocki` Gavki`e`%
&o+ Zc`ea` }kyk~`r øeki`gae~a cly ay~`nly pqa ilcken`e ile cly Ay~`nly Qeknly%
?%19% A}`cq`r cly ykbqkae~ay m`i~lrk`cay8
&`+ 2! &1:>:+
&o+ 2:!,=6! &0::+
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 115/846
Nr% Fëi~lr Pqa}anl Qrç`y
?'=0
&i+ 0:!,:! &9#=?6#6::+
?%1>% Yqzùeb`ya pqa f`{ 1: zaryle`y ilgzk~kaenl zlr 9 r`ebly nkmarae~ay# zrkgarl#
yabqenl { ~ariarl% ¼Iqàc ay ac eøgarl na rayqc~`nly na c`y 1: zaryle`y# yk c`y ~lg`gly
9 ` qe ~kagzl &ay naikr# na 9 ae 9+<
?%11% Ae ikar~` ilgz`ðç`# > ayirk~lrkly na yaira~`rk`y ya yk~ø`e ae cçea` ile~r` c` z`ran%
I`n` yaira~`rk` zqana yae~`rya ae iq`cpqkar o`eil na cly ayirk~lrkly% ¼Iqàe~ly rrabcly
ya zqanae f`iar z`r` yae~`r ` c`y yaira~`rk`y< &?>+
?%1=% Ae qe `cg`iëe f`{ 1 i`h`y `n{`iae~ay z`r` `cg`iae`r 1 loha~ly nkmarae~ay% Ac
nazùyk~l na i`n` loha~l zqana cg`iae`rya y`~kym`i~lrk`gae~a ae qe` i`h`% ¼Na iqàe~`y
g`ear`y zqanae `ykbe`rya 1 loha~ly ` 1 i`h`y<
?%12% Yqzùeb`ya pqa f`{ = z`r~ay nkmarae~ay z`r` yar cg`iae`n`y# zarl ylc`gae~a# f`{
> i`h`y nkyzlekocay% ¼Iqàe~`y zargq~`ikleay yle zlykocay< &9=:+
?%16% ¼Na iqàe~`y g`ear`y nkmarae~ay ya zqana ra`ckx`r qe` zrkgar`# yabqen`# ~ariar`
l iq`r~` yacaiikùe ae~ra 0? agzray`y `rraen`nlr`y na apqkzl na ile~rlc na
ile~`gke`ikùe `gokae~`c<
?%1;% Ile~ay~`r cl ykbqkae~a%
&`+ ¼Iqàc ay ac eøgarl na zargq~`ikleay na c`y ca~r`y # o# i# ay naikr# ~lg`n`y nly qe
~kagzl< &=+
&o+¼Iqàcay yle ay~`y ca~r`y< &`o# o`# `i# i`# oi# io+
?%=:% Qe gai`ekygl acai~rùekil na ile~rlc rapqkara na 1 ifkzy na gaglrk` knëe~kily%
¼Na iqàe~`y g`ear`y zqana kef`okck~`rya ay~a gai`ekygl ilcli`enl cly 1 ifkzy ae c`y 1
zlykocay zlykikleay nae~rl nac ile~rlc`nlr<
?%=0% Ya rapqkara yae~`r ` 1 flgoray { > gqharay ae qe` mkc` na ~`c g`ear` pqa c`y
gqharay liqzae cqb`ray z`ray% ¼Iqàe~ly `rrabcly f`{< &?66:+
?%=?% Qe `z`r`~l na yabqrkn`n na qe eablikl ile 0: ol~leay ya kef`okck~` iq`enl 9
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 116/846
Nr% Fëi~lr Pqa}anl Qrç`y
?'=?
ol~leay nkmarae~ay ya lzrkgae ae c` yaiqaeik` `zrlzk`n` &cly ol~leay el zqanae
lzrkgkrya nly }aiay+% Yk ac iùnkbl ilrrai~l ya lc}kn`# ¼Iqàc ay c` zrlo`okckn`n na
nay`rg`r ac `z`r`~l ` ~r`}ëy na lzrkgkr# `ca`~lrk`gae~a# 9 ol~leay<
?%=9% Ya y`i`e ? olca~ly na c` cl~arç` ae~ra ?: zlykocay z`r` ac zrkgarl { yabqenl
zragkly% ¼Iqàc ay c` zrlo`okckn`n na b`e`r ilgzr`enl qe olca~l< &0,96:+
?%=>% Ae qe` i`rrar` na 6 zarrly ya hqab` qe zragkl na av`i~`% Yk yacaiikle`gly 9
eøgarly na zarrly# ¼iqàc ay c` zrlo`okckn`n na `iar~`r ilgzr`enl qe ylcl olca~l<
?%=1% Ileyknëraya qe` i`rrar` na 0: i`o`ccly ile qe zragkl na av`i~` z`r` iq`cpqkar`
pqa zqan` yacaiikle`r ac lrnae av`i~l { na b`e`r nayna ac zrkgarl f`y~` ac nëikgl
cqb`r%
&`+ ¼Iqàe~`y zargq~`ikleay zlykocay f`{< &9#=?6#6::+
&o+ ¼Iqàc ay c` zrlo`okckn`n na b`e`r yk ya ilgzr` qe ylcl olca~l< &?%2v0:'2
+
&i+ ¼Iqàc ay c` zrlo`okckn`n na yacaiikle`r cly ~ray zrkgarly cqb`ray<
&0,0:Z9+
?%==% Qe` zrqao` ya ilgzlea na 0? zrabqe~`y na m`cyl { }arn`narl% ¼Na iqàe~`y
g`ear`y nkmarae~ay qe ay~qnk`e~a zqana g`ri`r ac z`zac ile qe` rayzqay~` z`r` i`n`
zrabqe~`<
?%=2% ¼Na iqàe~`y g`ear`y zqanae 9 mlily rlhly# > mlily `g`rkccly { ? mlily `xqcay yar
`rrabc`nly ae qe ilrnùe acëi~rkil ile ; zlr~`càgz`r`y< &0?=:+
?%=6% ¼Iqàc ay ac eøgarl na zargq~`ikleay na c` z`c`or` /ay~`nçy~ki`/<
?%=;% Ikeil i`eki`y rlh`y# ? i`eki`y oc`ei`y { 9 `xqcay ya `rrabc`e ae qe` mkc`% Yk ~ln`y
c`y i`eki`y yle nac gkygl ilclr# { el ya zqana nky~kebqkr qe` na c` l~r`# ¼iqàe~ly
`rrabcly zqanae f`iarya< &?>?:+
?%2:% ¼Na iq`e~`y g`ear`y zqanae 2 ikae~çmkily yar ykbe`nly qe iq`r~l ~rkzca { nly
iq`r~ly nlocay< @yqgkr rabc` na z`r~kikùe%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 117/846
Nr% Fëi~lr Pqa}anl Qrç`y
?'=9
?%20% Na qe brqzl na > pqçgkily { 9 mçykily# aeile~r`r ac eøgarl na ilgk~ëy pqa ya
zqanae mlrg`r ileyky~ae~ay na ? pqçgkily { 0 mçykil% Yqbaraeik`8 qy`r qe zrlnqi~l na
ilgoke`ikleay% &06+
?%2?% Qe apqkzl na ilcabkl hqab` 0? hqably nqr`e~a c` ~agzlr`n`% ¼Na iqàe~`y g`ear`y
zqana ac apqkzl ~argke`r c` ~agzlr`n` ile 2 hqably b`e`nly { 9 zarnknly< Yqbaraeik`
qy`r c` rabc` na z`r~kikùe na zargq~`ikleay%
?%29% Ileyknar`r qe brqzl na 1 zaryle`y ileyky~ae~ay na 9 flgoray { ? gqharay# ~lnly
zar~aeaikae~ay ` qe` lrb`ekx`ikùe% Ykaenl `yç# ile~ay~`r cl ykbqkae~a8
&`+ ¼Iqàe~ly ilgk~ëy na 9 zaryle`y zqanae mlrg`rya na ~lnl ac brqzl< &0:+
&o+ ¼Na iqàe~`y g`ear`y zqanae c`y ? zlykikleay# zrayknae~a { }kiazrayknae~a yar
mlrg`nly< &?:+
&i+ ¼Iqàc ay c` zrlo`okckn`n na pqa qe ilgk~ë na ? zaryle`y yacaiikle`n`y#
`ca`~lrk`gae~a# ileyky~kar`e na 0 flgora { 0 gqhar< &=,0:+
?%2>% ¼Iqàe~`y g`ely na 1 i`r~`y na mclr kgzark`c &c` mclr kgzark`c ileyky~a na y`i`r 0:#
yl~`# rake`# ra{# `y na qe ylcl z`cl# ay naikr# na ~rëolcay# ilr`xleay# nk`g`e~ay {
ayz`n`y+ yle zlykocay na qe` g`xl na 1? i`r~`y# ae c`y iq`cay ac lrnae el ay na
kgzlr~`eik`<
?%21% Yk pqaragly y`oar c` zrlo`okckn`n na y`i`r qe` mclr kgzark`c na qe g`xl na 1?
i`r~`y# ` y`okaen`y na pqa ya zqanae mlrg`r > mclray kgzark`cay &0:# yl~`# rake`# ra{# `y
na i`n` qe` na c`y iq`~rl mlrg`y# ay naikr# ~rëolcay# ayz`n`y# nk`g`e~ay# ilr`xleay+
ae~leiay# i`ciqc`r ay~` zrlo`okckn`n% &0%1>v0:'=
+
?%2=% Ae c` cl~arç` na av`y ya hqab`e 1> eøgarly { ya yacaiikle`e ylc`gae~a = na accly%
&`+ ¼Iqàe~`y ilgoke`ikleay ya zqanae f`iar<
&o+ ¼Iqàc ay c` zrlo`okckn`n na `iar~`r ilgzr`enl qe ylcl olca~l<
&i+ ¼Iqàc ay c` zrlo`okckn`n na `iar~`r ilgzr`enl qe gkccùe na olca~ly<
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 118/846
Nr% Fëi~lr Pqa}anl Qrç`y
?'=>
?%22% Yqzleb`gly pqa na ~lnly cly kenk}knqly pqa ilgzr`e qe` ilgzq~`nlr` zaryle`c#
=: keicq{ae qe zrlbr`g` na zrliay`nlr na z`c`or`y ae yq ilgzr`# >: keicq{a qe
zrlbr`g` na ayz`rikgkae~l na flh`y &CL^QY+ { 9: keicq{a `goly zrlbr`g`y%
&`+ ¼Iqàc ay c` zrlo`okckn`n na yacaiikle`r qe ilgzr`nlr pqa keicq{` qe zrlbr`g` na
zrliay`nlr na z`c`or`y# n`nl pqa keicq{` qe zrlbr`g` na CL^QY< Qy`r qe nk`br`g`
na ]aee% &:%21+
&o+ ¼Iqàc ay c` zrlo`okckn`n na pqa qe ilgzr`nlr keicq{` qe zrlbr`g` na CL^QY#
n`nl pqa keicq{` qe zrlbr`g` na zrliay`nlr na z`c`or`y<
&:%1+ ?%26% Qe` ra}ky~` na zqocki`ikleay zqocki` ~ray ilcqge`y ke~k~qc`n`y @r~a &@+#
Ckorly &O+# Ikeag` &I+% C` yacaiikùe `ca`~lrk` na qe ilgzr`nlr na ra}ky~`y# ile
rayzai~l ` ay~`y ~ray ilcqge`y ya n` `o`hl &ac`olr`ikùe zrlzk`+8
Caçn`y rabqc`rgae~a s @ O I @∫ O @ ∫ I O ∫ I @ ∫ O ∫ I
Zrlo`okckn`n s %01 %?> %>2 %:; %:6 %01 %:2
I`ciqc`r { f`iar qe nk`br`g` na ]aee z`r`8
&`+ C` zrlo`okckn`n na pqa ca` c` ra}ky~` @r~a &@+# n`nl pqa ca{ù c` ra}ky~` Ckorly &O+%
&o+ C` zrlo`okckn`n na caar c` ra}ky~` @r~a &@+# n`nl pqa ca{ù c`y ra}ky~`y Ckorly &O+ {
Ikeag` &I+%
&i+ C` zrlo`okckn`n na caar c` ra}ky~` @r~a &@+# n`nl pqa f`{` caçnl iq`enl gaely qe`%
?%2;% Yqzleb`gly pqa Z&@+ 7 %1# Z&O+ 7 %># Z&@ ∫ O+ 7 %?1% F`iar cly ykbqkae~ay
iùgzq~ly { qy`r qe nk`br`g` na ]aee%
&`+ Z&Os@+ &%1+
&o+ Z&O‛s@+ &%1+
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 119/846
Nr% Fëi~lr Pqa}anl Qrç`y
?'=1
&i+ Z&@sO+ &%=?1+
&n+ Z&@‛sO+ &%921+
&a+ Z&@ ∨ O+ &%;+
?%6:% Qe` mkrg` na ileyqc~lrç` `gokae~`c zrayae~` ckik~`ikleay z`r` c` iley~rqiikùe na
~ray zrl{ai~ly na zc`e~`y na ~r`~`gkae~l na `bq`y rayknq`cay% Nahagly @ 7 zrl{ai~l k
ilemarknl z`r` k 7 0# ?# 9% Yqzùeb`ya pqa8
Z&@0+ 7 %??
Z&@?+ 7 %?1
Z&@9+ 7 %?6Z&@0 ∫ @?+ 7 %00
Z&@0 ∫ @9+ 7 %:1
Z&@? ∫ @9+ 7 %:2
Z&@0 ∫ @?∫ @9+ 7 %:0
Aeile~r`r8
&`+ @0 ∨ @?
&o+ @‛ ∫ @?
Yqbaraeik`8 qy`r @‛ ∫ @? 7 &@0 ∨ @?+‛ 7 0 ' Z&@0 ∨ @?+
&i+ @0 ∨ @? ∨ @9
&n+ @0" ∫ @? ∫ @9
Yqbaraeik`8 qy`r 0 ' Z&@0 ∫ @? ∫ @9+
?%60% Ileyknëraya qe brqzl na 1 zaryle`y ileyky~ae~ay ae 9 flgoray { ? gqharay# ~lnly
cly iq`cay zar~aeaikae~ay ` qe` lrb`ekx`ikùe% Ykaenl `yç# aeile~r`r cly ykbqkae~ay
aeqeik`nly%
&`+ ¼Iqàe~ly ilgk~ëy na 9 zaryle`y zqanae mlrg`rya< &1I9+
&o+ Naikr na iq`e~`y g`ear`y zqanae mlrg`rya c`y zlykikleay na zrayknae~a {
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 120/846
Nr% Fëi~lr Pqa}anl Qrç`y
?'==
}kiazrayknae~a% &1Z?+
&i+ Naikr c` zrlo`okckn`n na pqa qe ilgk~ë na ? zaryle`y ileyky~kràe na 0 flgora { 0
gqhar% &T9I0²?I0W,1I?+
?%6?% ¼Iqàc ay c` zrlo`okckn`n na y`i`r qe` mclr ilrrkn`# ay naikr# 1 i`r~`y na qe` ylc`
naelgke`ikùe# pqa el keicq{`e nac 0: `c `y< ]ar Mkbqr` ?%=%
?%69% Ae ac hqabl na zùpqar na 1 i`r~`y# avky~ae qe ~l~`c na 1? i`r~`y pqa }`e nayna ac ?#
9# ># 1# =# 2# 6# ;# 0:# H# P# D# @y { i`n` qe` na ay~`y i`r~`y# ~kaeae > mkbqr`y# ay naikr#
~rëolcay# nk`g`e~ay# ayz`n`y { ilr`xleay% ^lg`enl ae ileyknar`ikùe ay~l# ¼Iqàc ay c`
zrlo`okckn`n na y`i`r qe` mclr kgzark`c# ay naikr# c`y i`r~`y 0:# H# P# D# @y# na qe` na c`y
iq`~rl mkbqr`y# ay naikr# ilr`xleay# nk`g`e~ay# ~rëolcay l ayz`n`y< Z`r` ay~l# }ar Mkbqr`
?%=% &=?>,?#1;6#;=:+
?%6>% ¼Iqàc ay c` zrlo`okckn`n na y`i`r > i`r~`y na c` gkyg` ic`ya# ay naikr# qe zldar<
Ay~l ay# iq`~rl ?# iq`~rl 9# iq`~rl >#……iq`~rl H# iq`~rl D# iq`~rl @y% Z`r` ay~l }ar
Mkbqr` ?%=%
?%61% ¼Iqàc ay c` zrlo`okckn`n na y`i`r qe` i`y` ccae` &mqcc flqya+# ay naikr# qe` ~arik` {
qe z`r< &%::0>>+
?%6=% Ae ac hqabl na o`r`h`y# ¼Iqàc ay c` zrlo`okckn`n na y`i`r qe` ~arik`<
?%62% Ae ac hqabl na e`kzay# ¼Iqàc ay c` zrlo`okckn`n na y`i`r qe z`r na qe g`xl
lrnke`rkl na 1? i`r~`y< ]ar Mkbqr` ?%=% &:%>?+
?%66% Ae qe ay~qnkl na fkbkaea kenqy~rk`c { yabqrkn`n# qe yqzar}kylr na qe brqzl na ?:
~r`o`h`nlray na c` kenqy~rk` naya` y`oar c` lzkekùe na accly# &` cly pqa yacaiikle`rà
`ca`~lrk`gae~a+# ylora ikar~l rabc`gae~l na yabqrkn`n rac`ikle`nl ile agkykleay na
b`yay nae~rl na c` màorki`% Yk 0? na accly ay~àe ` m`}lr nac eqa}l rabc`gae~l { cly l~rly
6 ay~àe ae ile~r`# ¼Pqë zrlo`okckn`n f`{ na pqa nly ~r`o`h`nlray yacaiikle`nly# zlr ac
yqzar}kylr# ya g`ekmkay~ae ae ile~r` nac eqa}l rabc`gae~l na yabqrkn`n< Yqbaraeik`8
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 121/846
Nr% Fëi~lr Pqa}anl Qrç`y
?'=2
qy`r c` rabc` gqc~kzcki`~k}` z`r` a}ae~ly kenazaenkae~ay# ay naikr8 Z&@∫ O+ 7 Z&@+
Z&O+%
?%6;% Iq`~rl e`kzay na qe gle~a na 1? i`r~`y &g`xl na i`r~`y `garki`el+ ya y`i`e ae
yqiaykùe &yke raagzc`xl+% Aeile~r`r c` zrlo`okckn`n na pqa c` zrkgar` i`r~` y`i`n` ya`
qe y4 c` yabqen` i`r~` y`i`n` ya` qe 6 na nk`g`e~ay4 c` ~ariar` i`r~` y`i`n` ya` iq`enl
gaely qe` rake` { c` iq`r~` i`r~` y`i`n` ya` qe = rlhl% Yqbaraeik`8 Nah`r pqa Z&@+ ya`
c` zrlo`okckn`n na y`i`r ac y# Z&O+ ya` c` zrlo`okckn`n na y`i`r ac 6 na nk`g`e~ay# Z&I+
ya` c` zrlo`okckn`n na y`i`r gaely qe` rake` { Z&N+ ya` c` zrlo`okckn`n na y`i`r ac =
rlhl% Qy`r nk`br`g`y na ]aee z`r` nael~`r c`y zrlo`okckn`nay { c`y ke~aryaiikleay na c`y
iq`~rl i`r~`y y`i`n`y% _amarkrya ` c` Mkbqr` ?%= na `o`hl%
Mkbqr` ?%=% Nk`br`g` aypqagà~kil gly~r`enl c`y 1? i`r~`y na hqabl na o`r`h`y% Ac
gle~a na i`r~`y agzkax` ile ac ? f`y~` ac 0:# ae i`n` qe` na yqy naelgke`ikleay {
~argke` ile iq`~rl i`r~`y `nkikle`cay# pqa yle c`y yl~`y# c`y rake`y# cly ra{ay { cly `yay%
@pqç# eù~aya pqa# c`y mkbqr`y na nk`g`e~ay { na ilr`xleay yle ykagzra nac ilclr rlhl {
c`y mkbqr`y na ~rëolcay { na ayz`n`y yle ykagzra na ilclr eabrl%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 122/846
Nr% Fëi~lr Pqa}anl Qrç`y
?'=6
Mqae~a8 f~~z8,,ae%kdkzank`%lrb,kdk,Kg`ba8Ya~\lm\zc`{keb\i`rny\1?%HZB
Mkbqr` ?%2% Nk`br`g` aypqagà~kil na c`y 1? i`r~`y kcqy~r`enl c` zrlo`okckn`n na y`i`r
qe @y l qe _a{% Mqae~a8 C`raeia C% C`zke% Y~`~ky~kiy mlr Glnare Oqykeayy Naikykle%
&0;6?+%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 123/846
Nr% Fëi~lr Pqa}anl Qrç`y
9'0
I@ZK^QCL 9
Nky~rkoqikleay Okelgk`c a fkzarbalgë~rki`
@zcki`ikleay baear`cay na c` nky~rkoqikùe okelgk`c%' _ac`ikùe ae~ra c`
nky~rkoqikùe elrg`c { c` nky~rkoqikùe okelgk`c%' _ac`ikùe ae~ra c` nky~rkoqikùe
okelgk`c { c` nky~rkoqikùe na Zlkyyle%' C` nky~rkoqikùe fkzarbalgë~rki`%'
Yqzlykikleay { zrlzkan`nay na c` nky~rkoqikùe fkzarbalgë~rki`%'
C` nky~rkoqikùe okelgk`c ay qe` na c`y nky~rkoqikleay na zrlo`okckn`n nkyira~`y gày
qy`n`y ae ay~`nçy~ki`% Ya zqana ileyknar`r ilgl qe ~kzl na `eàckyky na cùbki`
nanqi~k}`# zlrpqa }` nac ilehqe~l l ~l~`c ` c` z`r~a% Ya nark}` na qe zrliayl
ileliknl ilgl aey`{ly na Oarelqcck% Qe zrliayl Oarelqcck ay qe aey`{l na `cbøe
zrliayl l avzarkgae~l pqa zqana rayqc~`r ae# ylc`gae~a# qel na nly rayqc~`nly
gq~qly avicq{ae~ay# ay naikr# oke`rkly# ilgl zlr ahagzcl# –ëvk~l‗ { –mr`i`yl‗#
nlena c` zrlo`okckn`n na ëvk~l ya nael~` ilgl z% ^`gokëe ac avzarkgae~l
okelgk`c ya zqana ke~arzra~`r ilgl qe` yk~q`ikùe namai~qly` l el namai~qly`#
ilrrai~` l keilrrai~`# zrayae~a l `qyae~a# e`ikgkae~ly na ekðly l ekð`y# i`r`y l
àbqkc`y na qe` glean`# a~i% Na ay~` g`ear`# cly n`~ly na qe zrliayl okelgk`c
&oke`rkl+ ileyky~a# øeki`gae~a# na nly yk~q`ikleay l rayqc~`nly%
@zcki`ikleay baear`cay na c` nky~rkoqikùe okelgk`c
Qe` na c`y àra`y zrkeikz`cay na `zcki`ikùe na c` nky~rkoqikùe okelgk`c ay ae cly
i`gzly na c` kebaekarç` kenqy~rk`c# ay naikr# ae zrliayly kenqy~rk`cay# nlena ac
rayqc~`nl na qe zrliayl ay nkiù~lgl &zrlzlrikleay na qe loha~l namai~qlyl l el
namai~qlyl# na ëvk~l l mr`i`yl# a~i%+ ^`gokëe ya qy` ae `zcki`ikleay gënki`y &iqr`r
l el iqr`r+ { ae `zcki`ikleay gkck~`ray &zab`r l el zab`r na qe gçykc+% Kbq`cgae~a#
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 124/846
Nr% Fëi~lr Pqa}anl Qrç`y
9'?
ya qy` z`r` nael~`r ac eøgarl na farr`gkae~`y namai~qly`y zrlnqikn`y zlr qe`
gàpqke`# a~i%
C` nky~rkoqikùe okelgk`c ~`gokëe ya zqana `zcki`r ` c` kebaekarç` `gokae~`c%
Ilgl ya nkhl `e~ay# cly n`~ly nac zrliayl okelgk`c ileyky~ae na nly rayqc~`nly
nkyira~ly &oke`rkly+% Zlr ahagzcl# ae qe oklaey`{l# qe lrb`ekygl na zrqao` ay~à#
{` ya` }k}l l gqar~l# ay naikr# nayzqëy na yar avzqay~l ` c` ileiae~r`ikùe na `cbøe
naykemai~`e~a# ae mqeikùe na c` ileiae~r`ikùe { nac ~kagzl na avzlykikùe%
Kbq`cgae~a# ae ac i`yl na qe` nayi`rb` na `bq`y rayknq`cay nlgëy~ki`y l
kenqy~rk`cay# ëy~` zqana l el zqan` ay~`r nae~rl na cly cçgk~ay ay~kzqc`nly zlr c`y
ca{ay `gokae~`cay% @eàclbgae~a# ya zqana `zcki`r ` c` kebaekarç` `gokae~`c ae c`
pqa qe` kenqy~rk` iqgzca l el iqgzca ile c`y rabqc`ikleay `gokae~`cay nac `kra#
nac `bq`# na rqknl# na ile~`gke`ikùe na ~karr`# a~i% ^`gokëe ya zqana `zcki`r ` c`
kebaekarç` ik}kc ae ac àra` na iley~rqiikùe# a~i%
Namkekikùe8 Qe aey`{l Oarelqcck ay qe avzarkgae~l `ca`~lrkl pqa ~kaea ylcl nly
rayqc~`nly zlykocay# ay naikr# ëvk~l l mr`i`yl# nlena c` zrlo`okckn`n na ëvk~l ya
nael~` zlr z { ac mr`i`yl ya nael~` zlr p 7 0 ' z% Ac avzarkgae~l ileyky~a na e
aey`{ly raza~knly nlena cly aey`{ly yle kenazaenkae~ay% Na ay~` g`ear`# yk z ay c`
zrlo`okckn`n na pqa qe a}ae~l liqrrkrà# ae qe ylcl aey`{l &cc`g`nl `rok~r`rk`gae~a
ëvk~l+ {# c` rac`ikùe p ay c` zrlo`okckn`n na pqa ac a}ae~l m`cc`rà ae iq`cpqkar
aey`{l# ae~leiay# c` nky~rkoqikùe na zrlo`okckn`n na c` }`rk`oca `ca`~lrk` okelgk`c
V # ay kbq`c `c eøgarl na aey`{ly% Ay naikr# nlena ac rayqc~`nl ay qe ëvk~l ile
z`ràga~rly z { e 7 0# ?# 9# …# e# ay~l ay8
Z& V + 7 o&v4e#z+ 7 eIv zv
pe'v
7 e!,v!&e ’ v+! zv
&0 ’ z+e'v
&9'0+
Nlena8
e 7 yacaiikùe nac ~`g`ðl na c` gqay~r` ileyknar`n` ilgl aey`{ly kenazaenkae~ay
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 125/846
Nr% Fëi~lr Pqa}anl Qrç`y
9'9
raza~knly na Oarelqcck &kenazaenkae~ay zlrpqa el kgzlr~` iq`e~`y }aiay ya razk~` ac
avzarkgae~l c`y zrlo`okckn`nay na ëvk~l l mr`i`yl zarg`eaiae iley~`e~ay+%
v 7 :# 0# ?# 9#…%# e l ya` ac eøgarl av`i~l na ëvk~ly zlykocay ae e aey`{ly
z 7 zrlo`okckn`n na ëvk~l
eIv 7 e!,v!&e ’ v+! 7 ilamkikae~a okelgk`c ` ya` ac eøgarl na ilgoke`ikleay na e
loha~ly ~lg`nly ` qe ~kagzl r
p 7 0 ’ z 7 zrlo`okckn`n na mr`i`yl
Ay na }arya pqë# c` zrlo`okckn`n na el ëvk~l &l mr`i`yl+ ay pe# zlr cl ~`e~l# c`
zrlo`okckn`n na iq`enl gaely qe ëvk~l ay 0 ' pe%
C` nky~rkoqikùe na zrlo`okckn`n nkyira~` Oarelqcck ` }aiay ya ca cc`g`
nky~rkoqikùe okelgk`c zlrpqa cly }`clray na c` }`rk`oca `ca`~lrk` V zqanae yar v 7
:# 0# ?# 9#…%# e pqa ilrrayzlenae ` ~ërgkely yqiayk}ly na c` mùrgqc` okelgk`c l
avz`eykùe okelgk`c% Ay~l pqkara naikr pqa# c` nky~rkoqikùe okelgk`c nark}` yq
elgora nac faifl na pqa cly ~ërgkely e $ 0 ae c` avz`eykùe okelgk`c na c` mqeikùe
&p $ z+e ilrrayzlena ` }`rkly }`clray na o&v4e#z+# z`r` v 7 :# 0# ?# 9#…%# e% @yç# c`
avz`eykùe okelgk`c ay8
&p $ z+e
7 pe
$ eI0 pe'0
z $ eI? pe'?
z?
$ %% $ ze &9'?+
Nlena8
eI0# eI?#… ya cc`g`e cly ilamkikae~ay okelgk`cay
Cly ilamkikae~ay okelgk`cay ya zqanae ay~kg`r qy`enl ac ~rkàebqcl na Z`yi`c pqa ya
n` `o`hl%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 126/846
Nr% Fëi~lr Pqa}anl Qrç`y
9'>
Mkbqr` 9%:% ^rkàebqcl na Z`yi`c pqa ya qy` z`r` ay~kg`r cly ilamkikae~ay okelgk`cay%
Ae ay~a ~rkàebqcl ya el~` pqa# ac zrkgarl { ac øc~kgl eøgarl na i`n` raebcùe ay 0%
@nagày# i`n` l~rl eøgarl ae i`n` lrnae`ikùe zqana lo~aearya zlr gankl na
yqg`r cly nly eøgarly pqa `z`raiae nkrai~`gae~a `rrko`% &Ac`olr`ikùe zrlzk`+
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 127/846
Nr% Fëi~lr Pqa}anl Qrç`y
9'1
Mkbqr` 9%0% Bràmki`y gly~r`enl }`rk`y nky~rkoqikleay okelgk`cay ae mqeikùe na z {
na e% C` nky~rkoqikùe okelgk`c ay ra`cgae~a qe` m`gkck` na nky~rkoqikleay% I`n`
}`clr nkmarae~a na e l na z ayzaikmki` qe` nky~rkoqikùe nkmarae~a% C`y mkbqr`y na
`rrko` gqay~r`e# ilgl c` nky~rkoqikùe okelgk`c }`rç` z`r` nkmarae~ay }`clray na z {
na e &nlena z ay c` zrlo`okckn`n na ëvk~l { p ay c` zrlo`okckn`n na mr`i`yl {# nlena
ae e raza~kikleay na qe aey`{l na Oarelqcck# ac eøgarl na ëvk~ly zlykocay ay :# 0#
?#…# e+% Yke ago`rbl# yke kgzlr~`r ac }`clr na e# c` nky~rkoqikùe okelgk`c ayykgë~rki` iq`enl z 7 :%1% Zarl# iq`enl z 5 :%1# c` nky~rkoqikùe ay `ykgë~rki` { ac
zkil liqrra ` c` naraif` nac iae~rl% ^`gokëe# iq`enl z 3 :%1 c` nky~rkoqikùe ay
`ykgë~rki` { ac zkil liqrra ` c` kxpqkarn` nac iae~rl% &Ac`olr`ikùe zrlzk`+
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 128/846
Nr% Fëi~lr Pqa}anl Qrç`y
9'=
^@OC@ 9%:% ^`oc` gly~r`enl `cbqe`y zrlzkan`nay na c` nky~rkoqikùe Okelgk`c%
&Ac`olr`ikùe zrlzk`+
Zrlgankl ± 7 ez
δ 7 ez
]`rk`ex` ω?7 ezp
Nay}k`ikùe ay~àen`r ezp 7 ω
_ac`ikùe ae~ra c` nky~rkoqikùe okelgk`c { c` nky~rkoqikùe elrg`c
C` nky~rkoqikùe okelgk`c ya zqana `zrlvkg`r zlr c` nky~rkoqikùe elrg`c iq`enl eay br`ena {# iq`enl ek z ek ~`gzlil p ay~àe gq{ iari`e`y ` iarl% Ay~l ya naoa `
pqa# ac glnacl okelgk`c ay ke`zrlzk`nl iq`enl e ay av~rag`n`gae~a br`ena%
@mlr~qe`n`gae~a# c` `zrlvkg`ikùe elrg`c ay gày amki`x ` gankn` pqa e `qgae~`%
Ae c` zrài~ki`# c` `zrlvkg`ikùe na c` nky~rkoqikùe okelgk`c qy`enl c` nky~rkoqikùe
elrg`c ay `naiq`n` ykagzra { iq`enl ez ≡ 0: { ep ≡ 0:% Ae~leiay# yk ez 3 0: l
ep 3 0:# c` nky~rkoqikùe okelgk`c ay~à nag`yk`nl yayb`n`# z`r` n`r `zrlvkg`ikleay
y`~kym`i~lrk`y# ilgl ile c` iqr}` elrg`c pqa ay ykgë~rki`% Z`r` f`iar c`y
`zrlvkg`ikleay na c` okelgk`c qy`enl c` nky~rkoqikùe elrg`c ay ile c` }`rk`oca
`ca`~lrk` ay~`en`rkx`n` n`n` `o`hl%
X 7 &V ’ez+ , ezp ay naikr X 7 &V ’ ±+ , ω &9'9+
Nlena8 ez 7 ± { ezp 7 ω?l ya` ω 7 ezp
_ac`ikùe ae~ra c` nky~rkoqikùe okelgk`c { c` nky~rkoqikùe na Zlkyyle
Ile c` nky~rkoqikùe okelgk`c# yk e ay br`ena &e ≡ 1: aey`{ly na Oarelqcck+ { yk ac
zrlgankl ± 7 ez 3 1 & z iari`e` ` iarl { p iari`e` ` 0+ ae yagah`e~ay i`yly# c`
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 129/846
Nr% Fëi~lr Pqa}anl Qrç`y
9'2
nky~rkoqikùe okelgk`c ay~à gq{ iari`e` ` c` nky~rkoqikùe na Zlkyyle%
Ahagzcly na zrlocag`y rac`ikle`nly ile c` nky~rkoqikùe okelgk`c l
nky~rkoqikùe Oarelqcck
Ahagzcl )0% I`ciqc`r c`y ykbqkae~ay zrlo`okckn`nay okelgk`cay nkrai~`gae~a na c`
mùrgqc`# z`r` o&v4e#z+
&`+ O&946#%=+(
&o+ O&146#%=+
&i+ Z&9 ≮ V ≮ 1+ iq`enl e 7 6 { z 7 %=
&n+ Z&0 ≮ V + iq`enl e 7 0? { z 7 %0
&a+ o&v46#:%=+( nlena v 7 :
(Eù~aya c` nkmaraeik` ae~ra ac qyl na c` ca~r` g`{øyiqc` O { c` gkeøyiqc` o
Ylcqikùe8
&`+ O&946#:%=+ nkia pqa pqaragly V 7 9# e 7 6# z 7 %=
Z& V 7 9+ 7 6!,9!&6 ’ 9+! &:%=+9
&0 ’ :%=+6'9
7 :%0?>
@eàclb`gae~a# qy`enl c` ~`oc` okelgk`c na zrlo`okckn`nay kenk}knq`cay8 O&946#:%=+
7 :%0?>
Kbq`cgae~a# qy`enl c` mùrgqc` n`8
eIv zv
pe'v
7 6I9 &:%=+9
&:%>+6'9
7 &1=+&:%?0=+&:%:0+
7 :%0?>
&i+ Z&9 ≮ V ≮ 1+ 7 O&146#:%=+ ’ O&946#:%=+
7 :%?2; ’ :%0?>
7 :%011
Nlena8
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 130/846
Nr% Fëi~lr Pqa}anl Qrç`y
9'6
O&146#:%=+ 7 :%?2; &qy`enl c` ~`oc` okelgk`c na zrlo`okckn`nay kenk}knq`cay+
7 eIv zv
pe'v
7 6I1 &:%=+1
&:%>+6'1
7 &1=+&:%:26+&:%:=>+
7 :%?2; &qy`enl c` mùrgqc`+
&n+ Z& V ≡ 0+ ile e 7 0? { z 7 :%0% Ay~l nkia pqa pqaragly8
Z& V ≡ 0+ 7 0 ’ Z& V 3 0+ 7 0 ’ Z& V 7 :+
7 0 ’ :%::0
7 :%;;; &qy`enl c` ~`oc` na zrlo`okckn`nay kenk}knq`cay+
Z& V ≡ 0+ 7 0 ’ Z& V 7 :+ 7 0 ' eIv zv
pe'v
7 0 ' 6I: &:%=+:
&:%>+6':
7 0 ' &0+&0+&:%:::2+
7 :%;;;9 &qy`enl c` mùrgqc`+
Ahagzcl )?% F`iar cly gkygly zrlocag`y nac keikyl )0 zarl `flr` qy`enl c` ~`oc`
na c` nky~rkoqikùe okelgk`c% Ilgz`r`r cly rayqc~`nly% Ac cai~lr cl naoarà f`iar%
Ahagzcl )9% Qy`enl c` ~`oc` na c` nky~rkoqikùe okelgk`c ay~kg`r8
&`+ O&>40:#:%9+
&o+ O&=40:#:%2+
Ylcqikùe8
&`+ O&>40:#:%9+ nkia pqa qy`gly o&v4e#z+# nlena v 7 ># e 7 0: { z 7 :%9%
Ae~leiay# Z& V 7 >+ 7 O&>40:#:%9+% Z`r` ay~l# oqyi`gly ae c` ~`oc` na c` nky~rkoqikùe
okelgk`c na zrlo`okckn`nay kenk}knq`cay ac }`clr na e 7 0:#ϊ 7 z 7 %9:: { v 7 ` 7 >
{ ely n` :%?::%
Zlr cl ~`e~l# Z& V 7 >+ 7 O&>40:#:%9+ 7 :%?::
Eù~aya pqa `pqç ~`gokëe ya zqana qy`r c` mùrgqc` okelgk`c &9'0+# ay naikr#
Z& V + 7 o&v4e#z+ 7 eIv zv
pe'v
7 e!,v!&e ’ v+! zv
&0 ’ z+e'v
{ n` ac gkygl rayqc~`nl%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 131/846
Nr% Fëi~lr Pqa}anl Qrç`y
9';
&o+ Z`r` ay~a keikyl ya zrliana ae mlrg` ykgkc`r%
Ahagzcl )>% Qe` glean` fleay~` ya c`ex` = }aiay &pqa ay cl gkygl pqa c`ex`r
yaky glean`y ` c` }ax+% Cc`gagly c`y i`r`y qe ëvk~l% I`ciqc`r c`y ykbqkae~ay
zrlo`okckn`nay8
&`+ C` zrlo`okckn`n na pqa y`cb`e av`i~`gae~a ? i`r`y
&o+ C` zrlo`okckn`n na pqa y`cb`e iq`enl gaely > i`r`y
&i+ C` zrlo`okckn`n na el i`r`y# ay naikr# ~lnly mr`i`yly
Ylcqikùe8
&`+ @pqç qy`gly c` mùrgqc` na c` nky~rkoqikùe okelgk`c8
Z& V + 7 o&v4e#z+ 7 eIv zv
pe'v
Nlena8
eIv 7 ilamkikae~a okelgk`c 7 e! , v!&e ' v+!
e 7 eøgarl na aey`{ly
z 7 zrlo`okckn`n na pqa ac a}ae~l liqrr` ae qe ylcl aey`{l
p 7 0 ’ z 7 zrlo`okckn`n na pqa ac a}ae~l m`cca &mr`i`yl+
v 7 c` zrlo`okckn`n na pqa ac a}ae~l liqrr` ae :# 0# ?# …# e
eøgarl na ëvk~ly zlykocay
Eù~aya pqa c` zrlo`okckn`n na el ëvk~ly ay pe# zlr cl ~`e~l# c` zrlo`okckn`n na
iq`enl gaely qe ëvk~l ay 0 ' pe
@pqç e 7 =# z 7 :%1# p 7 0 ' :%1 7 :%1%
Ae~leiay# c` zrlo`okckn`n na pqa y`cb`e av`i~`gae~a ? i`r`y ay8
Z& V 7 ?+ 7 O&?4=#:%1+ 7 =I? &:%1+?
&:%1+='?
7 01,=>
&o+ C` zrlo`okckn`n na pqa y`cb`e iq`enl gaely > i`r`y & V ≡ >+ ay8
Z& V 7 > l 1 l =+ 7 O&>4=#:%1+ $ O&14=#:%1+ $ O&=4=#:%1+
7 =I>&:%1+>&:%1+
='>$ =I1 &:%1+
1&:%1+
='1$ =I= &:%1+
=&:%1+
='=
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 132/846
Nr% Fëi~lr Pqa}anl Qrç`y
9'0:
7 00,9?
Ahagzcl )1% Ae qe ay~qnkl na ~lvkilclbç`# c` zrlo`okckn`n na pqa qe aemargl ya
raiqzara na qe` ke~lvki`ikùe ay na :%>% Yk ya y`oa pqa qe` gqay~r` na 01 zaryle`y
ya f` ke~lvki`nl# i`ciqc`r c`y ykbqkae~ay zrlo`okckn`nay8
&`+ C` zrlo`okckn`n na iq`enl gaely 0: zaryle`y ylora}k}`e%
&o+ C` zrlo`okckn`n na pqa na 9 ` 6 zaryle`y &keicqyk}`gae~a+ ke~lvki`n`y
ylora}k}`e%
&i+ C` zrlo`okckn`n na pqa av`i~`gae~a 1 zaryle`y ke~lvki`n`y ylora}k}`e%
Ylcqikùe8
&`+ Nahagly pqa V ya` ac eøgarl na ke~lvki`nly pqa ylora}k}`e% @pqç# ac ~ërgkel
–iq`enl gaely 0:‗ ykbekmki` pqa ac }`clr na c` }`rk`oca `ca`~lrk` ay V ≡ 0:%
^`gokëe y`oagly pqa c` gqay~r` ay e 7 01% @pqç# zqnkër`gly qy`r c avzraykùe
okelgk`c o&v4e#z+ 7 eIv zv
pe'v
{ yqy~k~qkr cly }`clray na v 7 0:# 00# 0?# 09# 0># 01
ae c` mùrgqc` na `o`hl# { cqabl yqg`r ~lnly cly rayqc~`nly qy`enl c` avzraykùe na
`o`hl%
o&v401#:%>+ 7 01Iv &:%>+v &:%=+01'v
Yke ago`rbl# ay~a zrliankgkae~l yarç` gq{ c`rbl { ~anklyl% Ykaenl `yç# ay~l ya
ykgzckmki` gqifl yk ~lg`gly ac ilgzcagae~l na c` zrlo`okckn`n na 0
&`ilrnàenlya na pqa c` zrlo`okckn`n el zqana yar g`{lr pqa 0 l eab`~k}`+ {
qy`enl c` ~`oc` na c` nky~rkoqikùe okelgk`c%
Z& V ≡ 0:+ 7 0 ’ Z& V 3 0:+% Ay~l nkia pqa v 7 :# 0# ?# 9#…%%# ;
;
7 0 ' ∐ o&v401#:%>+ 7 0 ’ :%;==?v7:
7 :%:996 &qy`enl c` ~`oc` na c` nky~rkoqikùe okelgk`c+
Ac }`clr na :%;==? ya y`i` na c` ~`oc` okelgk`c# oqyi`enl ac }`clr na e 7 01# v 7 ;
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 133/846
Nr% Fëi~lr Pqa}anl Qrç`y
9'00
{ z 7 :%>% Ay~l ya caa ilgl :%;==?%
&o+ Ay~a zrlocag` nkia pqa# c` zrlo`okckn`n na pqa ya raiqzarae ae~ra 9 { 6
ke~lvki`nly# keicqyk}`gae~a# ay cl gkygl pqa naikr# Z&9 ≮ V ≮ 6+% Ay~l pqkara
naikr pqa cly }`clray na c` }`rk`oca `ca`~lrk` yle v 7 9# ># 1# =# 2# 6% Eqa}`gae~a# yk
el qy`gly c` ~`oc` okelgk`c# ac zrliankgkae~l ay gq{ c`rbl% Zlr ay~l }`gly `
r`xle`r ilgl ykbqa8
6 ?
Z&9 ≮ V ≮ 6+ 7 ∐ o&v401#:%>+ ' ∐ o&v401#:%>+v7: v7:
7 Z& V ≮ 6+ ’ Z& V ≮ ?+
7 :%;:1: ’ :%:?20
7 :%622; &qy`enl c` ~`oc` na c` nky~rkoqikùe okelgk`c+
&i+ C` zrlo`okckn`n na pqa av`i~`gae~a 1 ke~lvki`nly ylora}k}`e ay na v 7 1# e 7
01# z 7 :%>% Ay~l ya zqana f`iar na ~ray g`ear`y8 qy`enl c` ~`oc` na c`y
zrlo`okckn`nay kenk}knq`cay &c` mlrg` gày yaeikcc`+ l c` ~`oc` `iqgqc`n` l# okae# c`
mùrgqc`% Qy`enl c` ~`oc` okelgk`c kenk}knq`c# oqyi`gly ac }`clr na e 7 01 ile z 7
:%> { ile v 7 1 { n` :%06=%
1 >
Z& V 7 1+ 7 O&1401#:%>+ 7 ∐ o& v401#:%>+ ' ∐ o& v401#:%>+v7: v7:
7 :%>:9? ’ :%?029
7 :%061;
Yk qy`gly c` mùrgqc` yarç` c`rbl { ~anklyl# ilgl ya }a `o`hl%
Z& V 7 1+ 7 O&1401#:%>+ 7 01I1 &:%>+1
&:%=+01'1
7 01!,1!&01'1+! &:%::>0+&:%=+0:
7 :%061;
Ahagzcl )=% Yk ac ?: na cly ~lrekccly zrlnqiknly zlr qe` gàpqke` yle
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 134/846
Nr% Fëi~lr Pqa}anl Qrç`y
9'0?
namai~qlyly# na~argke`r c` zrlo`okckn`n pqa na > ~lrekccly yacaiikle`nly
`ca`~lrk`gae~a8
&`+ Qel ay~`rà namai~qlyl
&o+ Ekebqel ay~`rà namai~qlyl
&i+ @ cl gày ? ay~`ràe namai~qlyly
&n+ Iq`enl gaely qel ay~`rà namai~qlyl
Ylcqikùe8
&`+ @pqç# v 7 0# e 7 ># z 7 :%?:# p 7 :%6:
Z& V 7 0+ 7 >I0 &:%?+0
&:%6+>'0
7 :%>:;=
&o+ Z& V 7 :+ 7 >I: &:%?+:
&:%6+>':
7 :%>:;=
&i+ @pqç# ac ~ërgkel –` cl gày ?‗ ykbekmki` V ≮ ?# cl iq`c pqkara naikr pqa
pqaragly aeile~r`r Z&V 7 : l 0 l ?+% Ae~leiay8
Z& V ≮ ?+ 7 Z&:+ $ Z&0+ $ Z&?+
7 :%>:;= $ :%>:;= $ :%019= &na c` ~`oc` na zrlo`okckn`nay okelgk`cay
kenk}knq`cay+
7 :%;2?6
@pqç# ~`gokëe ya zqana qy`r Z& V ≮ ?+ 7 %;2> &na c` ~`oc` `iqgqc`n`+
&n+ Ac ~ërgkel –iq`enl gaely 0‗ ykbekmki V ≡ 0# cl iq`c pqkara naikr pqa v 7 0# ?#
9# >% Ae~leiay pqaragly i`ciqc`r Z&9+ { Z&>+ zlrpqa {` i`ciqc`gly Z&0+ { Z&?+%
L~rl r`xle`gkae~l yarç` ac na i`ciqc`r c` zrlo`okckn`n na pqa V 7 ># gaely c`
zrlo`okckn`n na V 7 :% Z`r` ay~l# qy`gly c` ~`oc` okelgk`c `iqgqc`n` oqyi`enl e
7 ># z 7 :%? { V 7 > { ca ray~`gly e 7 ># z 7 :%? { V 7 :% Ay naikr8
Z& V 7 >+ ’ Z& V 7 :+ 7 Z& V ≮ >+ ’ Z& V ≮ :+
7 0 ’ :%>0
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 135/846
Nr% Fëi~lr Pqa}anl Qrç`y
9'09
7 :%1;
Ahagzcly `zcki`nly ` c` kebaekarç` `gokae~`c
Ahagzcl )2% Yqzùeb`ya pqa ac >: na cly rçly na ikar~` rabkùe kenqy~rk`c na
Gëvkil ay~àe ile~`gke`nly ile oaeiael% Yk ~lg`gly qe` gqay~r` `ca`~lrk` na
~`g`ðl e 7 9:# i`ciqc`r cl ykbqkae~a8
&`+ Av`i~`gae~a 01 rçly ay~`ràe ile~`gke`nly ile oaeiael
&o+ Iq`enl gaely 01 rçly ay~`ràe ile~`gke`nly ile ay~a ilgzqay~l
lrbàekil i`eiarçbael# na qe` gqay~r` na e 7 ?1%
&i+ El gày na 0: rçly# zarl iq`enl gaely na 1 rçly ay~`ràe ile~`gke`nly na qe`
gqay~r` `ca`~lrk` na e 7 ?1%
Ylcqikùe8
Qy`gly c` nky~rkoqikùe okelgk`c# zlrpqa yle nly a}ae~ly gq~qly avicq{ae~ay l
oke`rkly# ay naikr# ay~àe l el ay~àe ile~`gke`nly cly rçly% Ae~leiay# cc`gagly
`rok~r`rk`gae~a# qe ëvk~l aeile~r`r qe rçl ile~`gke`nl {# qe mr`i`yl# el aeile~r`r
qe rçl ile~`gke`nl% Ya qy` c` mùrgqc` okelgk`c avzray`n` ilgl8
o&v4e#z+ 7 eIv zv &0 ’ z+e'v 7 e! , &e ’ v+! zv pe'v
&`+ @pqç# e 7 9:# v 7 01# z 7 :%>:# p 7 :%=:% C` gqay~r` na 9: ya zqana ke~arzra~`r
ilgl 9: aey`{ly raza~knly na Oarelqcck% @flr`# yqy~k~q{aenl cly }`clray ae c`
mùrgqc` na `rrko` n`8
O&0149:#:%>:+ 7 Z& V 7 01+
7 9:! , &9: ’ 01+! &:%>+01
&:%=+9:'01
7 :%:29
^`gokëe ya zqnkar` qy`r c` ~`oc` na c` nky~rkoqikùe okelgk`c na naeykn`n na
zrlo`okckn`n l na zrlo`okckn`nay kenk}knq`cay# pqa yle gày zraiky`y { gày màikcay
na qy`r pqa c` mùrgqc`% Ykaenl `yç# ile e 7 9: { z 7 :%>8
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 136/846
Nr% Fëi~lr Pqa}anl Qrç`y
9'0>
Z& V 7 01+7 :%:269 ≆ :%:6
Ac }`clr na :%:26 ≆ :%:6 nkia pqa f`{ iari` na 6 zlykokckn`nay ae~ra 0:: na
yacaiikle`r qe` gqay~r` na 9: rçly pqa ay~ëe ile~`gke`nly ile oaeiael% @pqç ya
}a pqa# ` gankn` pqa `qgae~` e# c` zrlo`okckn`n na ëvk~l ~`gokëe `qgae~`%
&o+ Iq`enl gaely 01 kenki V ≡ 01 { e 7 ?1# z 7 :%> { p 7 :%=:
@pqç ac ayz`ikl gqay~r`c ay na8
v 7 :# 0# ?# 9# ># 1# =# 2# 6# ;# 0:# 00# 0?# 09# 0># 01# 0=# ……# ?># ay naikr# na ?1
aey`{ly na Oarelqcck%
O&014?1#:%>+ 7 Z& V ≡ 01+ 7 0 ’ Z& V 3 01+ 7 0 ’ Z& V ≮ 0>+
0>
7 0 ' ∐ O&0>4?1#:%>+ 7 0 ' %;== 7 :%:9>v7:
&i+ @pqç# Z&1 ≮ V ≮ 0:+ 7 Z& V ≮ 0:+ ’ Z& V ≮ >+
Ahagzcl )6% Ae qe ay~qnkl na c`olr`~lrkl o`i~arklcùbkil na `bq`y ya `mkrg` pqa# ac
9%: na c`y ~lg`y nlgkikck`rk`y ile~kaeae c` o`i~ark` A% Ilck# ae ileiae~r`ikleay
`rrko` nac cçgk~a ay~kzqc`nl zlr c`y ca{ay `gokae~`cay% Yk ay~` `mkrg`ikùe ay
ilrrai~`# aeile~r`r c` zrlo`okckn`n na pqa# ac eøgarl na o`i~ark`y A% ilck# ae qe`
gqay~r` `ca`~lrk` na ?1 ~lg`y nlgkikck`rk`y# ya aeile~r`rà8
&`+ Ekebqe` o`i~ark`
&o+ Iq`enl gaely 0 o`i~ark`
&i+ Ae~ra 0 { 1 keicqyl
&n+ Gày na 1 o`i~ark`y
&a+ Gày na 1# zarl gaely na 0: o`i~ark`y
Ylcqikùe8
Qy`gly c` nky~rkoqikùe okelgk`c# zlrpqa yle nly a}ae~ly gq~qly avicq{ae~ay l
oke`rkly4 ya ile~kaea c` o`i~ark` &cc`gàenlc` ëvk~l `rok~r`rk`gae~a+ l el ya ile~kaea
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 137/846
Nr% Fëi~lr Pqa}anl Qrç`y
9'01
c` o`i~ark` &mr`i`yl+%
&`+ @pqç# e 7 ?1# z 7 %:9# p 7 %;2# V 7 :
Ac ~`g`ðl na gqay~r` e 7 ?1 kenki` pqa yle ?1 aey`{ly raza~knly na Oarelqcck# ay
naikr# pqa cly zlykocay }`clray na c` }`rk`oca `ca`~lrk` V yle na v 7 :# 0# ?# 9#
>#……%# ?>% Ae~leiay#
&o+ Iq`enl gaely 0 o`i~ark` kenki V ≡ 0 { ya avzray` ilgl8
Z& V ≡ 0+ 7 0 ’ Z& V 7 :+ 7 0 ’ Z& V 3 0+
7 0 ’ :%>=2:
7 :%199
&i+ Ae~ra 0 { 1 keicqyl ya avzray` ilgl8
1
Z&0 ≮ V ≮ 1+ 7 ∐ O&14?1#:%:9+ ’ Z& V 3 0+ 7 :%;;;; ’ :%>=2 7 %199v7:
Ahagzcl );% Ae qe rçl `n{`iae~a ` qe` xle` kenqy~rk`c# c` zrlo`okckn`n na i`n`
gqay~r` na `bq` y`i`n` nac rçl avian` ac cçgk~a na irlgl na 0: gb,C# ay na :%0:%
Yk ya yqzlea pqë# c`y gqay~r`y na `bq` yle kenazaenkae~ay ile rayzai~l ` c`
zrayaeik` na irlgl# ae~leiay8
&`+ Aeile~r`r c` zrlo`okckn`n na pqa ae qe` gqay~r` na ~`g`ðl e 7
06# av`i~`gae~a# ? avian`e ac cçgk~a na 0: gb,C na irlgl%
&o+ Aeile~r`r c` zrlo`okckn`n na pqa `c gaely > gqay~r`y avian`e ac cçgk~a%
&i+ Aeile~r`r c` zrlo`okckn`n na pqa iq`enl gaely 9 gqay~r`y# zarl gaely na 2
avian`e ac cçgk~a ay~kzqc`nl%
&n+ Aeile~r`r c` zrlo`okckn`n na pqa gày na 9 gqay~r`y# zarl gaely na 2 avian`eac cçgk~a ay~kzqc`nl na irlgl%
Ylcqikùe8
&`+ Nahagly pqa V 7 eøgarl na gqay~r`y na `bq` pqa avian`e ac cçgk~a ay~kzqc`nl
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 138/846
Nr% Fëi~lr Pqa}anl Qrç`y
9'0=
na 0: gb,C nac ~l~`c na c`y 06 loyar}`ikleay% Ae~leiay# V ay qe` }`rk`oca `ca`~lrk`
okelgk`c ile z 7 :%0 { e 7 06% Zlr ileykbqkae~a# qy`enl c` mùrgqc` okelgk`c
o&v4e#z+ 7 eIv zv
pe'v
{ yqy~k~q{aenl cly }`clray ilrrayzlenkae~ay n`8
O&?406#:%0+ 7 Z& V 7 ?+ 7 06! , ?!&06 ’ ?+! &:%0+?
&:%;+06'?
7 &019+&:%:0+&:%0619+
7 :%?6>
&o+ Z& V ≡ >+ 7 06Iv &:%0+ v
&:%;+06'v
Qy`enl ay~a aemlpqa# ~aenrç`gly pqa yqy~k~qkr cly }`clray na v 7 ># 1# =# 2#……# 06
ae c` mùrgqc` na `rrko` { cqabl yqg`rcly% ^`gokëe zqnkër`gly qy`r c` ~`oc`
okelgk`c na zrlo`okckn`nay kenk}knq`cay l na zrlo`okckn`n na mqeikùe na g`y` l
mqeikùe `iqgqc`n` {# cqabl# yqg`r cly rayqc~`nly% &¼Iqàc ay c` nkmaraeik` ae qy`r c`
~`oc` `iqgqc`n` { c` kenk}knq`c<+%
Na iq`cpqkar g`ear`# ay gqifl gày màikc qy`r ac a}ae~l ilgzcagae~`rkl# {` ya`
qy`enl c` avzraykùe na `o`hl l okae# c` ~`oc` okelgk`c%
Z& V ≡ >+ 7 0 ’ Z& V 3 >+ 7 0 ’ Z& V ≮ 9+
9
7 0 ' ∐ 06Iv &:%0+v
&:%;+06'v
v7:
7 0 ’ &:%01 $ :%9: $ :%?6> $ :%0=6+
7 0 ’ :%;:? 7 :%:;6
@flr`# yk qy`gly c` ~`oc` okelgk`c `iqgqc`n`# oqyi`gly ac }`clr na e 7 06# ile V
7 9 { z 7 :%0# z`r` y`i`r ac m`i~lr Z& V 3 >+ { n` %;:?% Zlr cl ~`e~l#
Z& V ≡ >+ 7 0 ' %;:? 7 :%:;6
&i+ @pqç ay~`gly oqyi`enl Z&9 ≮ V 3 2+% Ay~l ely cca}` `8
=
Z&9 ≮ V 3 2+ 7 ∐ 06Iv &:%0+v&:%;+
06'v
v79
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 139/846
Nr% Fëi~lr Pqa}anl Qrç`y
9'02
7 :%0=6 $ :%:2 $ :%:?? $ :%::1 7 :%?=1
L~rl r`xle`gkae~l yarç` ilgl ykbqa8 v 7 :# 0# ?# 9# ># 1# =…#06
Z& V ≮ =+ ’ Z& V ≮ 9+# ay naikr# z`r` cly }`clray na V 7 9# ># 1# =
@pqç# eqa}`gae~a# zlnagly qy`r c` ~`oc` okelgk`c na zrlo`okckn`nay kenk}knq`cay
{ yqg`r c`y iq`~rl zrlo`okckn`nay na v 7 9# ># 1# =% ^`gokëe ya zqana qy`r c` ~`oc`
okelgk`c `iqgqc`n`# ay naikr# oqyi`enl e 7 06# z 7 :%0# { V 7 = {# cqabl#
ray~àenlca ac }`clr na V 7 9%
=
&n+ Z&9 3 V 3 2+ 7 ∐ 06Iv &:%0+v&:%;+
06'v
v7>
Ahagzcl )0:% Ae qe` ke}ay~kb`ikùe na ile~`gke`ikùe `gokae~`c ya ay~qnk`rle
ikae~ly na kenqy~rk`y% Ya` V ac eøgarl na kenqy~rk`y pqa el iqgzcae ile c`y
rabqc`ikleay `gokae~`cay nac `kra { nac `bq` na qe` gqay~r` `c `x`r na 0:
kenqy~rk`y% Yk ya y`oa pqa ac }`clr na c` zrlo`okckn`n ay na z 7 :%1# i`ciqc`r c`y
ykbqkae~ay zrlo`okckn`nay%
&`+ C` zrlo`okckn`n na pqa# av`i~`gae~a# 1 kenqy~rk`y iqgzc`e ile cly cçgk~ay
`gokae~`cay%
&o+ C` zrlo`okckn`n na pqa el gày na ?# iqgzc`e ile ac rabc`gae~l%
&i+ C` zrlo`okckn`n na pqa iq`enl gaely ;# cl iqgzc`e%
&n+ C` zrlo`okckn`n na pqa gaely na 1 kenqy~rk`y iqgzc`e# zarl iq`enl gaely 9 yk
cl iqgzc`e%
Ylcqikùe8
&`+ Z& V 7 1+ 7 O&140:#:%1+
7 :%?>= &qy`enl c` ~`oc` okelgk`c+
&o+ Z& V ≮ ?+ 7 :%:11 &qy`enl c` ~`oc` okelgk`c+
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 140/846
Nr% Fëi~lr Pqa}anl Qrç`y
9'06
&i+ Z& V ≡ ;+ 7 0 ' %;6;
7 :%:00
&n+ Z&9 ≮ V 3 1+ 7 Z&>+ $ Z&9+ 7 %?:1 $ :%002
7 %9?? &na c` ~`oc` okelgk`c kenk}knq`c+
Ahagzcl )00% Ay~a zrlocag` kcqy~r` ac qyl na c` nky~rkoqikùe okelgk`c { yq
`zrlvkg`ikùe ile c` nky~rkoqikùe elrg`c% _ayqc~` pqë# c` nky~rkoqikùe elrg`c# ile
± 7 ez { ω?7 ezp zrlzlrikle` qe` oqae` `zrlvkg`ikùe ` c` okelgk`c iq`enl e↟
∕ {# iq`enl z ya `zrlvkg` ` : l ` 0% @yç# yqzùeb`ya pqa e 7 01# z 7 :%> {
pqaragly aeile~r`r Z& V 7 >+% Z`r` ay~l qy`r c` nky~rkoqikùe okelgk`c { c`
nky~rkoqikùe elrg`c ilgl qe` `zrlvkg`ikùe ` ay~` øc~kg`% Ilgz`r`r cly
rayqc~`nly%
Ylcqikùe8
Qy`enl c` nky~rkoqikùe okelgk`c ay~kg`gly ac zrlgankl# ay naikr# ± 7 ez 7
&01+&:%>+ 7 = { c` }`rk`ex`# ω?7 ezp 7 &01+&:%>+&:%=+ 7 9%=# c` iq`c n` qe`
nay}k`ikùe ay~àen`r na ω 7 0%6;2%
Aeyabqkn`# qy`enl c` nky~rkoqikùe okelgk`c `iqgqc`n` n`8
o&v4e#z+ 7 Z& V 7 >+ 7 O&>401#:%>+ ' O&9401#:%>+
7 Z& V ≮ >+ ’ Z& V ≮ 9+
7 %?02 ' %:;0 7 %0?=
Pqa ay cl gkygl pqa qy`r c` mùrgqc` l c` nky~rkoqikùe na zrlo`okckn`nay
kenk}knq`cay# ay naikr8
Z& V 7 >+ 7 01I> &:%>+>
&:%=+01'>
7 :%0?16
@pqç# ya }a pqë# qy`enl c` ~`oc` okelgk`c na zrlo`okckn`nay kenk}knq`cay ya caa
nkrai~`gae~a ile e 7 01 { z 7 :%> { n` :%0?=
@flr` okae# qy`enl c` nky~rkoqikùe elrg`c# ilgl qe` `zrlvkg`ikùe# qy`gly c`
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 141/846
Nr% Fëi~lr Pqa}anl Qrç`y
9'0;
}`rk`oca `ca`~lrk` elrg`c ay~àen`r zloc`ikle`c X# ay naikr8
X 7 & V ’ ±+ , ω { yq ay~kg`nlr gqay~r`c x 7 &V ’ V + , y
Yke ago`rbl# c` }`rk`oca `ca`~lrk` nkyira~` na V 7 ># ae mlrg` na }`rk`oca `ca`~lrk`
ile~keq`# ay~à ae~ra 9%1 { >%1% @nagày# ile ± 7 = { ω 7 0%6;2 f`iagly c`
~r`eymlrg`ikùe qy`enl c` }`rk`oca `ca`~lrk` elrg`c ay~àen`r X%
X9%1 7 &9%1 ’ =+ , 0%6;2 7 ' 0%9?
X>%1 7 &>%1 ’ =+ , 0%6;2 7 ' :%2;
Na g`ear` pqa# Z& V 7 >+ 7 Z&'0%9? 3 X 3 ':%2;+
7 :%?0>6 ’ :%:;9> &na c` ~`oc` na x+
7 :%0?0>
Mke`cgae~a# ac }`clr na :%0?0> ay~à o`y~`e~a na `iqarnl ile ac }`clr na :%0?16
lo~aeknl ile c` nky~rkoqikùe okelgk`c%
Ahagzcl )0?% Yqzùeb`ya pqa ya ~kaea qe` gqay~r` na ?: i`yly na gankikleay na
`eàckyky na nag`en` oklpqçgki` na lvçbael &NOL1+ zrl}aekae~ay na qe gqay~ral
na qe rçl# zrlianae~ay na ?: cqb`ray nkmarae~ay ` cl c`rbl na yq ~r`{ai~lrk`% Yk ya
y`oa pqa# c` zrlo`okckn`n na pqa c` ileiae~r`ikùe na c` nag`en` oklpqçgki` na
lvçbael na 1 &NOL1+ nç`y ay~à nae~rl na cly cçgk~ay ay~kzqc`nly zlr c`y ca{ay
`gokae~`cay ay na z 7 :%= &ëvk~l+# f`iar cly ykbqkae~ay iàciqcly8
&`+ I`ciqc`r ac zrlgankl { c` nay}k ikùe ay~àen`r na c }`rk`oca
`ca`~lrk` V okelgk`c%
&o+ Qy`enl c` nky~rkoqikùe okelgk`c i`ciqc`r c` zrlo`okckn`n na pqa av`i~`gae~a 0:
i`yly na NOL ay~ëe nae~rl nac cçgk~a ay~kzqc`nl%
&i+ F`iar cly gkygl pqa ae ac keikyl &o+ zarl qy`enl c` nky~rkoqikùe elrg`c%
&n+ F`iar qe` ~`oc` na cly }`clray na c` }`rk`oca `ca`~lrk` V ilrrayzlenkae~ay ` v 7
:# 0# ?# 9# >#…#0; ae mqeikùe na e 7 ?: { z 7 :%= qy`enl c` mùrgqc` { c` ~`oc`
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 142/846
Nr% Fëi~lr Pqa}anl Qrç`y
9'?:
okelgk`c% I`ciqc`r ~`gokëe c` zrlo`okckn`n `iqgqc`n`%
&a+ F`iar qe fky~lbr`g` na zrlo`okckn`nay okelgk`cay z`r` ac ~`g`ðl na c` gqay~r`
e 7 ?: { z 7 :%= ile qe` iqr}` elrg`c ylorazqay~`%
&m+ I`ciqc`r Z& V ≮ 1+# Z& V ≡ 0?+ { Z& V ≮ 0?+ qy`enl# `go`y c` nky~rkoqikùe
okelgk`c { c` nky~rkoqikùe elrg`c ilgl `zrlvkg`ikùe ` ay~` øc~kg`%
Ylcqikùe8
&`+ Ac zrlgankl# c` }`rk`ex` { c` nay}k`ikùe ay~àen`r okelgk`cay yle8
Zrlgankl 7 ± 7 ez 7 &?:+&:%=+ 7 0?
]`rk`ex` 7 ω?7 ezp 7 ±&:%>+ 7 >%6
Nay}k`ikùe ay~àen`r 7 ω 7 ∘ω?7 ?%0;
&o+ @pqç# c` `zrlvkg`ikùe na c` nky~rkoqikùe okelgk`c ` c` nky~rkoqikùe elrg`c ay
oqae`# zlrpqa ez 7 &?:+&:%=+ ≡ 0: { ep 7 &?:+&:%>+ ≡ 0:%
Z`r` i`ciqc`r c` zrlo`okckn`n na pqë# av`i~`gae~a# 0: i`yly ay~ëe nae~rl na c`y
elrg`y ay~kzqc`n`y ya f`ia qy`enl c` nky~rkoqikùe okelgk`c ile cly }`clray e 7 ?:#
z 7 :%= { p 7 0 ’ z 7 0 ’ :%= 7 :%>% @flr`# yqy~k~q{aenl cly }`clray ae c` mùrgqc`
okelgk`c ely n`8
o&v4e#z+ 7 eIv zv
pe'v
z`r` v 7 :# 0# ?# 9# ># 1# =# 2# 6# ;# 0:……# 0;
O&0:4?:#:%=+ 7 ?:I0: &:%=:+0:
&:%>+?:'0:
7 :%00=?
Yke ago`rbl# qy`enl c` ~`oc` okelgk`c na zrlo`okckn`nay `iqgqc`n`y ely n` c`
zrlo`okckn`n kenk}knq`c8
Z& V 7 0:+ 7 %?>>2 ' %0?21 7 %002?
^`gokëe# qy`enl c` ~`oc` okelgk`c na zrlo`okckn`nay na mqeikùe na g`y` ile e 7
?:# ϊ 7 %=:: { ` 7 0: n` qe }`clr na :%002%
Ilgl ya }a `rrko`# ac qyl na c` mùrgqc` okelgk`c ay c`rbl { ~anklyl% Yke
ago`rbl# yk qy`gly c` ~`oc` na c`y zrlo`okckn`nay okelgk`cay kenk}knq`cay l na
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 143/846
Nr% Fëi~lr Pqa}anl Qrç`y
9'?0
mqeikùe na g`y` l# okae# c` ~`oc` na zrlo`okckn`nay `iqgqc`n`y# cly iàciqcly ya
ykgzckmki`e na ylorag`ear`%
&i+ @flr`# qy`enl c` nky~rkoqikùe elrg`c ile }`rk`ocay `ca`~lrk`y ile~keq`y# ely n`8
&;%1 ≮ V ≮ 0:%1+ l ya` Z&'0%0> ≮ X ≮ '%=6+% Ay~l ya i`ciqc` qy`enl c` }`rk`oca
`ca`~lrk` ay~`en`rkx`n` X# ay naikr#
X 7 & V ’ ±+ , ω nlena V 7 ;%1 { 0:%1# ± 7 0?# y 7 ω 7 ?%0;
X;%1 7 &;%1 ’ 0?+ , ?%0; 7 ' 0%0>
X0:%1 7 &0:%1 ’ 0?+ , ?%0; 7 ' :%=6
Aeyabqkn`# qy`enl c` ~`oc` na c` nky~rkoqikùe elrg`c r`xle`gly ilgl8
Z&'0%0> ≮ X ≮ '%=6+ 7 Z&X 7 '%=6+ ’ Z&X 7 '0%0>+
7 &:%?>69 ’ :%0?20+
7 %0?0?
@c ilgz`r`r cly nly rayqc~`nly }agly pqa c` nky~rkoqikùe okelgk`c n` :%002? { c`
nky~rkoqikùe elrg`c n` :%0?0?% Ay~` `zrlvkg`ikùe yarç` gahlr ` gankn` pqa e mqar`
gày br`ena%
&n+ Z`r` f`iar qe` ~`oc` ile ~ln`y c`y zrlo`okckn`nay ilrrayzlenkae~ay ` v 7 :# 0# ?#
9# >#…# 0;% &]ar ^@OC@ 9%0 na `o`hl+%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 144/846
Nr% Fëi~lr Pqa}anl Qrç`y
9'??
^@OC@ 9%0% ^`oc` gly~r`enl c`y zrlo`okckn`nay kenk}knq`cay { `iqgqc`n`y ile e
kbq`c ` ?: { z kbq`c ` :%=% &Ac`olr`ikùe zrlzk`+
]`clr na c` }`rk`oca Z& V + kenk}knq`c Z& V + `iqgqc`n`
`ca`~lrk` V
: :%::: :%:::
0 :%::: :%:::
? :%::: :%:::
9 :%::: :%:::
> :%::: :%:::1 :%::? :%::?
= :%::> :%::=
2 :%:01 :%:?0
6 :%:9= :%:12
; :%:20 :%0?6
0: :%002 :%?>1
00 :%01; :%>:>
V 7 ± 7 0? &zrlgankl+ :%06: :%16>(
0> :%0?> :%;2>
01 :%:21 :%;>;
0= :%:91 :%;6>
06 :%::9 :%;;;
?: :%::: 0%:::
Ac `y~arkyil &(+ yað`c` c` cli`ckx`ikùe nac zrlgankl%
&a+ Z`r` ay~a keikyl c` Mkbqr` 9%? a `o`hl gqay~r` qe fky~lbr`g` na zrlo`okckn`n
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 145/846
Nr% Fëi~lr Pqa}anl Qrç`y
9'?9
okelgk`c z`r` e 7 ?:# z 7 :%=# ± 7 0? { ω 7 ?%0;# ile iqr}` elrg`c na
`zrlvkg`ikùe ylorazqay~`% @pqç# ya }a pqa# `qe iq`enl ac fky~lbr`g` na
zrlo`okckn`n ay~à qe zlil yayb`nl f`ik` c` kxpqkarn`# zlrpqa z 5 %=% C` iqr}`
elrg`c n` gq{ oqae` `zrlvkg`ikùe ` c` okelgk`c%
Mkbqr` 9%?% Bràmki` gly~r`enl qe fky~lbr`g` na zrlo`okckn`n okelgk`c z`r` e 7 ?:#
z 7 :%=# ± 7 0? { ω 7 ?%0;# ile iqr}` elrg`c na `zrlvkg`ikùe ylorazqay~`% @pqç#
ya }a ic`r`gae~a# pqë# `qe iq`enl ac fky~lbr`g` na zrlo`okckn`n ay~à qe zlil
yayb`nl f`ik` c` kxpqkarn`# &zlrpqa z 5 %=+# c` iqr}` elrg`c n` gq{ oqae`
`zrlvkg`ikùe ` c` okelgk`c% &Ac`olr`ikùe zrlzk`+
&m+ Z`r` i`ciqc`r cly }`clray na `o`hl qy`enl c` nky~rkoqikùe okelgk`c { c` elrg`c#
ya zrliana ilgl8
0% Z& V ≮ 1+
?% Z& V ≡ 0?+
9% Z& V ≮ 0?+
Qy`enl c` ~`oc` na c` nky~rkoqikùe okelgk`c ile e 7 ?:# z 7 :%= { p 7 :%> n` cly
ykbqkae~ay rayqc~`nly%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 146/846
Nr% Fëi~lr Pqa}anl Qrç`y
9'?>
Z`r`8 Z& V ≮ 1+ 7 %::?
Z`r`8 Z& V ≡ 0?+ 7 0 ’ Z& V 3 0?+
7 0 ' %>:>
7 %1;=
El~`8 Z`r` Z& V ≡ 0?+ ¼zlrpqë ac }`clr lo~aeknl el ya `zrlvkgù ` %1< Z`r` ile~ay~`r
ay~` zrabqe~` ramkër`ya ` c` Mkbqr` 9%? na `rrko`%
Z`r`8 Z& V ≮ 0?+ 7 0 ’ Z& V 5 0?+ 7 %1;=
@flr`# z`r` i`ciqc`r cly }`clray na `rrko` qy`enl c` nky~rkoqikùe elrg`c#
ilgl qe` `zrlvkg`ikùe ` c` okelgk`c# z`r` i`n` qel na c`y zrabqe~`y Z& V ≮ 1+#
Z&V ≡ 0?+ { Z& V ≮ 0?+ eaiayk~`gly ile}ar~kr c`y }`rk`oca `ca`~lrk`y nkyira~`y ` c`y
}`rk`ocay `ca`~lrk`y elrg`cay X qy`enl c` }`rk`oca `ca`~lrk` ay~`en`rkx`n` X ile ±
7 0? { ω 7 ?%0; { cqabl oqyi`r ac }`clr na X ae c` ~`oc` na c` nky~rkoqikùe elrg`c {
i`ciqc`r c` zrlo`okckn`n ilrrayzlenkae~a%
Qy`enl c` mqeikùe X 7 & V ’ ±+,ω { ay~`en`rkx`enl ely n`8
X1 7 &1 ’ 0?+,?%0; 7 ' 9%0;2
@flr` qy`enl c` ~`oc` na c` nky~rkoqikùe elrg`c oqyi`gly x 7 '9%0;2 { n` %:::2# l
ya` ≆ %::0% Ykgkc`rgae~a# ile Z& V ≡ 0?+ ile}ar~kgly V 7 0? ` }`clray na X ile ± 7
0? { ω 7 ?%0; { n`8
X 7 &0? ’ 0?+,?%0; 7 :
Pqa ilrrayzlena ` qe` zrlo`okckn`n na %1:::% C` gkyg` yk~q`ikùe liqrrkrç` ile
Z& V ≮ 0?+…… &Pqa ~`gokëe ya zqana caar na c` bràmki`+%
Ahagzcl )09% Yk ae c` m`orki`ikùe na `iiaylrkly z`r` qe yky~ag` na ile~rlc na
z`r~çiqc`y &ikicùe+ ya `ylik` ile qe zrliayl Oarelqcck# ile qe zrlgankl na z`r~ay
namai~qly`y na :%?:# ay~kg`r c` zrlo`okckn`n8
&`+ Na el aeile~r`r z`r~ay namai~qly`y nac yky~ag` na ile~rlc na qe` gqay~r`
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 147/846
Nr% Fëi~lr Pqa}anl Qrç`y
9'?1
`ca`~lrk` na 0: z`r~ay%
&o+ Na el aeile~r`r z`r~ay namai~qly`y na cly ikicleay m`orki`nly na qe` gqay~r`
na ?: z`r~ay%
Ylcqikùe8
&`+ Qy`enl c` mùrgqc` okelgk`c8 o&v4e#z+ 7 eIv zv
pe'v
{ yqy~k~q{aenl V 7 :# z 7 :%?
{ p 7 :%6 ely n`8
Z& V 7 :+ 7 O&:40:#:%?+ 7 0:I: &:%?+:
&:%6+0:':
7 :%0:2
Ay~a rayqc~`nl ~`gokëe ya zqana lo~aear qy`enl c` ~`oc` okelgk`c na
zrlo`okckn`nay kenk}knq`cay l na mqeikùe na g`y`# ay naikr# oqyi`enl e 7 0:# z 7
:%? { V 7 :%
&o+ Eqa}`gae~a qy`enl c` mùrgqc` okelgk`c { yqy~k~q{aenl n`8
Z& V 7 :+ 7 O&:4?:#:%?+7 ?:I: &:%?+:
&:%6+?:':
7 &0+&0+&:%:0?+
7 :%:0?
@eàclb`gae~a# ay~a gkygl rayqc~`nl ya zqana lo~aear qy`enl c` ~`oc` okelgk`c
`iqgqc`n` oqyi`enl e 7 ?:# z 7 :%? { V 7 : { n` :%:0?% @pqç# eù~aya pqa ~`gokëe
ya lo~kaea ac gkygl rayqc~`nl qy`enl c` ~`oc` okelgk`c na zrlo`okckn`nay
kenk}knq`cay%
Ahagzcl )0>% Yk ~aeagly qe` gqay~r` `ca`~lrk` na e 7 ?: &zaiay+ z`r` }`rkly
}`clray na z# zlnagly ay~kg`r c` zrlo`okckn`n na V gqar~ay na cly lrb`ekygly
ylga~këenlcly ` ikar~`y ileiae~r`ikleay ~ùvki`y zrl}aekae~ay na qe` nayi`rb`
kenqy~rk`c na qe rçl% Z`r` ay~l f`iar cly ykbqkae~ay iàciqcly8
&`+ I`ciqc`r ac zrlgankl ± { c` nay}k`ikùe ay~àen`r ω# na c` gqar~a na cly zaiay# yk
ac }`clr na z 7 :%:1
&o+ C` zrlo`okckn`n na pqa gqar` ` cl gày 0 lrb`ekygl
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 148/846
Nr% Fëi~lr Pqa}anl Qrç`y
9'?=
&i+ C` zrlo`okckn`n na pqa el gqar` ekebøe lrb`ekygl
&n+ C` zrlo`okckn`n na pqa gqar`e iq`enl gaely 9 lrb`ekygly
&a+ C` zrlo`okckn`n na Z& V 7 0:+
Ylcqikùe8
&`+ Zrlgankl 7 V 7 ± 7 ez 7 &?:+&%:1+ 7 0%:%
Nay}k`ikùe ay~àen`r 7 ω 7 ezp 7 &0%:+&%;1+ 7 %;1
&o+ Z& V ≮ 0+ 7 %29=
&i+ Z& V 7 :+ 7 %916
&n+ Z& V ≡ 9+ 7 0 ' Z& V 3 9+ 7 0 ' %;?>1 7 %:211
&a+ Z& V 7 0:+ 7 0%:
Ahagzcl )01% C` zlykokckn`n na pqa qe` gqay~r` na `kra ile~aeb` qe
gkirllrb`ekygl ca~`c ay na 0:% Yqzlekaenl pqa c`y gqay~r`y yle kenazaenkae~ay#
ile rayzai~l ` c` zrayaeik` nac gkirllrb`ekygl# aeile~r`r c` zrlo`okckn`n na pqa8
&`+ Ae c`y 06 ykbqkae~ay# av`i~`gae~a ? ile~aeb`e ac bargae%
&o+ @c gaely > gqay~r`y ile~aeb`e ac bargae%
&i+ C` zrlo`okckn`n na pqa gaely na 2 gqay~r`y na `kra ile~aeb`e ac bargae# zarl
iq`enl gaely 9 gqay~r`y ~`gokëe cl ~aeb`e# a%b%# Z&9≮ V 3 2+%
Ylcqikùe8
&`+ Ya` V ac eøgarl na gqay~r`y na `kra pqa ile~aeb`e ac bargae z`~ùbael ae c`y
06 gqay~r`y ykbqkae~ay `e`ckx`n`y% Ae~leiay# V ay qe` }`rk`oca `ca`~lrk` okelgk`c#
ile z 7 :%0 { e 7 06% Zlr ileykbqkae~a8
Z& V 7 ?+ 7 O&?406#:%0+ 7 06I? &:%0+?
&:%;+0=
7 :%?6>
06
&o+ Z& V ≡ >+ 7 Ω 06Iv &:%0+v
&:%;+06'v
# nlena 06Iv 7 06!,v!&06 ’ v+!v7:
@pqç# yke ago`rbl# ay gày màikc qy`r ac a}ae~l ilgzcagae~`rkl%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 149/846
Nr% Fëi~lr Pqa}anl Qrç`y
9'?2
Z& V ≡ >+ 7 0 ’ Z& V 3 >+
9
7 0 ’ Ω 06IV &:%0+v
&:%;+06'v
v7:
7 0 ’ &:%01 $ :%9:: $ :%?6> $ :%0=6+
7 :%:;6
=
&i+ Z&9 ≮ V 3 2+ 7 Ω 06Iv &:%0+v
&:%;+06'v
v79
7 :%0=6 $ :%:2 $ :%:?? $ :%::1
7 :%?=1
^`gokëe# Z& V ≮ =+ ’ Z& V ≮ ?+ 7 %;;69 ' %2996
7 %?=>1
Ahagzcl )0=% Ae qe ay~qnkl na fkbkaea kenqy~rk`c { yabqrkn`n cca}`nl ` i`ol ae
gqif`y g`pqkc`nlr`y kenqy~rk`cay# yqzùeb`ya pqa f`{ qe` zloc`ikùe br`ena na
~lg`nlray na ckilr { l~r` zloc`ikùe na `oy~agkly% Ae ay~a i`yl# c` zrlo`okckn`n na
ëvk~l l na ~lg`nlray ya `yqga pqa ay z kbq`c ` :%> {# c` zrlo`okckn`n na `oy~agkly
&zrlo`okckn`n na mr`i`yl+ ay na p kbq`c ` :%=% Yk y`i`gly qe` gqay~r` `c `x`r na e 7
0: lzar`nlray na c` g`pqkc`nlr`# ae~leiay# ac eøgarl na c` }`rk`oca `ca`~lrk` na V
~lg`nlray na ckilr ay na v 7 :# 0# ?# 9# ># 1# =# 2# 6# ;# 0:# ykaenl `yç8
&`+ Zraz`r`r qe` ~`oc` gly~r`enl c`y zrlo`okckn`nay kenk}knq`cay { }`clray na V %
&o+ Zraz`r`r qe` bràmki` ae mqeikùe na zrlo`okckn`nay okelgk`cay kenk}knq`cay ae c`
lrnae`n` { na v ae c` `oyiky`# nlena c`y o`rr`y kenkpqae c` zrlo`okckn`n na mqeikùe
na g`y` Z& V 7 v+%
&i+ I`ciqc`r ac zrlgankl { c` }`rk`ex` na ay~` nky~rkoqikùe%
&n+ Na c` bràmki` caar ~ln`y c`y zrlo`okckn`nay Z& V 7 v+%
&a+ Ay~kg`r Z& V 7 >+ a ke~arzra~`r ac rayqc~`nl
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 150/846
Nr% Fëi~lr Pqa}anl Qrç`y
9'?6
Ylcqikùe8
&`+ C` ~`oc` na zrlo`okckn`nay kenk}knq`cay# ile e 7 0: { z 7 :%> ya n` ae c`
^@OC@ 9%? na `o`hl% Ay~l ya f`ia ile ac zrlbr`g` Gkek~`o% Ac zrliankgkae~l
z`r` baear`r c`y zrlo`okckn`nay na mqeikùe na g`y` Z& V 7v+ ay8
I`ci ↟ Zrlo`okck~{ nky~rkoq~kley↟ Okelgk`c
Ae c` }ae~`e` na –Okelgk`c Nky~rkoq~kle‗ zqe~a`r –Zrlo`okck~{‗ a ke~rlnqikr ac
eøgarl na aey`{ly &0:+ { c` zrlo`okckn`n na ëvk~l &:%>+% @nagày# zqe~a`r –Kezq~
ilcqge‗# ke~rlnqikr cly }`clray na V # { ae c` }ae~`ekcc` na –Lz~kle`c y~lr`ba‗ zlear
Z& V 7v+ { cqabl LD% ^ln`y ay~`y lrnaeay baear`e c` cly }`clray na c` ^@OC@ 9%?%
^@OC@ 9%?% ^`oc` gly~r`enl c`y zrlo`okckn`nay
okelgk`cay kenk}knq`cay }y% }`clray na V %
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
Z& V7v+ ]`rk`oca `ca`~lrk` V
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
:%::=:>2 :
:%:>:900 0
:%0?:;9? ?
:%?0>;;0 9
:%?1:6?9 >
:%?::=16 1
:%000>22 =
:%:>?>=2 2
:%:0:=02 6
:%::0129 ;
:%:::0:1 0:
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
&o+ Z`r` f`iar c` bràmki` na Z& V 7v+ }y% }`clray na V qy`r ac zrlbr`g` Gkek~`o {
zrlianar na c` ykbqkae~a g`ear`8 Krya `8 Br`zf ↟ Yi`~~arzcl~% Ae c` }ae~`e` na
–Yi`~~arzcl~‗ pqa `z`raia# krya ` –Rk~f Ileeai~ Ckea‗ a ke~rlnqikr cly }`clray na
Z& V 7v+ { }`clray na c` }`rk`oca `ca`~lrk` V % Ae c` }ae~`e` na –Yi`~~arzcl~ N`~`
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 151/846
Nr% Fëi~lr Pqa}anl Qrç`y
9'?;
]ka‗ zqe~a`r –Y{golcy‗ { –Zrlhai~ Ckeay‗ { LD% Ay~l baear` c` bràmki` na c`y
zrlo`okckn`nay okelgk`cay na mqeikùe na g`y` Z& V 7v+# ae mqeikùe na cly }`clray na
c` }`rk`oca V gly~r`n` `o`hl% Ykaenl `yç# `e`ckx`r c` ilemkbqr`ikùe na cly rayqc~`nly
na c` br`mki` { naikr yk ay lockiq` ` c` naraif` l ` c` kxpqkarn` { avzcki`r zlrpqa
liqrra na ay` g`ear`% C` bràmki` ya gqay~r` `o`hl%
]`r k`oca `ca`~lrk` V
Z & V 7 v +
0:6=>?:
0:6=>?:
:%?1
:%?:
:%01
:%0:
:%:1
:%::
:%?1
:%?:
:%01
:%0:
:%:1
:%::
Br`mki` na Z&V7v+ }y% }`rk`oca `ca`~lrk` V
Mkbqr` 9%9% Bràmki` gly~r`enl Z& V 7 v+ ae mqeikùe na V % @pqç# naoknl ` pqa z 7
:%> 3 :%1# c` nky~rkoqikùe ay lockiq` f`ik` c` naraif`% &Ac`olr`ikùe zrlzk`+
&i+ Ac zrlgankl ±# c` }`rk`ex` ω?{ c` nay}k`ikùe ay~àen`r ω na ay~` nky~rkoqikùe
yle8 ± 7 ez 7 &0:+&:%>+ 7 >%:# ω? 7 ezp 7 &0:+&:%>+&:%=+ 7 ?%># ω 7 ∘?%> 7 0%111
&n+ Na c` Mkbqr` 9%9 ya zqanae caar ~ln`y c`y zrlo`okckn`nay Z& V 7 v+ gly~r`n`y ae
c` ^@OC@ 9%9 { ~`gokëe qy`enl c` ^@OC@ 9%?%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 152/846
Nr% Fëi~lr Pqa}anl Qrç`y
9'9:
^@OC@ 9%9% ^`oc` gly~r`enl cly }`clray na c` }`rk`oca `ca`~lrk` V z`r` ay~a
zrlocag`% &Ac`olr`ikùe zrlzk`+
Z& V 7 :+ 7 :%::=: Z& V 7 =+ 7 :%0001Z& V 7 0+ 7 :%:>:9 Z& V 7 2+ 7 :%:>?1
Z& V 7 ?+ 7 :%0?:; Z& V 7 6+ 7 :%:0:=
Z& V 7 9+ 7 :%?01: Z& V 7 ;+ 7 :%;;0=
Z& V 7 >+ 7 :%?1:6 Z& V 7 0:+ 7 :%:::0
Z& V 7 1+ 7 :%?::=
&a+ Z& V 7 >+ 7 :%?1:6 nkia pqë# yk yacaiikleàr`gly 0:: gqay~r`y na ~`g`ðl e 7 0:#
na qe` zloc`ikùe na lzar`nlray na c` kenqy~rk` g`pqkc`nlr` ayzar`rç`gly pqa ?1
na ay~`y gqay~r`y ~aenrç`e qe }`clr na V 7 > ~lg`nlray na ckilr%
Ahagzcl )02% C` z`r`ay~`~`c ZAGAV na Gëvkil ya `}liù ` f`iar zarmlr`ikleay ae
ac yqray~a na ^`o`yil% Z`r` }ar c` m`i~kokckn`n mke`eikar` na pqa mqar` ile}aekae~a
f`iar c`y zarmlr`ikleay# ZAGAV ile~r`~ù cly yar}kikly na qe` mkrg` na ay~qnkly
ay~`nçy~kily% Ya y`oa pqa# i`n` zlxl zarmlr`nl ya ic`ykmki` ilgl zrlnqi~k}l l el
zrlnqi~k}l% C` avzarkaeik` na ZAGAV ay pqa# ae ay~a ~kzl na avzclr`ikleay# ya
y`oa zlr avzarkaeik` pqa# ac 01 na cly zlxly zarmlr`nly yle zrlnqi~k}ly% Z`r` c`y
avzclr`ikleay za~rlcar`y ya yacaiikle`rle `ca`~lrk`gae~a 0? yk~kly% Ile ay~`
kemlrg`ikùe ae gae~a# f`iar cly ykbqkae~ay iàciqcly%
&`+ ¼Iqàc ay c` zrlo`okckn`n na pqa cly 0? zlxly pqa ya zarmlrae ae i`n` qel na cly
0? yk~kly# ya`e zrlnqi~k}ly<
&o+ ¼Iqàc ay c` zrlo`okckn`n na pqa ekebøe zlxl zarmlr`nl ya` zrlnqi~k}l<
&i+ ¼Iqàc ay c` zrlo`okckn`n na pqa av`i~`gae~a qe zlxl ya` zrlnqi~k}l<&n+ Z`r` f`iar rae~`oca `c z`çy# iq`enl gaely ~ray na cly zlxly na avzclr`ikùe
naoae yar zrlnqi~k}ly% Ykaenl `yç# ¼Iqàc ay c` zrlo`okckn`n na pqa ac eablikl ya`
rae~`oca<
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 153/846
Nr% Fëi~lr Pqa}anl Qrç`y
9'90
Yqbaraeik`8 Qy`r Z& V 7 0?+# Z& V 7 :+# Z& V 7 0+# Z& V ≡ 9+# a~i%
Nky~rkoqikùe Fkzarbalgë~rki`
C` mqeikùe fkzarbalgë~rki` ay qe` nky~rkoqikùe nkyira~` na zrlo`okckn`n# c` iq`c
ay~à ay~raif`gae~a ckb`n` ` c` nky~rkoqikùe okelgk`c% C` g`ear` gày ykgzca na }ar
c` nkmaraeik` ae~ra c`y nly nky~rkoqikleay r`nki` ae c` mlrg` pqa ya f`ia ac
gqay~ral% C` nkmaraeik` ae~ra ay~`y nly nky~rkoqikleay ay pqa# ae c` nky~rkoqikùe
okelgk`c# cly ke~ae~ly yle kenazaenkae~ay# zlrpqa f`{ raagzc`xl ae c` yacaiikùe na
c` gqay~r`% Yke ago`rbl# ae ac i`yl na c` nky~rkoqikùe fkzarbalgë~rki`# f`{
nazaenaeik`# zlrpqa c` yacaiikùe na c` gqay~r` ya f`ia yke raagzc`xl { c`
zrlo`okckn`n na ëvk~l i`gok` na qe ke~ae~l ` l~rl%
Ac glnacl fkzarbalgë~rkil ay `zrlzk`nl# iq`enl ac gqay~ral ay yke
raagzc`xl na qe` zloc`ikùe mkek~` {# iq`enl ya rapqkara c` zrlo`okckn`n na qe
eøgarl ayzaiçmkil na ëvk~ly {,l mr`i`yly%
Yqzlykikleay { zrlzkan`nay na c` nky~rkoqikùe fkzarbalgë~rki`
0% Qe` gqay~r` `ca`~lrk` na ~`g`ðl e ya yacaiikle` yke raagzc`xl na E ç~agy%
?% d na cly E ç~agy zqanae yar ic`ykmki`nly ilgl ëvk~ly {# E ’ d ay ic`ykmki`nl
ilgl mr`i`yly%
9% C` zloc`ikùe l ilehqe~l na c` gqay~r` ileyky~a na E kenk}knqly# loha~ly l
acagae~ly &qe` zloc`ikùe mkek~`+%
>% I`n` kenk}knql zqana yar i`r`i~arkx`nl ilgl qe ëvk~l l qe mr`i`yl { f`{ d
ëvk~ly ae c` zloc`ikùe%
1% Qe` gqay~r` na e kenk}knqly ya yacaiikle` yke raagzc`xl &f`{ nazaenaeik`# ae
ile~r`y~a ile c` okelgk`c ae c` pqa f`{ kenazaenaeik`+ ae mlrg` `ca`~lrk`%
Namkekikùe na c` nky~rkoqikùe fkzarbalgë~rki`
Ae c` nky~rkoqikùe na zrlo`okckn`n na qe` }`rk`oca `ca`~lrk` fkzarbalgë~rki` V# ac
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 154/846
Nr% Fëi~lr Pqa}anl Qrç`y
9'9?
eøgarl na ëvk~ly ae qe` gqay~r` `ca`~lrk` na ~`g`ðl e# yacaiikle`n` na E ç~agy#
na cly iq`cay d ya cc`g`e ëvk~ly { E ’ d ya cc`g`e mr`i`yly ay8
d Iv E'd Ie'v f& V 4E#e#d+ 7 ┊┊┊┊┊┊┊┊┊ v 7 :# 0# ?# 9#%%%# e &9'>+
EIe
Nlena8
d 7 ëvk~ly ae e ke~ae~ly# ay naikr# c` i`e~kn`n na acagae~ly
knae~kmki`nly ilgl ëvk~l ae c` zloc`ikùe
E ’ d 7 mr`i`yly
e 7 ~`g`ðl na c` gqay~r` `ca`~lrk` l i`e~kn`n na acagae~ly ae c`
zloc`ikùe
E 7 eøgarl na ç~agy &~`g`ðl na c` zloc`ikùe+
Nlena v el zqana avianar na d { &e ’ v+ el zqana avianar na &E ’ d+
Loyar}`ikleay8
EIe _azrayae~` c` i`e~kn`n na mlrg`y ae c`y pqa ya zqana
yacaiikle`r qe` gqay~r` na ~`g`ðl e na qe` zloc`ikùe na
na ~`g`ðl E
d Iv _azrayae~` c` i`e~kn`n na g`ear`y ae c`y pqa ya zqana
yacaiikle`r v ëvk~ly na qe ~l~`c na d ëvk~ly na c`
zloc`ikùe
E'd Ie'v _azrayae~` c` i`e~kn`n na g`ear`y ae c`y pqa ya zqana
yacaiikle`r e ’ v mr`i`yly na qe ~l~`c na E ’ d mr`i`yly ae c` zloc`ikùe
@zcki`ikleay na c` nky~rkoqikùe fkzarbalgë~rki`
C`y `zcki`ikleay na ay~` nky~rkoqikùe ya aeiqae~r`e ae c`y zrqao`y acai~rùeki`y4
`yabqr`ex` na i`ckn`n4 yacaiikùe na nk`g`e~ay kenqy~rk`cay# `cbqely na cly iq`cay
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 155/846
Nr% Fëi~lr Pqa}anl Qrç`y
9'99
yle na i`ckn`n yqzarklr ` cly l~rly4 ae zrlocag`y na gqay~raly na naic`r`ikleay na
kgzqay~ly ylora kebrayly# nlena d ae~ra E naic`r`ikleay `rifk}`n`y ile~kaeae
nanqiikleay iqay~kle`ocay% Kbq`cgaea# c` nky~rkoqikùe fkzarbalga~rki` ~kaea c`y
gkyg`y `zcki`ikleay ` c` kebaekarç` `gokae~`c# pqa ile c` okelgk`c# ile c`
nkmaraeik` pqa ile c` fkzarbalgë~rki` ac gqay~ral ay yke raagzc`xl%
I`r`i~arçy~ki`y na c` nky~rkoqikùe fkzarbalgë~rki`
Yk e ay rac`~k}`gae~a zapqaðl ile rayzai~l ` E# c` zrlo`okckn`n z`r` i`n` ke~ae~l
i`gok` ckbar`gae~a# cl pqa kenki` pqa ya ~kaea qe avzarkgae~l okelgk`c% Ay~`
yk~q`ikùe zqana `zrlvkg`rya ` c` nky~rkoqikùe fkzarbalgë~rki` qy`enl c`
nky~rkoqikùe okelgk`c ile z 7 d,E% @nagày# ac zrlgankl { c` }`rk`ex` na c`
nky~rkoqikùe fkzarbalgë~rki` ya zqanae `zrlvkg`r gank`e~a c`y mùrgqc`y8
± 7 ez 7 ed,E &9'1+
ω?7 ezp 7 e&d,E+&0 ’ d,E+ &9%=+
_ac`ikùe ae~ra c` nky~rkoqikùe fkzarbalgë~rki` { c` nky~rkoqikùe okelgk`c
F`{ qe` rac`ikùe ke~aray`e~a ae~ra c` nky~rkoqikùe okelgk`c { c` nky~rkoqikùe
fkzarbalgë~rki`% Ilgl ya nkhl `e~ay# yk e ay zapqað` ilgz`r`n` ile E# c`
e`~qr`cax` na E ç~agy i`gok` gq{ zlpqk~l ae i`n` gqay~ral% Zlr cl ~`e~l# c`
i`e~kn`n d,E hqab` ac z`zac nac z`ràga~rl z na c` nky~rkoqikùe okelgk`c% Ilgl
rayqc~`nl# c` nky~rkoqikùe okelgk`c zqana yar }ky~` ilgl qe` ankikùe zloc`ikle`c
br`ena na c` nky~rkoqikùe fkzarbalgë~rki`%
@yç# iq`enl f`{ qe avzarkgae~l fkzarbalgë~rkil# ae ac iq`c el ya n` ac
}`clr na d nkrai~`gae~a# zarl yk ile }`clray n`nly na E { na c` zrlo`okckn`n z &l
ae ~ërgkely na zlriae~`ha+# ac }`clr na d ya zqana i`ciqc`r qy`enl c` rac`ikle z 7
d,E%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 156/846
Nr% Fëi~lr Pqa}anl Qrç`y
9'9>
Ahagzcly qy`enl c` nky~rkoqikùe fkzarbalgë~rki`
Ahagzcl )06% Qe ilgk~ë na ~`g`ðl 1 ay yacaiikle`nl `ca`~lrk`gae~a# na 9
Pqçgkily { 1 Mçykily% Aeile~r`r c` nky~rkoqikùe na zrlo`okckn`n z`r` ac eøgarl na
Pqçgkily ae ac ilgk~ë% F`iar qe` bràmki` l fky~lbr`g` pqa }`{` ae mqeikùe na c`
}`rk`oca `ca`~lrk` V { na Z& V +%
Ylcqikùe8
@pqç# E 7 6# e 7 1# d 7 9% Ya qy` c` mùrgqc` &9'>+ na c` nky~rkoqikùe
fkzarbalgë~rki`8
d Iv E'd Ie'v
f&v4E#e#d+ 7 ┊┊┊┊┊┊┊┊┊ v 7 :#0#?#9…%#e
EIe
Yqy~k~q{aenl cly }`clray ae c` mùrgqc` na `rrko` ely n` c` mlrg` oyki` cky~` z`r`
yqy~k~qkr cly }`clray na c` }`rk`oca `ca`~lrk` V %
9Iv 6'9I1'v
f&v46#1#9+7 ┊┊┊┊┊┊┊┊┊ 6I9
Zlr cl ~`e~l# nac ayz`ikl gqay~r`c v 7 :# 0# ?# 9# > { yqy~k~q{aenl ay~ly }`clray ae c`
avzraykùe na `rrko` n` cly ykbqkae~ay aeqeik`nly8
Z& V 7 :+ 7 f&:46#1#9+ 7 9I: 1I1 , 6I1 7 &0+&0+,1= 7 0,1= 7 :%:06
Z& V 7 0+ 7 f&046#1#9+ 7 9I0 1I> , 6I1 7 &9+&1+,1= 7 01,1= 7 :%?=6
Z& V 7 ?+ 7 f&?46#1#9+ 7 9I? 1I9 , 6I1 7 &9+&0:+,1= 7 9:,1= 7 :%19=Z& V 7 9+ 7 f&946#1#9+ 7 9I9 1I? , 6I1 7 &0+&0:+,1= 7 0:,1= 7 :%02;
Z& V 7 >+ 7 f&>46#1#9+ 7 9I> 1I0 , 6I1 7 &:+&1+,1= 7 :
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 157/846
Nr% Fëi~lr Pqa}anl Qrç`y
9'91
^@OC@ 9%>% ^`oc` gly~r`enl c` ~`oqc`ikùe na c` nky~rkoqikùe fkzarbalgë~rki`%
]`rk`oca `ca`~lrk` V s : 0 ? 9 >
f&v46#1#9+ s 0,1= 01,1= 9:,1= 0:,1= :
Z`r` f`iar ac fky~lbr`g`# øeki`gae~a ya br`mki`e cly }`clray na c` }`rk`oca
`ca`~lrk` v 7 :# 0# ?# 9# > ae c` `oyiky` { cly }`clray na f&v46#1#9+ ae c` lrnae`n`%
Ahagzcl )0;% _amkrkëenlya `c zrlocag` `e~arklr# i`ciqc`r c`y ykbqkae~ay
zrlo`okckn`nay8
&`+ C` zrlo`okckn`n na pqë# av`i~`gae~a# 0 Pqçgkil ya` yacaiikle`nl%&o+ C` zrlo`okckn`n na pqë# iq`enl gaely 0 Pqçgkil ya` yacaiikle`nl
&i+ C` zrlo`okckn`n na pqë# ae~ra 0 { 9 &keicqyl+ Pqçgkily ya`e yacaiikle`nly%
Ylcqikùe8
&`+ Yqy~k~q{aenl cly }`clray na E 7 6# e 7 1# d 7 9 ae c` mùrgqc` fkzarbalgë~rki`8
Z&V 7 v+ 7 f&v4E#e#d+ 7 d Iv E'd Ie'v , EIe
Z&V 7 0+ 7 f&046#1#9+ 7 9I0 1I> , 6I1
7 &9+&1+ , 1= 7 :%?=6
&o+ Z& V ≡ 0+ 7 0 ’ Z& V 7 :+ 7 0 ’ T&9I: 1I1+,6I1W
7 0 ’ &0+&0+ , 1=
7 0 ’ :%:06 7 :%;6?
&i+ Z&0 ≮ V ≮ 9+ 7 Z& V ≮ 9+ ’ Z& V 7 :+
7 Z& V 7 0+ $ Z& V 7 ?+ $ Z& V 7 9+
7 F&046#1#9+ $ F&?46#1#9+ $ F&9#6#1#9+
7 &9I0 1I>+,1=+ $ &9I? 1I9+,1=+ $ &9I9 1I?+,1=+
7 &&9+&1+,1=+ $ &&9+&0:+,1=+ $ &&0+&0:+,1=+
7 &:%?=6+ $ &:%19=+ $ &:%02;+ 7 :%;69
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 158/846
Nr% Fëi~lr Pqa}anl Qrç`y
9'9=
Ahagzcl )?:% Qe ago`rpqa na ?: ilgzq~`nlr`y ile~kaea 1 pqa ay~àe namai~qly`y%
Yk 0: na ay~`y ilgzq~`nlr`y ya yacaiikle`e `ca`~lrk`gae~a# z`r` yq keyzaiikùe#
¼Iqàc ay c` zrlo`okckn`n na pqa ? na c`y 0: ay~ëe namai~qly`y<
Ylcqikùe8
@pqç# V 7 ?# e 7 0:# d 7 1 { E 7 ?:% @flr` yqy~k~q{aenl ay~ly }`clray ae c`
mùrgqc` fkzarbalgë~rki` n`8
Z&V 7 ?+ 7 F&?4?:#0:#1+ 7 1I? 01I6 , ?:I0: 7 &0:+&=>91+,06>21=
7 :%9>6
Eù~aya c` nkmaraeik` ae~ra ac qyl na c` ca~r` g`{øyiqc` F { c` ca~r` gkeøyiqc` f%
Ahagzcl )?0% _azk~`gly ac ahagzcl `e~arklr# zarl `flr` ile qe cl~ay na 0::
ilgzq~`nlr`y# ?1 na c`y iq`cay ay~àe namai~qly`y# na c` ykbqkae~a g`ear`%
&`+ Qy`enl c` mùrgqc` fkzarbalgë~rki`
&o+ Qy`enl c` mùrgqc` okelgk`c ilgl qe` `zrlvkg`ikùe ` c` nky~rkoqikùe
fkzarbalgë~rki`%
Ylcqikùe8
&`+ Yqy~k~q{aenl v 7 ?# e 7 0:# d 7 ?1# E 7 0:: ae c` mùrgqc` n`8
Z& V 7 ?+ 7 F&?40::#0:#?1+ 7 ?1I? 21I6 , 0::I0:
7 &9::+&0%=62v0:0:
+ , 0%290v0:09
7 :%?;?
@pqç }agly pqa cly n`~ly yle gq{ c`rbly { ~anklyly% Yke ago`rbl# qy`enl c`
nky~rkoqikùe okelgk`c# ilgl qe` `zrlvkg`ikùe# o`yàenlely ae ac faifl na pqa ac
}`clr na E 7 0:: ay br`ena ile rac`ikùe ` e 7 0:# ae~leiay# zlnagly qy`r c`
okelgk`c ilgl qe` `zrlvkg`ikùe ` c` fkzarbalgë~rki` { n`8
Qy`enl v 7 ?# e 7 0:# z 7 d,E 7 ?1,0:: 7 %?1% Zlr cl ~`e~l#
Qy`enl c` mlrgqc` okelgk`c# o&v#e#z+ 7 eIv zv
pe'v
,v!
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 159/846
Nr% Fëi~lr Pqa}anl Qrç`y
9'92
Z& V 7 ?+ 7 O&?#0:#:%?1+ 7 0:I? &:%?1+?
&:%21+6
7 &>1+&:%:=?1+&:%0::+ 7 :%?609
El~`% Loyër}aya pqa c` nkmaraeik` ae~ra cly nly }`clray ay na ylcl %:0% Ae baear`c#
ay zlykoca nagly~r`r pqa c` nky~rkoqikùe fkzarbalgë~rki`# f&v4E#e#d+ ya `zrlvkg` `
c` nky~rkoqikùe okelgk`c# o&v4e#z+# ile z 7 d,E% Zlr rabc` baear`c# zqana qy`rya c`
nky~rkoqikùe okelgk`c ilgl qe` `zrlvkg`ikùe ` c` nky~rkoqikùe fkzarbalgë~rki`# yk
e 3 E,0:%
Gày ahagzcly na zrlocag`y na c` nky~rkoqikùe okelgk`c qy`enl ac zrlbr`g`
na ilgzq~`nlr` Gkek~`o
@orkr ac zrlbr`g` Gkek~`o a krya `8
I`ci ↟ Zrlo`okck~{ Nky~rkoq~kley↟ Okelgk`c…
Ay~l f`ia pqa `z`raxi` c` }ae~`e` na –Okelgk`c Nky~rkoq~kley‗% Ae ay~` }ae~`e`
zqe~a`r –Zrlo`okck~{‗% Ae c` }ae~`ekcc` na –Eqgoar lm ^rk`cy‗ zlear ac }`clr na e
yacaiikle`nl &~`g`ðl na c` gqay~r`+% @ykgkygl# ae c` }ae~`ekcc` na –Zrlo`okck~{ lm
Yqiiayy‗ zlear c` zrlo`okckn`n l ac zlriae~`ha &ae mlrg` naikg`c+ naya`nl% Ae c`
}ae~`ekcc` na –Kezq~ Ilcqgey‗ zlear c` ilcqge` I0 l ya` c` ilcqge` ile cly n`~ly
pqa ya pqkar`e a}`cq`r% Ae c` }ae~`ekcc` na –Lz~kle`c Y~lr`ba‗ ya zlenràe cly n`~ly
baear`cay pqa ya `cg`iae`r`e% Cqabl zlear –LD‗% Aeyabqkn`# z`r` baear`r c`y
zrlo`okckn`nay `iqgqc`n`y nae~rl na c` gkyg` }ae~`e` na –Okelgk`c
Nky~rkoq~kley‗ zqe~a`r c` –Iqggqc`~k}a Zrlo`okck~{‗ { zrlianar `eàclb`gae~a#
ilgl `rrko`%
@eàclb`gae~a# z`r` f`iar bràmki`y krya `8
Br`zf ↟ Yi`~~arzcl~ ↟ Rk~f Ileeai~ ckea# a~i%
Ae c` }ae~`e` na –Yi`~~arzcl~y Rk~f Ileeai~ Ckea‗# zlear I? l I9 ae [ {# I0 ae
V % &Ykagzra pqa ya ~aeb` `cbqe` nqn`# ileyqc~`r c` }ae~`ekcc` na –Facz‗%+
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 160/846
Nr% Fëi~lr Pqa}anl Qrç`y
9'96
Ahagzcl )??% Qe m`orki`e~a na zraikzk~`nlray acai~rly~à~kily `mkrg` pqë# ac = na
ay~a apqkzl z`r` ile~rlc`r c`y z`r~çiqc`y ile~`gke`e~ay nac `kra# ay~à namai~qlyl% Yk
ay~` `mkrg`ikùe ay ilrrai~`# aeile~r`r c`y zrlo`okckn`nay na pqa ac eøgarl na
`z`r`~ly namai~qlyly y`i`nly na qe` gqay~r` na 0: ay~ëe ae g`c ay~`nl%
&`+ Av`i~`gae~a nly `z`r`~ly ay~`ràe namai~qlyly
&o+ Iq`enl gaely nly `z`r`~ly ay~`ràe namai~qlyly
&i+ Gaely pqa qe `z`r`~l ay~`rà namai~qlyl
&n+ Ae~ra ? { 1 keicqyl { avicqyl
&a+ Z&Y +
&m+ F`iar bràmki`y na zrlo`okckn`n na mqeikùe na g`y` Z& V7v+ { na zrlo`okckn`n
`iqgqc`n`# Z& V ≮ v+
Ylcqikùe8
Z`r` lo~aear cly rayqc~`nly `za~aiknly qy`r c` ~`oc` baear`n`# pqa keicq{a c`
}`rk`oca `ca`~lrk` V &ae c` ilcqge` I0+ { c`y zrlo`okckn`nay okelgk`cay
kenk}knq`cay { c`y zrlo`okckn`nay okelgk`cay `iqgqc`n`y &ae c`y ilcqge`y I? {
I9+%
&`+ Z& V 7 ?+ 7 :%:;66 &na c` ilcqge` I?+
&o+ Z& V ≡ ?+ 7 0 ’ Z& V 3 ?+
&i+ Z& V 3 0+ 7 Z& V 7 :+ 7 :%196=
&n+ Z&? ≮ V ≮ 1+ 7 Z& V ≮ 1+ ’ Z& V ≮ 0+
7 Z& V 7 ?+ $ Z& V 7 9+ $ Z& V 7 >+ $ Z& V 7 1+
7 :%002=
Z&? 3 V 3 1+ 7 Z& V 7 >+ $ Z& V 7 9+ &na c` ilcqge` I?+
7 :%:0=6 $ :%::0;
7 :%:062
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 161/846
Nr% Fëi~lr Pqa}anl Qrç`y
9'9;
&a+ Z&Y + 7 0 &Ay~a }`clr ya lo~kaea na c` yqg`~lrk` na ~ln`y c`y zrlo`okckn`nay na
mqeikùe na g`y` l zrlo`okckn`nay okelgk`cay kenk}knq`cay+# ay~l ay8
Z&Y + 7 &:%196=01 $ :%9>92;2 $ :%:;621: $ :%:00=6:; $ :%::0626 $ :%:::0>> $
:%:::::6+ ≆ 0
&m+ C`y Mkbqr`y 9%> &`+ { &o+ gqay~r`e cly bràmkily na c`y zrlo`okckn`nay okelgk`cay
kenk}knq`cay# ae mqeikùe na c` }`rk`oca `ca`~lrk V {# c`y zrlo`okckn`nay okelgk`cay
`iqgqc`n`y# ae mqeikùe na c` }`rk`oca `ca`~lrk V# rayzai~k}`gae~a%
^@OC@ 9%1% ^`oc` gly~r`enl cly }`clray na c` }`rk`oca `ca`~lrk` v &ilcqge` I0+#
c` zrlo`okckn`nay okelgk`cay kenk}knq`cay Z& V 7v+ { c` zrlo`okckn`nay okelgk`cay
`iqgqc`n`y Z& V ≮ v+ &ilcqge` I9+%
&`+ &o+
] rk oca ca ~lrk V
Z & V 7 v +
=1>9?0:
=1>9?0:
:%=
:%1
:%>
:%9
:%?
:%0
:%:
:%=
:%1
:%>
:%9
:%?
:%0
:%:
Br mki gly~r enl c zrlo okckn n Z&V37v+ }y% } rk oca ca ~lrk V
Mkbqr` 9%>% Aypqag`y gly~r`enl cly rayqc~`nly na ay~a ahagzcl% C` mkbqr` &`+
gqay~r` c` bràmki` na Z&V7v+ }y% }`rk`oca `ca`~lrk` V {# c` mkbqr` &o+# gqay~r` c`
bràmki` na Z&V 37 v+% &Ac`olr`ikùe zrlzk`+
] rk oca ca ~lrk v
Z r l o ` o k c k n ` n a y ` i q g q c ` n ` y
=1>9?0:
0%:
:%;
:%6
:%2
:%=
:%1
Mkbqr gly~r enl c br mki na Z&V7v+ }aryqy } rk oca ca ~lrk v
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 162/846
Nr% Fëi~lr Pqa}anl Qrç`y
9'>:
Ahagzcly `zcki`nly ` c` nky~rkoqikùe fkzarbalgë~rki` qy`enl ac zrlbr`g` na
ilgzq~`nlr` Gkek~`o
@orkr ac zrlbr`g` Gkek~`o a krya `8
I`ci ↟ Zrlo`okck~{ Nky~rkoq~kley↟ F{zarbalga~rki
Ay~` g`eklor` `ora c` }ae~`e` –F{zarbalga~rki Nky~rkoq~kle‗% Ae ay~` }ae~`e`
zqe~a`r –Zrlo`okck~{‗% Z`r` i`ciqc`r c`y zrlo`okckn`nay f{zarbalgë~rki`y na
mqeikùe na g`y` ae c` }ae~`ekcc` na –Zlzqc`~kle ykxa &E+‗ zlear ac }`clr na c`
zloc`ikùe gqay~ra`n` &E+% Ae c` }ae~`ekcc` na –Yqiiayy ke Zlzqc`~kle &G+‗ zlear
ac eøgarl na ëvk~ly &d+% Ae c` }ae~`ekcc` na –Ykgzca Ykxa &e+‗ zlear c` gqay~r`
yacaiikle`n` &e+% Ae c` }ae~`ekcc` na –Kezq~ ilcqgey‗ zlear cly }`clray pqa ya
}`{`e ` a}`cq`r &v+% Ae c` }ae~`ekcc` na –Lz~kle`c Y~lr`ba‗ ke~rlnqikr c` ilcqge`
nlena ya }`{`e ` `cg`iae`r cly }`clray baear`nly%
Ahagzcl )?9% @yøg`ya pqa qe` zloc`ikùe na 0: ganknlray na zF &pqa gknae c`
`iknax { c` `ci`ckekn`n na ylcqikleay pqçgki`y+ ile~kaea > qekn`nay namai~qly`y
&ëvk~ly `rok~r`rk`gae~a+% Yk qe` gqay~r` na 9 ganknlray ya yacaiikle` `c `x`r# yke
raagzc`xl# aeile~r`r c`y ykbqkae~ay zrlo`okckn`nay8
&`+ Av`i~`gae~a 0 `z`r`~l na zF ay~`rà namai~qlyl
&o+ Nly `z`r`~ly ay~`ràe namai~qlyly
&i+ ^ray `z`r`~ly na zF rayqc~`r`e namai~qlyly na c` gqay~r` yacaiikle`n`
&n+ @ cl gày ? `z`r`~ly ay~`ràe namai~qlyly
&a+ F`iar bràmki`y z`r` Z& V 7 v+ { Z& V ≮ v+
Ylcqikùe8
0% Zrkgarl ya ke~rlnqiae cly }`clray na c` }`rk`oca `ca`~lrk` &v+ ae c` ilcqge` I0
?% Aeyabqkn`# ~aeagly pqa knae~kmki`r c`y }`rk`ocay pqa ya ke~rlnqikràe ae ac
glnacl fkzarbalgë~rkil% @pqç# d 7 ># E 7 0:# e 7 9# v 7 :# 0# ?# 9# >%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 163/846
Nr% Fëi~lr Pqa}anl Qrç`y
9'>0
9% @flr`# ke~rlnqikr cly }`clray na E# e { d# ilgl ya kenkiù `rrko` { ac zrlbr`g`
Gkek~`o baear` c` ^@OC@ 9%1 na `o`hl%
^@OC@ 9%1% ^`oc` gly~r`enl cly }`clray na c`y nky~rkoqikleay fkzarbalga~rki`y
kenk}knq`cay { `iqgqc`n`y ae mqeikùe na c` }`rk`oca `ca`~lrk V %
>% Z`r` raylc}ar cly keikyly &`+'&a+# ay~l ya zqana f`iar qy`enl ac glnacl
fkzarbalgë~rkil# f&v8E#e#d+ 7 d Iv E'd Ie'v , EIe# cly n`~ly na c` ^@OC@ 9%1 l c`y
bràmki`y% Zlr ahagzcl yk ya qy` c` mùrgqc` ya yqy~k~q{ae cly }`clray na d# E { e {
cqabl ya yqy~k~q{ae cly }`clray na v ae c` mùrgqc` fkzarbalgë~rki`8
f&v40:# 9#>+ 7>I
v
0:'>I
9'v,
0:I
9
Qe` }ax faifl ay~l ya yqy~k~q{a cly }`clray na v 7 :# 0# ?# 9# >% Ay~a zrliankgkae~l#
yke ago`rbl# ay gq{ c`rbl { ~anklyl% Zarl yk qy`gly cly }`clray na c` ^@OC@ 9%1
ay~l ya ykgzckmki` na ylorag`ear`%
>% C`y ylcqikleay yle8
&`+ Z& V 7 0+ 7 >I0 =I? , 0:I9 7 :%1:: &l na c` ilcqge` I?+
&o+ Z& V 7 ?+ 7 >I? =I0 , 0:I9 7 :%9:: &l na c` ilcqge` I?+
&i+ Z& V 7 9+ 7 >I9 =I: , 0:I9 7 :%:99 &l na c` ilcqge` I?+
?
&n+ Z& V ≮ ?+ 7 Ω f&v40:#9#>+ 7 :%1::: $ :%0==2 $ :%9::: 7 :%;==2 &l na I9+v7:
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 164/846
Nr% Fëi~lr Pqa}anl Qrç`y
9'>?
&a+ Ae iq`e~l ` c baear`ikùe na c`y mkbqr`y rapqarkn`y zlr ac zrlocag`# ykbqkaenl
c`y key~rqiikleay `e~arklray ya baear`e c`y Mkbqr`y 9%1 &`+ { &o+ yað`c`n`y `o`hl%
&`+ &o+
]`rk`oca `ca~lrk` v
Z & V 7
v +
9%:?%1?%:0%10%::%1:%:
:%1
:%>
:%9
:%?
:%0
:%:
Mkbqr` gly~r`enl c` br`mki` na Z&V7v+ }y% }`rk`oca `ca`~lrk` v
]`rk`oca `ca~lrk` v
Z & V 7 3
v +
9%:?%1?%:0%10%::%1:%:
0%:
:%;
:%6
:%2
:%=
:%1
:%>
:%9
:%?
:%0
Mkbqr` gly~r`enl c` br`mki` na Z&V37v+ }y% }`rk`oca `ca`~lrk` v
Mkbqr` 9%1% C` mkbqr` &`+ gqay~r` c` nky~rkoqikùe fkzarbalgë~rki` na }`rk`ocay
kenk}knq`cay Z& V 7v+ }y% }`rk`oca `ca`~lrk` v { c` mkbqr` &o+ gqay~r` c`y }`rk`ocay
`iqgqc`n`y na c` nky~rkoqikùe fkzarbalgë~rki` Z& V ≮ v+ }y% }`rk`oca `ca`~lrk` V %
&Ac`olr`ikùe zrlzk`+%
Ahagzcl )?>% Ae qe` aeiqay~` qek}aryk~`rk` faif` ` ?> ay~qnk`e~ay nac qc~kgl `ðl
ra}ac` pqa i`yk ac 1: na ay` zloc`ikùe na ay~qnk`e~ay railgkaen` ~lg`rya iq`enl
gaely qe` l nly iar}ax`y nk`rk`gae~a# z`r` ay~qnk`r gahlr% Yk ya yacaiikle`e
`ca`~lrk`gae~a 00 na ay~ly ay~qnk`e~ay { ya cay zrabqe~` pqa lzke`e na ay~l# ay~kg`r
cl ykbqkae~a8
&`+ C` zrlo`okckn`n na pqa# ylc`gae~a# > ay~qnk`e~ay ya`e na ay~a z`raiar%
Ylcqikùe8
Qy`enl c` nky~rkoqikùe fkzarbalgë~rki` ile E 7 ?># e 7 00# d 7 Ez 7 &:%1:+&?>+ 7
0? { V 7 > { yqy~k~q{aenl cly }`clray ae c` mlrgqc` fkzarbalgë~rki`8
f&v4E#e#d+ 7 d Iv E'd Ie'v , EIe#
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 165/846
Nr% Fëi~lr Pqa}anl Qrç`y
9'>9
ely n`8 F&>4?>#00#0?+ 7 0?I> ?>'0?I00'> , ?>I00
7 &>;1+&2;?+,?#>;=#0>>
7 :%?=
El~`8 F`{ gë~lnly z`r` baear`r ~`oc`y na nky~rkoqikleay okelgk`cay
&`iqgqc`n`y a kenk}knq`cay+ qy`enl ac zrlbr`g` ilgzq~`rkx`nl Gkek~`o
Avky~ae gë~lnly z`r` baear`r ~`oc`y okelgk`cay `iqgqc`n`y l ~`oc`y okelgk`cay
kenk}knq`cay z`r` iq`cpqkar ~`g`ņl na e { na zrlo`okckn`nay z% Ay~l ya f`ia#
zlrpqa ay kgzrài~kil zlear ~lnly cly }`clray na e { na z ae c`y ~`oc`y na c`y
nkmarae~ay nky~rkoqikleay% Qy`enl ac Gkek~`o# ya zqanae baear`r ~`oc`y na
zrlo`okckn`nay okelgk`cay `iqgqc`n`y z`r` iq`cpqkar ~`g ņl na e { na z% Z`r`
ay~l# zrlianar ilgl8
I`ciqc`~lr ↟ Zrlo`okck~{ Nky~rkoq~kley↟ Okelgk`c…
Ae c` }ae~`e` na –Okelgk`c Nky~rkoq~kle‗ pqa `z`raia zqe~a`r –Iqggqc`~k}a
zrlo`okck~{‗% &Ay~l ya f`ia nayzqëy na ke~rlnqikr cly }`clray na c` }`rk`oca `ca`~lrk`
V ae c` ilcqge` I0+% Ae c`y }ae~`ekcc`y na –Eqgoar lm ~rk`cy‗ { –Zrlo`okck~{ lm
yqiiayy‗ zlear ac ~`g`ņl na e { ac }`clr na z# rayzai~k}`gae~a% Ay~l baear`rà c`y
zrlo`okckn`nay okelgk`cay `iqgqc`n`y# gkyg`y pqa ya `cg`iaearàe ae –Lz~kle`c
y~lr`ba‗ l ae c` ilcqge` I?%
@eàclb`gae~a# ya zqana baear`r qe` ~`oc` na zrlo`okckn`nay okelgk`cay
kenk}knq`cay z`r` iq`cpqkar ~`g`ðl na e l na z% Z`r` ay~l zrlianar ilgl8
I`ciqc`~lr ↟ Zrlo`okck~{ Nky~rkoq~kley↟ Okelgk`c…
Ae c` }ae~`e` na –Okelgk`c Nky~rkoq~kle‗ pqa `z`raia# zqe~a`r –Zrlo`okck~{‗% &Ay~l
ya f`ia nayzqëy na ke~rlnqikr cly }`clray na c` }`rk`oca `ca`~lrk` V ae c` ilcqge`
I0+% Ae c`y }ae~`ekcc`y na –Eqgoar lm ~rk`cy‗ { –Zrlo`okck~{ lm yqiiayy‗ zlear ac
~`g`ðl naya`nl na e { na z# rayzai~k}`gae~a% Ay~l baear`rà c`y zrlo`okckn`nay
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 166/846
Nr% Fëi~lr Pqa}anl Qrç`y
9'>>
okelgk`cay kenk}knq`cay# gkyg`y pqa ya `cg`iaearàe ae –Lz~kle`c y~lr`ba‗ l ae c`
ilcqge` I9# ay~l ay# yk {` ya qyù c` ilcqge` I? z`r` ac `cg`iae`ha na c`y
zrlo`okckn`nay okelgk`cay `iqgqc`n`y%
Ahagzcl )?1% ^aùrki`gae~a# ikar~` mlrg` na nayeq~rkikùe liqrra ae ac 01 na
zaryle`y yke pqa ya nae iqae~` na ayl% Ay~` mlrg` na nayeq~rkikùe el ya naoa ` pqa
el ya ilg` cl yqmkikae~a# ykel ` yk~q`ikleay ae pqa ac iqarzl el `ykgkc` cly
eq~rkae~kay# yke kgzlr~`r iq`e~l l ilgl ya ilg`% Ay~l ay naoknl ` c` `c~ar`ikùe
pqçgki` na c` y`ebra zlr }kn` `e~ke`~qr`c% Ykaenl `yç na~argke`r c`y ykbqkae~ay
zrlo`okckn`nay z`r` qe` gqay~r` na 1 zaryle`y%
&`+ Ekebqe` zaryle` cl ~kaea
&o+ Iq`enl gaely ? zaryle`y cl ~kaeae
&i+ Ae~ra ? { > cl ~kaeae# keicqyk}`gae~a
Ylcqikùe8
Qy`enl ac Gkek~`o ya baear` c` ~`oc` na `o`hl%
^@OC@ 9%=% ^`oc` gly~r`enl c`y zrlo`okckn`nay `iqgqc`n`y a kenk}knq`cay%
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ ]`rk`oca ca`~lrk` V Zrlo`okckn`nay okelgk`cay iqgqc`n`y Zrlo`okckn`nay okelgk`cay kenk}knq`cay%
: :%1;:>; :%1;:>;
0 :%;061> :%9?6:1
? :%;;0>> :%:2?;:
9 :%;;;1> :%::60:
> :%;;;;; :%:::>1
1 0%::::: :%::::0
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
&`+ Z&V 7 :+ 7 :%1;:>;
&o+ Z&V ≡ ?+ 7 0 ’ :%;061> 7 :%:601
&i+ Z&? ≮ V ≮ >+ 7 :%;;;; ' :%;061> 7 :%:601
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 167/846
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 168/846
Nr% Fëi~lr Pqa}anl Qrç`y
9'>=
&`+ C` zrlo`okckn`n na pqa na c`y ykbqkae~ay ?: m`cc`y `c gaely 0: m`cc`y ya nao`e `c
arrlr nac lzar`nlr%
&o+ C` zrlo`okckn`n na pqë# el gày na > na ?: m`cc`y ya nao`e ` arrlr nac lzar`nlr%
9%1% Na `iqarnl ile qe razlr~a zqocki`nl ae c` ra}ky~` Z`r`na# qe` aeiqay~` ` ek}ac
e`ikle`c na c` Qek}arykn`n na Gkifkb`e ` ay~qnk`e~ay qek}aryk~`rkly nac øc~kgl `ðl#
ra}ac` pqa i`yk ac 1: mqg`e g`rkfq`e`% Yk ya yacaiikle`e 0? ay~qnk`e~ay
`ca`~lrk`gae~a { ya cay zkna yq lzkekùe `c rayzai~l# aeile~r`r c` zrlo`okckn`n na
pqa ac eøgarl pqa mqg`e g`rkfq`e` ~lnly cly nç`y ya`8
&`+ Ae~ra 2 { ; keicqyl &:%9=6+
&o+ @ cl gày 1 &:%962?+
&i+ El gaely na 6 &0 ' Z& V ≮ 2++
9%=% Ae qe ay~qnkl na fkbkaea kenqy~rk`c { yabqrkn`n# ya y`oa pqa c` zrlo`okckn`n na
pqa qe lzar`nlr na c`y zc`e~`y na ra`i~lray eqica`ray ya` `nki~l ` c`y nrlb`y
farlki`y ay na :%:1% ¼Iqàc ay c` zrlo`okckn`n na pqa av`i~`gae~a 1 na cly
ykbqkae~ay 0:: lzar`nlray ya`e `nki~ly ` cly e`riù~kily< Qy`r c` nky~rkoqikùe
okelgk`c { c` elrg`c z`r` raylc}ar ay~a kgzlr~`e~a { nacki`nl zrlocag`%
9%2% Qe ay~qnkl na fkbkaea kenqy~rk`c av`gkeù c`y `i~k~qnay na cly ~r`o`h`nlray
kenqy~rk`cay `iari` na cly `e~knazrayk}ly% Ay~` ke}ay~kb`ikùe ra}acù pqa#
`zrlvkg`n`gae~a# ac 2: na cly ~r`o`h`nlray ae~ra}ky~`nly iraae pqa cly
`e~knazrayk}ly# ae ra`ckn`n# el iqr`e e`n`# ykel pqa ylcl aeiqorae ac zrlocag` ra`c
{ el `{qn`e ` raylc}ar cly zrlocag`y na ~r`o`hl% Na `iqarnl ` ay~` ke}ay~kb`ikùe#
¼Iqàc ay c` zrlo`okckn`n na pqa `c gaely 9 na cly ykbqkae~ay 1 ~r`o`h`nlray
yacaiikle`nly# `ca`~lrk`gae~a# ya`e na ay~` lzkekùe< &%692+
9%6% Ile rayzai~l `c zrlocag` `e~arklr 9%2# yk V razrayae~` ac eøgarl na
~r`o`h`nlray na c` kenqy~rk` pqa iraa pqa cly `e~knazrayk}ly el `{qn`e ` raylc}ar
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 169/846
Nr% Fëi~lr Pqa}anl Qrç`y
9'>2
cly zrlocag`y aglikle`cay nac ~r`o`hl# ykel pqa n`e ylc`gae~a qe` ylcqikùe
z`ck`~k}` `c zrlocag` na c`y nazraykleay aglikle`cay4 ykaenl `yç# ae~leiay#
aeile~r`r ac zrlgankl { c` }`rk`ex`# iq`enl ya yacaiikle`e `ca`~lrk`gae~a 1
zaryle`y na qe` gqay~r` na ?:%
9%;% Ae qe` ke}ay~kb`ikùe na fkbkaea kenqy~rk`c { yabqrkn`n# ac kebaekarl aei`rb`nl
nac naz`r~`gae~l na yabqrkn`n `mkrg` pqa# ylcl ac >: na ~lnly cly ~r`o`h`nlray
qy`e i`yily na yabqrkn`n iq`enl `cgqarx`e ae ac cqb`r nac ~r`o`hl% Yqzlekaenl
pqa ay~` `mkrg`ikùe ya` ilrrai~`# aeile~r`r c` zrlo`okckn`n na pqa > na cly
ykbqkae~ay = ~r`o`h`nlray na c` kenqy~rk`# acabknly# `ca`~lrk`gae~a# qyae cly i`yily
na yabqrkn`n# gkae~r`y ilgae ae ac cqb`r nac ~r`o`hl% &:%096+
9%0:% Qe` ilgz`ðç` iley~rqi~lr` na zraikzk~`nlray acai~rly~à~kily y`oa pqa# ae
zrlgankl# ac ?; na ay~a apqkzl na ile~rlc na z`r~çiqc`y rapqarkràe na
raz`r`ikleay nayzqëy na qe `ðl na qy`rya% Yk ya yacaiikle`e# `ca`~lrk`gae~a# ?:
zraikzk~`nlray acai~rly~à~kily# na c` zrlnqiikùe ~l~`c# aeile~r`r c` zrlo`okckn`n pqa8
&`+ @c gaely 1 zraikzk~`nlray rapqkar`e na raz`r`ikleay nayzqëy na qe `ðl%
&o+ Av`i~`gae~a 1 na ay~`y qekn`nay na ile~rlc na c` ile~`gke`ikùe `~glymërki`
rapqkar`e raz`r`ikùe nayzqëy na qe `ðl%
9%00% Ae qe ay~qnkl na `flrrl na aearbç`# ya `rbqgae~` pqa# ae ac >: na c`y
i`cam`iikleay `i~k}`n`y ile aearbç` ylc`r# c` iqae~` zlr yar}kikl o`h`
ileyknar`ocagae~a% Na `iqarnl ` ay~a `rbqgae~l# ¼Iqàc ay c` zrlo`okckn`n na pqa
c` iqae~` na yar}kikl o`ha# ae iq`enl gaely 1 na qe` gqay~r` na 1: i`cam`iikleay<
F`iar ay~a zrlocag` qy`enl c` nky~rkoqikùe okelgk`c { nayzqëy c` nky~rkoqikùe
elrg`c% Ilgz`r`r cly rayqc~`nly% &Okelgk`c 7 :%;;6# elrg`c 7 :%;;62+
9%0?% F`iar ac gkygl zrlocag` &9%00+ zarl qy`enl 1: ile e 7 ?1 { Z& V ≡ 1+%
9%09% Ya n`e cly ykbqkae~ay n`~ly8 e 7 01# z 7 :%>% I`ciqc`r c` zrlo`okckn`n na pqa ac
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 170/846
Nr% Fëi~lr Pqa}anl Qrç`y
9'>6
}`clr na c` }`rk`oca `ca`~lrk` V ya` av`i~`gae~a kbq`c ` >% F`iar ay~l8
&`+ Qy`enl c` nky~rkoqikùe okelgk`c% &:%0?=6+
&o+ Qy`enl c` nky~rkoqikùe elrg`c ilgl `zrlvkg`ikùe% &:%0?0>+
9%0>% Ya `rbqgae~` pqa ae ac =: na c`y key~`c`ikleay na i`cam`iikùe ylc`r c`
iqae~` zlr ileiaz~l na yar}kikl ya ranqia ae `c gaely qe ~arikl% Ae ileyle`eik`
ile ay~l# ¼Iqàcay yle c`y zrlo`okckn`nay na pqa c` iqae~` na yar}kikl ya ranqxi` ae
`c gaely qe ~arikl ae<8
&`+ Iq`~rl na ikeil key~`c`ikleay%
&o+ @c gaely iq`~rl na ikeil key~`c`ikleay%
9%01% Ae ay~qnkly na kebaekarç` ik}kc# yk c` zrlo`okckn`n na pqa ikar~` ilcqge` na
`c` `eif` i`kb` o`hl qe` i`rb` `vk`c n`n` ay na :%:1# i`ciqc`r c` zrlo`okckn`n f`{
na pqa ae~ra 0= ilcqge`y na aya ~kzl8
&`+ ¼I`kb`e iq`enl gày nly< &:%;120+
&o+ ¼I`kb`e `c gaely iq`~rl< &%::2:+
9%0=% C` zrlo`okckn`n na pqa ikar~` ic`ya na ilgzleae~a rayky~` qe` zrqao` na
iflpqa ay na :%11% Aeile~r`r c` zrlo`okckn`n na pqa ylora}k}`e# av`i~`gae~a# ? na
cly ykbqkae~ay > ilgzleae~ay pqa ya zrqaoae%
9%02% C` zrlo`okckn`n na pqa qe z`ikae~a ya raiqzara na qe zrlocag` i`rnk`il ay
na :%>% Yk ya yacaiikle` `ca`~lrk`gae~a qe` gqay~r` na 01 z`ikae~ay ile yçe~lg`y
na zrlocag`y i`rnç`ily# ¼Iqàc ay c` zrlo`okckn`n na8
&`+ Z& V ≡ 0:+ &:%:996+
&o+ Z&9 ≮ V ≮ 6+ &:%061;+
&i+ Z& V 7 1+ &:%061;+
&n+ Z&9 3 V 3 6+ &:%=;=>+
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 171/846
Nr% Fëi~lr Pqa}anl Qrç`y
9'>;
9%06% C` zrlnqiikùe nk`rk` na 61: z`r~ay m`orki`n`y ile~kaea 1: z`r~ay pqa el
iqgzcae ile cly rapqarkgkae~ly nac ickae~a% Nac cl~a ya ayilbae ? z`r~ay#
`ca`~lrk`gae~a# yke raagzc`xl% Ya` V ac eøgarl na z`r~ay na c` gqay~r` pqa el
iqgzcae ile cly rapqarkgkae~ly% Ykaenl `yç# ¼Iqàc ay c` mqeikùe na c` nky~rkoqikùe
`iqgqc`n` na V <
9%0;% C` a~`z` na qe` ~ariar` `car~` na yglb ae c` ikqn`n na Gavkil f` yknl n`n`#
ae c` iq`c ya ke}lcqir` ` 1: kenqy~rk`y ile~`gke`e~ay% Qe keyzai~lr na Z_LMAZ@
}kyk~`rà 0: kenqy~rk`y yacaiikle`n`y `ca`~l`rk`gae~a# z`r` keyzaiikle`rc`y zlr c`y
}klc`ikleay ` c`y cabkyc`ikleay `gokae~`cay# pqa c`y kenqy~rk`y zqnkar`e ay~`r
ilga~kaenl% ¼Iqàc ay c` zrlo`okckn`n na pqa 01 na c`y kenqy~rk`y ke}lcqir`n`y ay~ëe
}klc`enl# iq`enl gaely# qe` cabkyc`ikùe `gokae~`c< &?1I V ² 1:'?1I0:' V , 1:I0:+
9%?:% Qe m`orki`e~a na cc`e~`y z`r` `q~ly razlr~` pqa ae~ra qe i`rb`gae~l na =#:::
cc`e~`y na c` g`ri` Blln{a`r ragk~kn`y ` qe nky~rkoqknlr cli`c# 0?: cc`e~`y na ay~`
g`ri` ay~àe qe zlil namai~qly`y% Yk qe gl~lrky~` ilgzr`# `c `x`r# 0: na ay~`y
cc`e~`y# ¼Iqàc ay c` zrlo`okckn`n na pqa > na ay~`y cc`e~`y ay~ëe qe zlil n`ð`n`y<
F`iar ay~a zrlocag` qy`enl c` nky~rkoqikùe fkzarbalgë~rki` { c` okelgk`c%
9%?0% Qe m`orki`e~a na `z`r`~ly na glek~lral `gokae~`c &IL+ ile~kaena pqa ylcl ac
0: na ay~ly `z`r`~ly rapqkarae na raz`r`ikùe nae~rl nac zarklnl na b`r`e~ç` na qe
`ðl% Yk ya y`i` qe` gqay~r` `c `x`r na 0: na ay~ly `z`r`~ly# ae~leiay# ykaenl `yç#
i`ciqc`r cly ykbqkae~ay aeqeik`nly8
&`+ C` zrlo`okckn`n na pqa iq`enl gaely 9 na cly 0: `z`r`~ly rapqkar`e na
raz`r`ikùe nae~rl nac zarklnl na b`r`e~ç`% &:%:2:?+
&o+ Yk 1 na cly 0: `z`r`~ly rapqkrkarle na raz`r`ikùe ae ac zrkgar `ðl# ¼`zl{`rç`
ay~l l ramq~`rç` c` ile~aeikùe nac m`orki`e~a< ¼Pqë ykbekmki`nl ~kaea c` zrlo`okckn`n
i`ciqc`n`# &ae iq`e~l ` c` ile~aeikùe nac m`orki`e~a na pqa ylcl ac 0: na cly
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 172/846
Nr% Fëi~lr Pqa}anl Qrç`y
9'1:
`z`r`~ly rapqkarae na raz`r`ikùe nae~rl na qe `ðl+# iq`enl c` zrlo`okckn`n na pqa
iq`cpqkar` na cly `z`r`~ly rapqkar` na raz`r`ikùe ae ac zarklnl na b`r`e~ç`# ay na
:%0:< &Ac rayqc~`nl ay kbq`c ` :%::0= { n`nl pqa c` zrlo`okckn`n ay gq{
zapqað` ya raif`x` c` ile~aeikùe nac m`orki`e~a+
9%??% Ay~a ay qe ahagzcl ~aùrkil na qe zrlocag` na nayeq~rkikùe# pqa yqmrkrç`e
gqif`y zaryle`y aemarg`y# yke n`rya iqae~` pqa ay~àe nayeq~rkn`y% @pqç# yke
ago`rbl# ac `q~lr na ay~a ahagzcl ya ramkara ` qe ~kzl na eq~rkikùe namai~qly` pqa
el ay~à rac`ikle`nl ile c` nayeq~rkikùe ~r`nkikle`c naoknl ` c` m`c~` na `ckgae~ly%
Ae ay~a ile~av~l# ac `q~lr ya ramkara ` qe ~kzl na nayeq~rkikùe zlil ileliknl zlr c`
gankike` ~r`nkikle`c &ykel zlr c` e`~qrlz`~ç`+# ilgl ae ac i`yl nac iàeiar baeërkil#
ae ac iq`c c` zaryle` `mai~`n` ay~à nayeq~rkn`# el zlr el ilgar# ykel zlrpqa ac
lrb`ekygl el zqana `ykgkc`r cly `ckgae~ly# yke kgzlr~`r iq`e~l l ilgl ya ilg`%
&Ay~l ~`c }ax ya nao` ` pqa c`y zaryle`y pqa yqmrae na cly yçe~lg`y nac iàeiar
ykagzra ay~àe i`ey`n`y+% Na `iqarnl ` c` e`~qrlz`~ç`# ay~l liqrra ilgl rayqc~`nl
na c` `c~ar`ikùe pqçgki` na c` y`ebra# li`ykle`nl zlr }kn` `e~ke`~qr`c% Ykaenl `yç#
ae ay~a ahagzcl }`gly ` ileyknar`r qe i`yl fkzl~ë~kil na nayeq~rkikùe el
~r`nkikle`c# rac`ikle`nl ile zaryle`y pqa yqmrae aemargan`nay baeërki`y# { pqa ae
ay~a zrlocag`# ay~a ~kzl na nayeq~rkikùe ay nac ;:% Ykaenl `yç na~argke`r c`y
zrlo`okckn`nay z`r` cly ykbqkae~ay i`yly na nayeq~rkikùe# z`r` qe ~`g ņl `ca`~lrkl
na 9: zaryle`y# yk8
&`+ ^ln`y c`y zaryle`y aemarg`y ~kaeae zrlocag`y na nayeq~rkikùe%
&o+ Iq`enl gaely ?2 zaryle`y aemarg`y ay~àe nayeq~rkn`y%
&i+ O`hl ay~`y ilenkikleay ekebqe` zaryle` ay~à nayeq~rkn`%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 173/846
Nr% Fëi~lr Pqa}anl Qrç`y
9'10
Yqbaraeik`8 Z`r` raylc}ar ay~a zrlocag`# baear`r qe` ~`oc` na zrlo`okckn`nay
`iqgqc`n`y a kenk}knq`cay# zqay~l pqa cly }`clray na e { na z# na ay~a ahagzcl# yle
br`enay# { el `z`raiae ae c`y ~`oc`y okelgk`cay n`n`y zlr cly ~av~ly na ay~`nçy~ki`%
9%?9% Qe` aeiqay~` ` ikar~` qek}arykn`n# na qe z`çy nac fagkymarkl elr~a faif` ` ?:
ay~qnk`e~ay nac øc~kgl `ðl ra}ac` pqa# i`yk ac >: na ay` zloc`ikùe na ay~qnk`e~ay
`zrqao` ac ileyqgl nk`rkl na c` g`rkfq`e`% Yk ya yacaiikle`e `c `x`r 0: na ay~ly
ay~qnk`e~ay { ya cay zkna yq lzkekùe `c rayzai~l# i`ciqc`r cl ykbqkae~a8
&`+ ¼Iqàc ay c` zrlo`okckn`n na pqë# ylc`gae~a# 9 na cly ay~qnk`e~ay ya`e na ay~`
lzkekùe< &:%?>+
&o+ ¼Iqàc ay c` zrlo`okckn`n na pqa iq`enl gaely 0# zarl gaely na 9 ay~qnk`e~ay
ya`e na ay~` lzkekùe< &:%:21+
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 174/846
Nr% Fëi~lr Pqa}anl Qrç`y
>'0
I@ZK^QCL >
Nky~rkoqikùe na Zlkyyle
@zcki`ikleay na c` nky~rkoqikùe na Zlkyyle%' Ilenkikleay pqa ya rapqkarae z`r`
`zcki`r c` nky~rkoqikùe na Zlkyyle%' Mqeikleay zrlo`okcçy~ki`y na c` mqeikùe na
Zlkyyle%' @zcki`ikùe na c` nky~rkoqikùe na Zlkyyle nae~rl na yqy zrlzkly
~ërgkely { ilgl qe` `zrlvkg`ikùe ` c` nky~rkoqikùe okelgk`c%' Zrlzkan`nay
na c` nky~rkoqikùe na Zlkyyle%' Zrlocag`y na c` nky~rkoqikùe na Zlkyyle qy`enl
ac zrlbr`g` Gkek~`o%
C` nky~rkoqikùe Zlkyyle ay qe` nky~rkoqikùe na zrlo`okckn`n nkyira~`# zlrpqa ya
mlrg` ile~`enl `cbl% C` nky~rkoqikùe na Zlkyyle mqa nay`rrlcc`n` zlr ac mr`eiëy
Ykgale Naeky Zlkyyle# pqkëe c` nayirkokù ae 0692%
C` nky~rkoqikùe na Zlkyyle ya zqana ileyknar`r ilgl qe` cùbki` na
zrlo`okckn`n nanqi~k}`# ae mlrg` `eàclb` ` c` nky~rkoqikùe okelgk`c# zlrpqa ae ac
iàciqcl na c`y zrlo`okckn`nay ya }` nac ~l~`c ` c` z`r~a% Ay~l ay# zlrpqa ykagzraileliagly c` zrlo`okckn`n nac ayz`ikl gqay~r`c# c` iq`c ykagzra ay kbq`c ` 0 &ac
~l~`c l ilehqe~l+%
C` nky~rkoqikùe na Zlkyyle ~`gokëe zqana yar aemli`n` ilgl qe` mlrg`
ckgk~`e~a na c` nky~rkoqikùe okelgk`c# ay naikr# ilgl qe` `zrlvkg`ikùe na c`
okelgk`c# ay~l ay# iq`enl cly iàciqcly okelgk`cay yle gq{ c`rbly { ~anklyly% Zarl#
gày kgzlr~`e~a ~ln`}ç`# c` nky~rkoqikùe na Zlkyyle# ~`gokëe zqana yar aemli`n`
nae~rl na yqy zrlzkly ~ërgkely l naraifly%
C` nky~rkoqikùe na Zlkyyle ~kaea `zcki`ikleay ` qe` br`e }`rkan`n na
zrliayly mçykily4 ilgl rayqc~`nl na ay~l# ae c` gkyg` mlrg` pqa c` nky~rkoqikùe
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 175/846
Nr% Fëi~lr Pqa}anl Qrç`y
>'?
elrg`c { c` okelgk`c# c` nky~rkoqikùe na Zlkyyle ay qe` na c`y nky~rkoqikleay gày
qy`n`y% C` nky~rkoqikùe na Zlkyyle `zcki` ` c` liqrraeik` na `cbøe a}ae~l `ca`~lrkl
V # ylora qe ke~ar}`cl ayzaikmki`nl# nlena ac ke~ar}`cl zqana yar ~kagzl# nky~`eik`#àra`# }lcqgae# a~i%
Ae iq`e~l ` c`y nkmaraeik`y ae~ra c` nky~rkoqikùe na Zlkyyle { c` nky~rkoqikùe
okelgk`c# c` nky~rkoqikùe okelgk`c ay `mai~`n` zlr ac ~`g`ðl na c` gqay~r` e { c`
zrlo`okckn`n z# gkae~r`y pqa# c` nky~rkoqikùe na Zlkyyle ay `mai~`n` zlr ac
zrlgankl ±% @nagày# c` nky~rkoqikùe okelgk`c ~kaea }`clray zlykocay na v 7 :# 0# ?#
9#%%%# e# gkae~r`y pqa c` Zlkyyle ~kaea }`clray zlykocay na v 7 :# 0# ?# 9#%%%%`n
kemkek~qg# ay naikr yke ekebøe cçgk~a yqzarklr% C` Mkbqr` >%: na `o`hl gqay~r` c`
nky~rkoqikùe na Zlkyyle%
Mkbqr` >%:% Bràmki` gly~r`enl c` nky~rkoqikùe na Zlkyyle% &Ac`olr`ikùe zrlzk`+
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 176/846
Nr% Fëi~lr Pqa}anl Qrç`y
>'9
@zcki`ikleay na c` nky~rkoqikùe na Zlkyyle
0% C`y `zcki`ikleay na c` nky~rkoqikùe zqanae yar aemli`n`y ` ay~qnk`r ac eøgarl na
~ùvkily aeile~r`nly ae qe }lcqgae na `kra agk~knl zlr qe` kenqy~rk`
&ile~`gke`ikùe nac `kra+% L~r`y `zcki`ikleay yle ae c` ga~alrlclbç`# z`r` aeile~r`r
c` mraiqaeik` kgzra}ky~` na ~agzay~`nay# ikicleay# ~lre`nly# br`ekx`n`y#
keqen`ikleay# mqably mlray~`cay# a~i%# ae ikar~`y rabkleay nac gqenl%
?% ^`gokëe ya qy` ae oklclbç` z`r` ile~`r ac eøgarl na o`i~ark`y ae qe zc`~l na
zrqao`% Ya qy` ~`gokëe ae c` mçyki` z`r` ile~`r ac eøgarl na z`r~çiqc`y agk~kn`y na
qe` yqy~`eik` r`nk`i~k}`# ilgl zlr ahagzcl# iq`enl qe` yqy~`eik` r`nk`i~k}` agk~a z`r~çiqc`y `cm`# oa~` l b`gg`% @pqç c`y z`r~çiqc`y yle agk~kn`y# `c `x`r# ylora qe
c`rbl zarklnl na ~kagzl# { c` liqrraeik` na qe` agkykùe ay kenazaenkae~a na l~r`y
agkykleay%
9% Kbq`cgae~a# c` nky~rkoqikùe Zlkyyle ya qy` z`r` ac ile~rlc ay~`nçy~kil na i`ckn`n l
z`r` ile~`r ac eøgarl na ç~agy namai~qlyly &iq`enl ay nkmçikc qy`r c` nky~rkoqikùe
okelgk`c+%
>% L~r`y `zcki`ikleay kgzlr~`e~ay na c` nky~rkoqikùe na Zlkyyle yle z`r` aeile~r`r
ac eøgarl na `iiknae~ay# ae~ra cly ~r`o`h`nlray# ilgl zlr ahagzcl# ae qe`
kenqy~rk`# ae ay~qnkly na fkbkaea kenqy~rk`c { yabqrkn`n%
1% @nagày# l~r`y `zcki`ikleay yle c`y zrlo`okckn`nay na c`y nag`en`y na qe
zrlnqi~l { nag`en`y na yar}kikly% C` nky~rkoqikùe na Zlkyyle ~`gokëe ya qy` z`r`
aeile~r`r c` zrlo`okckn`n na pqa f`orà qe eøgarl ayzaiçmkil na raic`gly na
`iiknae~ay na `q~ly# ae qe` ilgz`ðç` na yabqrly nqr`e~a qe zarklnl na ~kagzl%
Ay~` nky~rkoqikùe ay kbq`cgae~a ø~kc z`r` aeile~r`r c` zrlo`okckn`n na qe eøgarl
ayzaiçmkil na liqrraeik`y pqa ~lg`e cqb`r zlr qe ~kagzl n`nl l ae qe` rabkùe
ayzaiçmki`%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 177/846
Nr% Fëi~lr Pqa}anl Qrç`y
>'>
=% @eàclb`gae~a# qe zrliayl na zrlnqiikùe ile~keq`# pqa m`orki` qe ikar~l loha~l
ae br`enay i`e~kn`nay# nlena qe loha~l namai~qlyl liqrra# `ca`~lrk`gae~a# ile
zrlo`okckn`n zapqað` a kenazaenkae~a# ~`gokëe zqana yar ileyknar`nl qe zrliaylZlkyyle%
2% Cly `iiknae~ay ae qe` màorki` br`ena zqanae liqrrkr# `c `x`r# ile qe` zapqað`
zrlo`okckn`n { yar kenazaenkae~ay na i`n` qel na cly l~rly ylora qe ~kagzl
ile~keql# ae iq{l i`yl# ay~a zrliayl ykbqa ` c` nky~rkoqikùe Zlkyyle% @nagày# ay~`
nky~rkoqikùe `zcki` z`r` aeile~r`r ac eøgarl na `iiknae~ay ae qe na~argke`nl
~r`gl i`rra~arl nqr`e~a qe zarklnl nkb`gly na 9 gayay%
6% @ykgkygl# c` nky~rkoqikùe na Zlkyyle ya qy` z`r` y`oar ac z`~rùe na ccab`n`y na
`}kleay ` qe `arlzqar~l4 ac eøgarl na namai~ly ylora c` yqzarmkika na qe` gay`4 ac
eøgarl na arrlray na kgzrae~` na qe ckorl# a~i%
Ilenkikleay pqa ya rapqkarae z`r` `zcki`r c` nky~rkoqikùe na Zlkyyle
0% Qe avzarkgae~l ileyky~a ae ile~`r ac eøgarl na }aiay na pqa qe ikar~l a}ae~l
liqrr` & v+# nqr`e~a qe` qekn`n na ~kagzl l ayz`ikl%
?% C` zrlo`okckn`n na pqa qe a}ae~l liqrr` ay c` gkyg` z`r` i`n` qekn`n na
~kagzl l ayz`ikl%
9% Ac eøgarl na a}ae~ly pqa liqrr`e ae qe` qekn`n na ~kagzl l ayz`ikl ay
kenazaenkae~a nac eøgarl na a}ae~ly pqa liqrrae ae c`y l~r`y yqylnkif`y qekn`nay%
>% ^aùrki`gae~a# qe eøgarl kemkek~l na liqrraeik`y nac a}ae~l naoae yar zlykocay
ae ac ke~ar}`cl%
Mqeikleay zrlo`okcçy~ki`y na c` nky~rkoqikùe Zlkyyle
Iq`enl c` nky~rkoqikùe na Zlkyyle ay `zrlzk`n`# c` zrlo`okckn`n na loyar}`r
av`i~`gae~a v eøgarl na liqrraeik`y zlr qekn`n na gankikùe &flr`y# gkeq~ly#
iae~çga~rly iøokily# zàbke`y# a~i%+# ay naikr# ac eøgarl na rayqc~`nly pqa liqrrae
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 178/846
Nr% Fëi~lr Pqa}anl Qrç`y
>'1
ae qe ke~ar}`cl na ~kagzl n`nl l ae qe` rabkùe ayzaiçmki`# ya aeiqae~r` qy`enl c`y
aiq`ikleay na `o`hl8
Z& V + 7 m& v+ 7 ± v
a'±
, v! &>'0+
Nlena8
± 7 zrlgankl na liqrraeik`y zlr ke~ar}`cl
7 ez
Nlena8 e 7 ~`g`ðl na c` gqay~r`
z 7 c` zrlo`okckn`n
a 7 ?%206?6%%% &O`ya na cly clb`rk~gly Eazark`ely+ v 7 :# 0# ?#%%%%%# ∕# ay naikr# cly }`clray na c` }`rk`oca `ca`~lrk V # ay~l ay# ac eøgarl
na rayqc~`nly pqa liqrrae ae qe ke~ar}`cl na ~kagzl%
Na `iqarnl ` c` mùrgqc` na `rrko`# c` nky~rkoqikùe na Zlkyyle ~kaea qe ylcl
z`ràga~rl ykgolckx`nl zlr c` ca~r` brkab` ±% Yk ileliagly ay~a }`clr nac zrlgankl
± zlnagly ayirkokr c` nky~rkoqikùe na zrlo`okckn`n ilgzca~`% Ay~a z`ràga~rl ±
zqana yar ke~arzra~`nl ilgl ac zrlgankl na c`y liqrraeik`y# zlr ke~ar}`cl na
~kagzl l ayz`ikl pqa i`r`i~arkx` ac zrliayl baear`nl zlr c` nky~rkoqikùe na
Zlkyyle%
L~r` g`ear` na }ar c` nky~rkoqikùe na Zlkyyle ay qy`enl c` mqeikùe n`n`
`o`hl8
&δ + v a'δ
z& v4δ + 7 ┊┊┊┊┊┊┊┊ &>'?+
v!
Nlena8
δ 7 ez ay qe` iley~`e~a n`n`% Ay ac eøgarl zrlgankl na
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 179/846
Nr% Fëi~lr Pqa}anl Qrç`y
>'=
rayqc~`nly zlr qekn`n na ~kagzl l rabkùe% @pqç# naoknl ` pqa δ ay zlyk~k}` z`r`
~lnly cly zlykocay }`clray na V# ae~leiay8
∕
Ω z&v4 δ + 7 0# cl iq`c ay c` ileyaiqaeik` nac nay`rrlccl na aδ ae c` yarka kemkek~` nav7:
G`ic`qrke n`n` ae ~lnly cly ~av~ly na iàciqcl# c` iq`c ya avzray` ilgl8∕
aδ
7 0 $ δ $ δ ?,?! $ δ 9,9! $ … $ 7 Ω δ v,v! &>'?`+v7:
Ay~` aiq`ikùe nagqay~r` pqa c` mqeikùe z&v4δ + y`~kym`ia c` yabqen` ilenkikùe
eaiay`rk` z`r` ayzaikmki`r qe` mqeikùe na zrlo`okckn`n na g`y` &zgm+%
± 7 ez c` iq`c ya zqana ke~arzra~`r ilgl ac eøgarl zrlgankl na ëvk~ly zlr ac
~`g`ðl na c` gqay~r` e%
a 7 ?%206?6%%%
v 7 eøgarl na rayqc~`nly pqa liqrrae ae qe ke~ar}`cl na ~kagzl &:# 0# ?# 9%%%%`n
kemkek~qg+
@zcki`ikùe na c` nky~rkoqikùe na Zlkyyle nae~rl na yqy zrlzkly ~ërgkely {
ilgl qe` `zrlvkg`ikùe ` c` nky~rkoqikùe okelgk`c
Ilgl ya gaeikleù `c zrkeikzkl# c` nky~rkoqikùe na Zlkyyle zqana avzcki`rya nayna
nly àebqcly8 nae~rl na yqy zrlzkly naraifly { ilgl qe` `zrlvkg`ikùe na c`
nky~rkoqikùe oklelgk`c% Ay~l liqrra# zlrpqa gqif`y }aiay yk ya `zcki` c`
nky~rkoqikùe okelgk`c ` ikar~ly zrlocag`y# cly iàciqcly yle gq{ av~aeyly# ae iq{l
i`yl ya zqana `zcki`r c` nky~rkoqikùe na Zlkyyle# c` iq`c n` cly gkygly rayqc~`nly#
zarl gqifl gày màikc na i`ciqc`rcly% Zlr ahagzcl# iq`enl c` nky~rkoqikùe na
Zlkyyle ya qy` ilgl qe` `zrlvkg`ikùe ` c` nky~rkoqikùe okelgk`c# ay~l ay
`zcki`oca# iq`enl e ay br`ena { c` zrlo`okckn`n# z ay zapqað`% &_ailrn`enl pqë#
ile c` nky~rkoqikùe okelgk`c# c` nky~rkoqikùe na Zlkyyle ya qy` ilgl qe`
`zrlvkg`ikùe ` c` nky~rkoqikùe elrg`c iq`enl e ay br`ena { iq`enl z l p ay~àe
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 180/846
Nr% Fëi~lr Pqa}anl Qrç`y
>'2
iari`e`y ` :+% C` `zrlvkg`ikùe na c` nky~rkoqikùe na Zlkyyle ` c` nky~rkoqikùe
okelgk`c ay `zrlzk`n` iq`enl z ≮ :%:1 { e ≡ ?:% Ae }arn`n# ac zlriae~`ha na arrlr
na cly rayqc~`nly lo~aeknly qy`enl c` nky~rkoqikùe na Zlkyyle# ilgl qe``zrlvkg`ikùe ` c` nky~rkoqikùe okelgk`c# ay na 0 ae ?2: l iari` na :%>% C`
^@OC@ 0%: gqay~r` c`y ilgz`r`ikleay na c` nky~rkoqikùe okelgk`c { c` Zlkyyle%
^@OC@ 0%:% ^`oc` gly~r`enl c`y ilgz`r`ikleay na rayqc~`nly na ahagzcly
`ca`~lrkly qy`enl c` nky~rkoqikùe okelgk`c { c` nky~rkoqikùe na Zlkyyle%
&Ac`olr`ikùe zrlzk`+
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
Nky~rkoqikùe Okelgk`c Nky~rkoqikùe Zlkyyle
Z& V ≡ 0+ 7 0 ’ M&:+ Z& V ≡ 0+ 7 0 ’ M&:+
7 0 ’ :%0?0= 7 :%626> 7 0 ’ :%091 7 :%6=1
Z& V ≮ ?+ 7 M&?+ 7 :%=2=; Z& V ≮ ?+ 7 M&?+ 7 :%=22
Z& V ≡ 9+ 7 0 ’ M&?+ Z& V ≡ 9+ 7 0 ’ M&?+
7 0 ’ :%=2=; 7 0 ’ :%=22
7 :%9?90 7 :%9?9
Zlr l~r` z`r~a# iq`enl c` nky~rkoqikùe na Zlkyyle ay avzcki`n` nae~rl na yqy
zrlzkly gërk~ly# ay~` nky~rkoqikùe rayqc~` na c`y liqrraeik`y pqa zqanae yar
nayirk~`y zlr qe` }`rk`oca `ca`~lrk` nkyira~`% Ay~` }`rk`oca nael~`n` zlr V # zqana
~lg`r }`clray na v 7 :# 0# ?# %%% &Nlena cly zqe~ly yqyzaeyk}ly nael~`e `n
kemkek~qg+# ay~l# ae ile~r`y~a ile c` nky~rkoqikùe okelgk`c nlena cly }`clray na v
yle na :# 0# ?# 9#%%%%# e%
Ahagzcly na c` `zcki`ikùe na c` nky~rkoqikùe na Zlkyyle nae~rl na yqy zrlzkly
~ërgkely l naraifly { ilgl `zrlvkg`ikùe ` c` nky~rkoqikùe okelgk`c
Ahagzcl )0% @yøg`ya pqa qe` nky~rkoqikùe na Zlkyyle ya n` zlr c` mqeikùe na
`o`hl8
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 181/846
Nr% Fëi~lr Pqa}anl Qrç`y
>'6
z& v+ 7 T&:%2?+ v
a':%2?
W , v!
Aeile~r`r8
&`+ z&:+&o+ z&0+
&i+ z&?+
&n+ z&9+
Ylcqikùe8
&`+ z&:+ 7 T&:%2?+:
a':%2?
W , :! 7 :%>6=6
&o+ z&0+ 7 T&%2?+0
a':%2?
W , 0! 7 :%91:1
&i+ z&?+ 7 T&%2?+?
a':%2?
W , ?! 7 :%0?=?
&n+ z&9+ 7 T&:%2?+9
&:%>6=6W , 9! 7 :%:9:
Ahagzcl )?% Qe ay~qnkl na fkbkaea kenqy~rk`c { yabqrkn`n faifl ` c`rbl zc`xl na
cly `iiknae~ay ae qe` màorki`# cca}ù ` c` baraeik` ` ileicqkr pqa ac eøgarl na
`iiknae~ay zlr ~r`o`h`nlr# nqr`e~a qe `ðl & V + ykbqa ` qe` nky~rkoqikùe Zlkyyle% Yk
ac eøgarl zrlgankl na `iiknae~ay zlr ~r`o`h`nlr zlr `ðl mqa na :%9# ay~kg`r cl
ykbqkae~a8
&`+ ¼Iqàc ay c` zrlo`okckn`n na pqa qe ~r`o`h`nlr yacaiikle`nl# `ca`~lrk`gae~a# el
~aenrà qe `iiknae~a nqr`e~a ac `ðl ykbqkae~a<
&o+ ¼Iqàc ay c` zrlo`okckn`n na pqa qe agzca`nl yacaiikle`nl# `ca`~lrk`gae~a#
~aenrà iq`enl gaely 0 `iiknae~a nqr`e~a ac ykbqkae~a `ðl<
&i+ ¼Iqàc ay c` zrlo`okckn`n na pqa qe ~r`o`h`nlr ~aenrà# av`i~`gae~a# 0 `iiknae~a<
&n+ ¼Iqàc ay c` zrlo`okckn`n na pqa qe ~r`o`h`nlr yacaiikle`nl `c `x`r na c` màorki`~aenrà ae~ra ? { > `iiknae~ay# keicqyk}`gae~a# ac zrùvkgl `ðl<
Ylcqikùe8
Ay~a zrlocag` ya zqana raylc}ar qy`enl c` aiq`ikùe na Zlkyyle { ~`gokëe qy`enl
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 182/846
Nr% Fëi~lr Pqa}anl Qrç`y
>';
c` ~`oc` na c` nky~rkoqikùe Zlkyyle%
@pqç# δ 7 :%9# V 7 :
Qy`enl c` aiq`ikùe &>'?+# z&v4δ + 7 δ
v
a'δ
, v! { yqy~k~q{aenl }`clray n`8&`+ z& V + 7 Z& V 7 :+ 7 &:%9+
:a
':%9, :! 7 :%2>0
Ay~l nkia pqa# ac `ðl ykbqkae~a# na i`n` 0:: ~r`o`h`nlray# 2> el ~aenràe ekebøe
`iiknae~a { ?= yk cl zqnkar`e ~aear%
Ilgl ya nkhl `e~arklrgae~a# ay~a zrlocag` ~`gokëe ya zqana raylc}ar
qy`enl c` ~`oc` na c` nky~rkoqikùe na Zlkyyle% Z`r` ay~l# oqyi`gly ae c` ~`oc` na
zrlo`okckn`nay `iqgqc`n`y l kenk}knq`cay ileδ 7 :%9 { ile v 7 : { n` %2>:
&o+ C` zrlo`okckn`n na pqa qe ~r`o`h`nlr ~aeb` iq`enl gaely qe `iiknae~a ya
zqana f`iar qy`enl c`y ~`oc`y na Zlkyyle%
Z& V ≡ 0+ 7 0 ' Z& V 7 :+ 7 0 ' :%2>0 7 :%?1;
&i+ C` zrlo`okckn`n na pqa# ac ~r`o`h`nlr ~aeb` av`i~`gae~a# qe `iiknae~a ya zqana
f`iar qy`enl c` mùrgqc` l c`y ~`oc`y na zrlo`okckn`nay kenk}knq`cay l `iqgqc`n`y%
Qy`enl c` mùrgqc` n` &>'0+# Z&V+ 7 m & v+ 7 ± v
a'±
, v!# ile ± 7 :%9 { V 7 0
m&0+ 7 &:%9+0 a':%9,0!
7 :%????
Qy`enl c` ~`oc` na c` nky~rkoqikùe na Zlkyyle na zrlo`okckn`nay kenk}knq`cay ely `8
Z& V 7 0+ 7 Z& V ≡ 0+ ' Z& V 7 :+ 7 :%;=9 ' :%2>0 7 :%???
&n+ C` zrlo`okckn`n na pqa qe ~r`o`h`nlr ~aeb` ae~ra ? { > `iiknae~ay# keicqyl# ay8
Z&? ≮ V ≮ >+ 7 Z&?+ $ z&9+ $ Z&>+
7 :%:999 $ %::99 $ :%:::?7 :%:9=6 &qy`enl c`y zrlo`okckn`nay kenk}knq`cay
na c` nky~rkoqikùe na Zlkyyle+
Ahagzcl )9% Z`r` ay~a zrlocag` qy`r c` nky~rkoqikùe na Zlkyyle { c` okelgk`c%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 183/846
Nr% Fëi~lr Pqa}anl Qrç`y
>'0:
Ykaenl `yç# yk ac eøgarl na namai~ly# zlr zka iq`nr`nl na c` ~ac` na qe apqkzl na
ile~rlc &mkc~rly+ g`eqm`i~qr`nl zlr ikar~` kenqy~rk` ykbqa ` qe zrliayl Zlkyyle# ile
δ 7 :%:6# ae~leiay# yk qe zka iq`nr`nl na c` ~ac` ay keyzaiikle`nl na qe` gqay~r``ca`~lrk` na 1:# ¼iqàc ay c` zrlo`okckn`n na pqa ac eøgarl na namai~ly loyar}`nly
ya`<8
&`+ Ekebøe namai~l
&o+ Iq`enl gaely 0 namai~l
&i+ Av`i~`gae~a# ? namai~ly
Ylcqikùe8
Zrkgar`gae~a# }`gly ` qy`r c` nky~rkoqikùe na Zlkyyle# ilgl qe` `zrlvkg`ikùe `
c` okelgk`c%
&`+ Zrlo`okckn`n na ekebøe namai~l%
Qy`enl c` mùrgqc` &>'?+# ile ± 7 δ 7 %:68
Z& V 7 :+ 7 δ v a'δ
, v!
7 &%:6+:
&a'%:6
+ , :!
7 %;?9
@c~are`~k}`gae~a# zlnagly y`i`r ac gkygl rayqc~`nl qy`enl c` ~`oc` na Zlkyyle
`iqgqc`n`% Ay~l ya f`ia oqyi`enl ac }`clr na δ 7 %:6 ile V 7 : { n` %;?9
&o+ Iq`enl gaely qe namai~l% @pqç# qy`enl# eqa}`gae~a# c` mùrgqc` &>'?+ na
Zlkyyle ile δ 7 %:6 { yqoy~k~q{aenl n`8
Z& V ≡ 0+ 7 0 ’ Z& V 7 :+ 7 0 ' %;?9 7 %:22
&i+ Av`i~`gae~a ? namai~ly%Z& V 7 ?+ 7 &%:6+
?a
'%:6, ?!
7 &:%::=>+ &:%;?+ , ?
7 :%::?;1
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 184/846
Nr% Fëi~lr Pqa}anl Qrç`y
>'00
@flr`# okae# yk ya qy`rà c` nky~rkoqikùe okelgk`c# ay~l yarç` c`rbl { ~anklyl# zlrpqa
e ay br`ena% Yke ago`rbl# z`r` qy`r c` rac`ikùe o& v4e#z+ 7 e!, v!&e ’ v+! z v
pe' v
#
eaiayk~`gly i`ciqc`r ac }`clr z# ay naikr# qy`enl ± 7 δ 7 ez% Ile δ 7 %:6 { e 7 1:n`8 %:6 7 &1:+&z+ { z 7 %::0= u %::?%
&`+ Qy`enl c` mùrgqc` okelgk`c o&v4e#z+ 7 e!,v!&e ’ v+! zv
pe'v
{ yqy~k~q{aenl cly
}`clray n`8
o& v41:#%:6+ 7 1:!, v!&1: ’ :+! &%::?+v
&%;;6+1:'v
O&:41:#%:6+ 7 1:!,:!&1: ’ :+! &%::?+:
&%;;6+1:':
7 &0+ &0+ &:%;:1+ 7 :%;:1
O&041:#%:6+ 7 1:!,0!&1: ’ 0+! &%::?+0
&:%;;6+1:'0
7 >;&%::?+&:%;:2+ 7 %:;6
O&?41:#:%:6+ 7 1:!,?!&1: ’ ?+! &%::?+?
&%;;6+1:'?
7 0??1 &%:::::>+&:%;:6+ 7 :%::>1
&o+ Iq`enl gaely qe namai~l ay8
Z& V ≡ 0+ 7 0 ’ Z& V 7 :+ 7 0 ' :%;:1
7 :%:;1
&i+ Av`i~`gae~a# ? namai~ly%
Ay~l ely cca}` ` Z& V 7 ?+ 7 O&?41:#:%:6+
7 1:I?&%::?+?&%;;6+
1:'?
7 &0??1+ &%:::::>+&:%;0+
7 :%::>1
Ilgl ya }a `rrko`# `c qy`r c` nky~rkoqikùe okelgk`c# ac zrliayl ay c`rbl {ilgzcki`nl# zlr cl pqa ay gahlr qy`r c` nky~rkoqikùe na Zlkyyle ilgl qe`
`zrlvkg`ikùe ` c` okelgk`c% Ae ay~a key~`e~a# ac cai~lr naoarà qy`r c` nky~rkoqikùe
Zlkyyle { ilgz`r`r cly rayqc~`nly lo~aeknly%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 185/846
Nr% Fëi~lr Pqa}anl Qrç`y
>'0?
Ahagzcl )>% Ae qe ay~qnkl na fkbkaea kenqy~rk`c { yabqrkn`n# qe` zloc`ikùe na
~r`o`h`nlray na qe brqzl na kenqy~rk`y pqa g`eah`e zrliayly# nlena f`{ rqknl# ac
1 yqmrae na zrlocag`y aglikle`cay pqa ke~armkarae ile yq ~r`o`hl% Yk ya y`i` qe`gqay~r` `ca`~lrk` na =: ~r`o`h`nlray# ¼Iqàc ay c` zrlo`okckn`n nac eøgarl na
~r`o`h`nlray# pqkaeay yqmrae nky~qrokly aglikle`cay< F`iar ay~a zrlocag` ile c`
nky~rkoqikùe okelgk`c {# cqabl# ile c` nky~rkoqikùe Zlkyyle ilgl qe` `zrlvkg`ikùe
` c` okelgk`c# { ilgz`r`r rayqc~`nly%
&`+ Gày na ? ~r`o`h`nlray yqmr`e na nky~qrokly aglikle`cay
&o+ Iq`enl gaely >
&i+ 1 l gày
Ylcqikùe8
Qy`enl c` nky~rkoqikùe okelgk`c ile z 7 %:1# e 7 =:# V 5 ?
o& v4=:#%:1+ 7 =:Iv &%:1+v
&%;1+=:'v
O&:4=:#%:1+ 7 =:I: &%:1+:
&%;1+=:':
7 &0+ &0+ &:%:>=+
7 :%:>=0
O&04=:#%:1+ 7 =:I0 &%:1+0
&%;1+=:'0
7 =: &%:1+&:%:>;+
7 :%0>11
O&?4=:#%:1+ 7 =:I? &%:1+?
&:%;1+=:'?
7 &=:+&1;+,? &%::?1+&:%:10+
7 :%??1;O&94=:#%:1+ 7 =:I9 &%:1+
9&%;1+
=:'9
7 &=:+&1;+&16+,= &%:::09+&%:19+
7 :%??;6
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 186/846
Nr% Fëi~lr Pqa}anl Qrç`y
>'09
O&>4=:#%:1+ 7 =:I> &%:1+>
&%;1+=:'>
7 &=:+&1;+&16+&12+,?> &%:::::=9+&:%:12+
7 :%02?> Eù~aya pqa ~lnly ay~ly }`clray ~`gokëe ya zqanae y`i`r qy`enl c` ~`oc` na Zlkyyle
na zrlo`okckn`nay kenk}knq`cay# ay naikr# oqyi`enlδ 7 :%:1 { V 7 :# 0# ?# 9# >%
&`+ Gày na nly ~r`o`h`nlray yqmr`e na nky~qrokly aglikle`cay
Z& V 5 ?+ 7 0 ’ Z& V ≮ ?+ 7 0 ’ Z&:+ $ Z&0+ $ Z&?+
7 0 ’ &:%:>=0 $ :%0>11 $ :%??1;+
7 :%16?1
&o+ Iq`enl gaely > nkia8
Z& V ≡ >+ 7 0 ’ Z& V 3 >+ 7 0 ’ Z&:+ $ Z&0+ $ Z&?+ $ Z&9+
7 0 ' %=191
7 :%9>=1
&i+ 1 l gày nkia8
Z& V ≡ 1+ 7 0 ’ Z& V ≮ >+ 7 0 ’ Z&:+ $ Z&0+ $ Z&?+ $ Z&9+ $ Z&>+
7 0 ' %6?617 :%0201
@flr`# qy`enl c` nky~rkoqikùe na Zlkyyle nae~rl na yqy zrlzkly naraifly
eaiayk~`gly i`ciqc`r δ # ay naikr# ± 7 ez 7 &=:+&%:1+ 7 9%:# zarl zrkgarl }`gly `
i`ciqc`r c`y zrlo`okckn`nay z`r` v 7 :# 0# ?# 9# > `e~ay na i`ciqc`r ac keikyl &`+%
Qy`enl c` mqeikùe &>'0+# Z& V + 7 m& v+ 7 ± v
a'±
, v! { yqy~k~q{aenl cly }`clray n`8
Z& V 7 :+ 7 9%::
&a'9%:
+ , :! 7 :%:>;6
Z& V 7 0+ 7 9%:0%:
&:%:>;6+,0! 7 :%0>;>
Z& V 7 ?+ 7 9%:?%:
&:%:>;6+,?! 7 :%??>:
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 187/846
Nr% Fëi~lr Pqa}anl Qrç`y
>'0>
Z& V 7 9+ 7 9%:9%:
&:%:>;6+,9! 7 :%??>:
Z& V 7 >+ 7 9%:>%:
&:%:>;6+,>! 7 :%0=6:
&`+ Gày na ? ~r`o`h`nlray yqmr`e nky~qrokly aglikle`cay
Z&v+ 7 Z& V 5 ?+ 7 0 ’ Z& V ≮ ?+ 7 0 ’ Z& v 7 :# 0# ?+
7 0 ’ :%>?9
7 :%122:
&o+ Iq`enl gaely > ~r`o`h`nlray
Z&V ≡ >+ 7 0 ’ Z& V ≮ 9+ 7 0 ’ :%=>2?
7 :%91?6
&i+ 1 l gày ~r`o`h`nlray
Z& V ≡ 1+ 7 0 ’ Z& V ≮ >+ 7 0 ’ :%601?
7 :%06>6
Ahagzcl )1% Na cly ç~agy zrlnqiknly zlr qe` m`i~lrç`# ac 9 ay~àe namai~qlyly%
Qe` gqay~r` na ?1 ç~agy ya yacaiikle` z`r` qe` keyzaiikùe% Qy`r c` nky~rkoqikùe
okelgk`c { c` Zlkyyle { ilgz`r`r cly rayqc~`nly na cly ykbqkae~ay8&`+ Av`i~`gae~a > ç~agy ay~`ràe namai~qlyly
&o+ 9 l gày loha~ly ay~`ràe namai~qlyly
Ylcqikùe8
Qy`enl c` nky~rkoqikùe okelgk`c8
&`+ :%::1>
&o+ :%:96
Qy`enl c` nky~rkoqikùe na Zlkyyle8
&`+ :%::=
&o+ :%:>0
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 188/846
Nr% Fëi~lr Pqa}anl Qrç`y
>'01
Ahagzcl )=% Qe zrlgankl na 9 `q~ly `rrko`e ` c` i`ya~` na ilorl na qe` i`rra~ar`
i`n` gkeq~l% Yk ay~` ~`y` ay `zrlvkg`n` zlr qe zrliayl Zlkyyle# ¼iqàc ay c`
zrlo`okckn`n na pqë# av`i~`gae~a# 1 `q~ly `rrko`ràe ae qe zarklnl na qe gkeq~l<Ylcqikùe8
@pqç# δ 7 ± 7 9# v 7 1
Qy`enl c` aiq`ikùe m& v+ 7 ± v
a'±
, v! { yqy~k~q{aenl cly }`clray lo~aeagly8
Z& V 7 1+ 7 &9+1
&a+'9
, 1! 7 T&?>9+&%:>;6+W , 0?: 7 %0::6
Ac }`clr na %0::6 ay c` zrlo`okckn`n na pqa 1 `q~ly `rrkoae ae qe gkeq~l
Eù~aya pqa ay~a zrlocag` ~`gokëe ya zqana raylc}ar qy`enl c` ~`oc` na
zrlo`okckn`nay na Zlkyyle# ay naikr# z`r` }`clray ayzaiçmkily na ± { na v pqa n`e
qe` ylcqikùe gày màikc { zraiky`% Z`r` ay~l# oqyi`gly ac }`clr na ± 7 9 ile v 7 1 {
n` :%;0=0# zarl ilgl c` ~`oc` n` c`y yqg`~lrk`y `iqgqc`n`y# ca ray~`gly 0% Zlr cl
~`e~l# Z& V 7 1+ 7 0 ' :%;0=0 7 %:6 u %0 &]ar ~`oc` na }`clray yacai~ly na c`
nky~rkoqikùe `iqgqc`n` na Zlkyyle+
Ahagzcl )2% Ac 0: na c`y farr`gkae~`y zrlnqikn`y ae ikar~l zrliayl na
g`eqm`i~qr` yle namai~qly`y% Aeile~r`r c` zrlo`okckn`n na pqë# ae qe` gqay~r` na
0: farr`gkae~`y yacaiikle`n`y# `ca`~lrk`gae~a# av`i~`gae~a# ? farr`gkae~`y ya`e
namai~qly`y% F`iar ay~l qy`enl8
&`+ C` nky~rkoqikùe na Zlkyyle
&o+ C` nky~rkoqikùe okelgk`c%
Ylcqikùe8
@pqç zleagly e 7 0: farr`gkae~`y% Ae~leiay# zrlo`okckn`n na qe` farr`gkae~`namai~qly` ay# z 7 0: 7 :%0: { ez 7 &0:+&:%0:+ 7 0%:# v 7 ?
&`+ Qy`enl c` aiq`ikùe na Zlkyyle8
z& v+ 7 &± v
a'±
+ , v! l okae Z& v+ 7 ± v
a'±
, v!
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 189/846
Nr% Fëi~lr Pqa}anl Qrç`y
>'0=
Nlena8 z 7 :%0# ± 7 ez 7 &0:+&:%0+ 7 0%:
Zrtna ? farr`gkae~`y namai~qly`y ae 0:w 7 &0%:+?
&a'0
+ , ?!
7 0,?a 7 :%069;&o+ Qy`enl c` aiq`ikùe na Oarelqcck
Z& V 7 ?+ 7 eIv z v
pe' v
Nlena8 e 7 0:
V 7 ?
z 7 :%0
p 7 0 ' z 7 0 ' :%0 7 :%;
Z& V 7 ?+ 7 0:I? &:%0+?&:%;+
0:'?7 0:! , T?!&0:'?+!W 7 :%0;
Ahagzcl )6% Ae ay~a zrlocag` `zcki`r c` mqeikùe ay~`nçy~ki` gày `zrlzk`n`% Ykaenl
`yç# yk ac 9%: na cly mlily acëi~rkily g`eqm`i~qr`nly zlr qe` ilgz`ðç` ay~àe
namai~qlyly# ae~leiay# aeile~r`r c` zrlo`okckn`n na pqë# ae qe` gqay~r` na 0::
mlily8
&`+ Ekebøe mlil ay~ë namai~qlyl
&o+ 0 mlil ay~ë namai~qlyl
&i+ ? mlily ay~ëe namai~qlyly
&n+ 9 mlily ay~ëe namai~qlyly
&a+ > mlily ay~ëe namai~qlyly
&m+ 1 mlily ay~ëe namai~qlyly
Ylcqikùe8
@pqç ay gày `zrlzk`nl qy`r c` nky~rkoqikùe na Zlkyyle# zlrpqa e ay br`ena% Ykaenl`yç# z 7 :%:9# e 7 0::# ± 7 ez 7 &0::+&:%:9+ 7 9%:# v 7 :# 0# ?# 9# ># 1
&`+ Qy`enl c` mùrgqc` Zlkyyle z& v#±+ 7 ± v
a'±
, v!# ile a'9
7 :%:>;2;
z& v#±+ 7 ± v
a'±
, v!
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 190/846
Nr% Fëi~lr Pqa}anl Qrç`y
>'02
z&:#9+ 7 &9%:+:
&a+'9%:
, :! 7 &0+& :%:>;2;+ 7 :%:>;2;
&o+ Z&0#9+ 7 &9+0&a+
'9%:, 0! 7 &9+&:%:>;2;+,0 7 :%0>;>
&i+ Z&?#9+ 7 &9+?
&a+'9%:
, ?! 7 &;+&:%:>;2;+ , ? 7 :%>>600&n+ Z&9#9+ 7 &9+
9&a+
'9%:, 9! 7 &?2+&:%:>;2;+ , = 7 :%??>0
&a+ Z&>#9+ 7 &9+>
&a+'9%:
, >! 7 &60+&:%:>;2;+ , ?> 7 :%0=6:
&m+ Z&1#9+ 7 &9+1
&a+'9%:
, 1! 7 &?>9+&:%:>;2;+ , 0?: 7 :%0::6
Zrlzkan`nay na c` nky~rkoqikùe na Zlkyyle
^@OC@ >%0% ^`oc` gly~r`enl `cbqe`y zrlzkan`nay na c` nky~rkoqikùe na Zlkyyle%
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
Zrlgankl ± 7 δ
]`rk`ex` ω?7 δ
Nay}k`ikùe ay~àen`r ω 7 ∘ δ
Glgae~l nac ilamkikae~a nac yaybl ΰ9 7 0,∘ δ
Glgae~l nac ilamkikae~a na dqr~lyky ΰ> 7 9 $ 0,δ
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
&Mqae~a8 Yzkabac# 0;=0+
Ahagzcl );% C` zrlo`okckn`n na pqa qe` zaryle` gqar` na qe `rray~l i`rnk`il# zlr
mqg`r ae aviayl# ay na :%::?% Aeile~r`r c` zrlo`okckn`n na pqa gaely na 1
zaryle`y# na c`y ykbqkae~ay ?#:::# glrkràe na qe yçe~lg` nac ilr`xùe% Aeile~r`r#
~`gokëe# ac zrlgankl { c` }`rk`ex`%
Ylcqikùe8Zrkgarl i`ciqc`gly ac zrlgankl { c` }`rk`ex`% C`y mùrgqc`y z`r` ay~l yle8
± 7 ez 7 &?:::+&:%::?+ 7 >%:
ω?7 ezp 7 &?#:::+&:%::?+&:%;;6+ 7 9%;;?
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 191/846
Nr% Fëi~lr Pqa}anl Qrç`y
>'06
Qy`enl c` ~`oc` na Zlkyyle { ykbqkaenl ay~a r`xle`gkae~l n`8
Z& V 3 1+ 7 Z& V ≮ >+
7 :%=?66 &na c` ~`oc` na Zlkyyle+Ahagzcly kcqy~r`enl ilgl br`mki`r cly n`~ly na c` }`rk`oca `ca`~lrk` V
Ahagzcl )0:% Yqzùeb`ya pqa ae qe ay~qnkl na ile~`gke`ikùe `gokae~`c ya key~`c`
qe` ran na 9#6>: yaeylray na `c~l }lcqgae z`r` gankr c`y ileiae~r`ikleay na
z`r~çiqc`y `~glymërki`y# gaelray pqa 0: gkir`y% Yk c` zrlo`okckn`n na pqa
iq`caypqkar` na ay~ly gqay~ra`nlray m`cca ay na %:::69 nqr`e~a qe `ðl# ae~leiay#
na~argke`r c`y zrlo`okckn`nay na pqa :# 0# ?# 9# >#… na cly gqay~ra`nlray m`ccae
nqr`e~a ac `ðl ae iqay~kùe% F`iar qe` bràmki` qy`enl z`zac yagkclb`rk~gl%
Ylcqikùe8
@pqç ya zqnkar` qy`r c` nky~rkoqikùe okelgk`c# zlrpqa f`oc` na qe` yk~q`ikùe
oke`rk`# ay naikr# m`cc`r l el m`cc`r% Yke ago`rbl# naoknl ` pqa e ay gq{ br`ena { z
ay zapqað`# c` nky~rkoqikùe Zlkyyle ay `zcki`oca% Ykaenl `yç# zrkgarl i`ciqc`gly ac
}`clr na δ %
δ 7 ez 7 &96>:+&:%:::69+ 7 9%?
Aeyabqkn`# ay~`ocaiagly eqay~rl zqe~l na z`r~kn` ile c` }`rk`oca `ca`~lrk V # ilgl
}`rk`oca kenazaenkae~a%
m& v+ 7 z& v49%?+ 7 &9%?+v
a'9%?
, v!
Cqabl yqy~k~qkgly cly }`clray na c` }`rk`oca `ca`~lrk V ae c` mùrgqc` na `rrko`
z&:49%?+ 7 9%?:
&:%:>0+,:! 7 :%:>0
z&049%?+ 7 9%?0
&:%:>0+,0! 7 :%09: z&?49%?+ 7 9%?
?&:%:>0+,?! 7 :%?:;
z&949%?+ 7 9%?9
&:%:>0+,9! 7 :%??9
z&>49%?+ 7 9%?>
&:%:>0+,>! 7 :%026
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 192/846
Nr% Fëi~lr Pqa}anl Qrç`y
>'0;
z&149%?+ 7 9%?1
&:%:>0+,1! 7 :%00>
z&=49%?+ 7 9%?=
&:%:>0+,=! 7 :%:=0
z&249%?+ 7 9%?2
&:%:>0+,2! 7 :%:?6 z&649%?+ 7 9%?
6&:%:>0+,6! 7 :%:00
z&;49%?+ 7 9%?;
&:%:>0+,;! 7 :%::9;2
z&0:49%?+ 7 9%?0:
&:%:>0+,0:! 7 :%::09
Z`r` br`mki`r cly n`~ly na c` }`rk`oca `ca`~lrk` V &`oyiky`+ { na c`
zrlo`okckn`n m& v+ 7 z& v4±+ &lrnae`n`+# ya qy` z`zac yagkclb`rç~gkil% Zlr ahagzcl#
c` Mkbqr` >%0 gqay~r` ac qyl na z`zac yagkclb`rç~gkil qy`nl z`r` br`mki`r cly
}`clray na c` }`rk`oca `ca`~lrk` V &ae c` `oyiky`+ { na c` zrlo`okckn`n m& v+ 7 z& v4±+%
C` bràmki` ile ay~ly }`clray ya gqay~r` `o`hl% Na c` Mkbqr` >%0 ay~kg`r c`y
ykbqkae~ay zrlo`okckn`nay% &Ac ay~qnk`e~a cl naoarà f`iar+%
&`+ C` zrlo`okckn`n na pqa m`ccae &keicqyk}`gae~a+# ae~ra 9 { ; gqay~ra`nlray
&o+ C` zrlo`okckn`n na pqa m`ccae gày na 6 gqay~ra`nlray
&i+ C` zrlo`okckn`n na pqa m`ccae &avicqyk}`gae~a+# ae~ra > { = gqay~ra`nlray
&n+ C` zrlo`okckn`n na pqa m`ccae gày na 0: gqay~ra`nlray
&a+ C` zrlo`okckn`n na pqa m`ccae ~lnly cly gqay~ra`nlray
&m+ C` zrlo`okckn`n na pqa el m`cca ekebøe gqay~ra`nlr
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 193/846
Nr% Fëi~lr Pqa}anl Qrç`y
>'?:
Mkbqr` >%0% Mkbqr` gly~r`enl ac qyl nac z`zac yagkclb`rk~gl br`mki`enl cly }`clray
na c` }`rk`oca `ca`~lrk` V &ae c` `oyiky`+ { na z& v4±+ ae c` lrnae`n`% &Ac`olr`ikùe
zrlzk`+
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 194/846
Nr% Fëi~lr Pqa}anl Qrç`y
>'?0
Zrlocag`y na c` nky~rkoqikùe na Zlkyyle qy`enl ac zrlbr`g` na ilgzq~`nlr`
Gkek~`o
Ahagzcl )00%Yqzùeb`ya pqa ac eøgarl V na fqr`i`eay loyar}`nly ae c` rabkùe nacI`rkoa# nqr`e~a cly øc~kgly 9 `ðly ~kaea qe` nky~rkoqikùe na Zlkyyle ile qe
zrlgankl na δ 7 ± 7 6% I`ciqc`r c`y ykbqkae~ay zrlo`okckn`nay8
&`+ C` zrlo`okckn`n na pqa liqrr`e ` cl gày 6 fqr`i`eay%
&o+ C` zrlo`okckn`n na pqa liqrr`e av`i~`gae~a 6 fqr`i`eay%
&i+ C` zrlo`okckn`n na pqa liqrr`e iq`enl gaely ; fqr`i`eay%
&n+ C` zrlo`okckn`n na pqa liqrr`e ae~ra 1 { 6 fqr`i`eay keicqyl%
&a+ C` zrlo`okckn`n na pqa liqrr`e ae~ra 1 { 6 fqr`i`eay avicqyl%
&m+ C` zrlo`okckn`n na pqa liqrr`e ` cl gày 6 fqr`i`eay# zarl gày na 1%
&b+ C` zrlo`okckn`n na pqa liqrr`e gày na ? fqr`i`eay%
&f+ F`iar bràmki`y na Z& V 7 v+ { Z& V ≮ v+ ae mqeikùe na v%
Ylcqikle8
Zrliankgkae~l8 Zrkgar`gae~a# qy`enl ac zrlbr`g` Gkek~`o oqyi`gly c`y
}ae~`ekcc`y yað`c`n`y `o`hl# ay naikr zrliankaenl ilgl8
I`ci 5 Zrlo`okck~{ nky~rkoq~kle 5 Zlkyyle%%
Ae c` }ae~`e` na –Zlkyyle nky~rkoq~kle‗ z`r` c` zrkgar` ilrrkn` zqe~a`gly ae
–Zrlo`o`okck~{‗ {# z`r` c` yabqen` ilrrkn` zleagly ac zqe~l ae –Iqggqc`~k}a
Zrlo`okck~{‗% Ae c` }ae~`e` na –Ga`e‗ zleagly ac }`clr nac zrlgankl δ l ± kbq`c `
6% Ae c` }ae~`e` na –Kezq~ ilcqge‗ zleagly I0 &cly }`clray na c` }`rk`oca
`ca`~lrk` v 7 :# 0# ?#%%%e+% @pqç# ay ile}aekae~a key~rqkr `c zrlbr`g` na pqa zleb`cly }`clray na c`y zrlo`okckn`nay na mqeikùe na g`y` &zrlo`okckn`nay kenk}knq`cay
Z& V 7 v+ ae c` ilcqge` I?% @ykgkygl# ya key~rq{a `c zrlbr`g` pqa zleb` c`y
zrlo`okckn`nay `iqgqc`n`y Z& V ≮ v+ ae c` ilcqge` I9% Qe` }ax# pqa ya ilrra ac
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 195/846
Nr% Fëi~lr Pqa}anl Qrç`y
>'??
zrlbr`g`# ya baear` c` ~`oc` na `o`hl pqa gqay~r` cly rayqc~`nly%
^@OC@ >%?% ^`oc` gly~r`enl c` }`rk`oca `ca`~lrk` v ae mqeikùe na c` zrlo`okckn`n#
Z& V7v+ { na c` zrlo`okckn`n `iqgqc`n`# Z& V ≮ v+%
Ylcqikùe8
&`+ Z& V ≮ 6+% @pqç# ay~a zrlocag` ya zqana f`iar na nly g`ear`y% Zrkgarl# ya
zqana f`iar yqg`enl c`y zrlo`okckn`nay na mqeikùe na g`y` Z& V 7 v+# ay naikr# na
Z&V 7 :+ f`y~` Z& V 7 6+ na cly }`clray na c` ilcqge` I? na c` ~`oc`% El loy~`e~a#ay~a zrliankgkae~l ay gq{ c`rbl a kgzrài~kil% Yke ago`rbl# yk qy`gly c`y
zrlo`okckn`nay `iqgqc`n`y na c` ilcqge` I9 l na c` Mkbqr` >%9# ac rayqc~`nl ay
zraiky`gae~a :%1;?11%
&o+ Z& V 7 6+% Ay~a iàciqcl cl f`iagly ca{aenl v 7 6 ae c` ilcqge` I? na c` ~`oc` {
n` :%09;=%
&i+ Z& V ≡ ;+% Ay~a iàciqcl ya f`ia ~lg`enl ac ilgzcagae~l% Ay naikr#
Z& V ≡ ;+ 7 0 ’ Z& V 3 ;+ 7 0 ’ :%1;?1 7 :%>:21%
&n+ Z&1 ≮ V ≮ 6+ 7 :%>;?
&a+ Z&1 3 V 3 6+ 7 :%?10
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 196/846
Nr% Fëi~lr Pqa}anl Qrç`y
>'?9
&m+ Z&1 3 V ≮ 6+ 7 Z& V 7 =+ $ Z& V 7 2+ $ Z& V 7 6+ 7 :%901; &na I?+
&b+ 0 ’ Z& V ≮ ?+ 7 :%;6=9
&f+ ]ar Mkbqr` >%? na `o`hl% Z`r` ay~l# qy`r Br`zf 5 zcl~% Ae c` }ae~`e` na –Br`zf}`rk`ocay‗ zlear I9 ae [ { I0 ae V % Ae –Ank~ `~~rkoq~`oca‗ zlear n`yf {# cqabl#
n`yf ae –Ckea ~{za‗# a~i%
V
Z & V 7 v +
0=0>0?0:6=>?:
:%0>
:%0?
:%0:
:%:6
:%:=
:%:>
:%:?
:%::
Yi`~~arzcl~ lm Z&V7v+ }y V
Mkbqr` >%?% Bràmki` gly~r`enl c` zrlo`okckn`n# Z& V 7 v+ ae mqeikùe na c` }`rk`oca
`ca`~lrk` v%
V
Z & V 3 7 v +
0=0>0?0:6=>?:
0%:
:%6
:%=
:%>
:%?
:%:
Yi`~~arzcl~ lm Z&V37v+ }y V
Mkbqr` >%9% Bràmki` gly~r`enl c` zrlo`okckn`n `iqgqc`n`# Z&v ≮ V + ae mqeikùe na
c` }`rk`oca `ca`~lrk` v%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 197/846
Nr% Fëi~lr Pqa}anl Qrç`y
>'?>
Aharikikly I`zç~qcl >
>%0% Yqzùeb`ya pqa V ~kaea qe` nky~rkoqikùe Zlkyyle ile zrlgankl na >% I`ciqc`r
c`y ykbqkae~ay zrlo`okckn`nay8
`% Z& V 7 :+ &:%:069+
o% Z& V ≮ ?+ &:%?92;+
i% Z& V 7 >+ &:%0;19+
n% Z& V 7 6+ &:%0;19+
a% Z& V ≡ ?+ &:%;;62+
>%?% Yk c` zrlo`okckn`n na pqa qe ifapqa ya` na}qac~l zlr ac o`eil ay na :%:::9 {
0:#1:: ifapqay ya i`gok`e# ae~leiay# ¼iqàc ay ac eøgarl zrlgankl &δ l ±+ na
ifapqay mr`qnqcae~ly<
>%9% C` zrlo`okckn`n na pqa qe kenk}knql yqmr` na qe` g`c` ra`iikùe na qe`
ke{aiikùe ay na %::0% Na~argke`r c` zrlo`okckn`n pqa na ?#::: kenk}knqly#
av`i~`gae~a 9 yqmr`e qe` g`c` ra`iikùe% F`iar ay~a zrlocag` qy`enl c`
nky~rkoqikùe okelgk`c { c` Zlkyyle% &Qy`enl c` nky~rkoqikùe na Zlkyyle 7 :%06;94
qy`enl c` nky~rkoqikùe okelgk`c 7 :%060+
>%>% Ac eøgarl zrlgankl &δ ~+ na z`r~çiqc`y r`nk`i~k}`y pqa z`y`e ` ~r`}ëy na qe
ile~`nlr# nqr`e~a qe gkckyabqenl# ae qe avzarkgae~l na c`olr`~lrkl ay na 9% ¼Iqàc
ay c` zrlo`okckn`n na pqa ae~rae = z`r~çiqc`y ae qe gkckyabqenl na~argke`nl<
Yqbaraeik`8 qy`r z& v4δ ~+ 7 a'δ ~
&δ ~+ v, v!
>%1% Qe ay~qnkl ae qe` màorki` na `z`r`~ly acai~rùekily cca}ù `c barae~a ` ileicqkr
pqa ac eøgarl na `iiknae~ay# zlr zaryle`# nqr`e~a ikar~l `ðl ykbqa ` c` nky~rkoqikùe
Zlkyyle% C` avzarkaeik` nagly~rù pqa ac eøgarl zrlgankl na `iiknae~ay zlr
zaryle` mqa na :%9% ¼Iqàc ay c` zrlo`okckn`n na pqa qe agzca`nl el ~aenrà qe
`iiknae~a nqr`e~a ac ykbqkae~a `ðl< &:%2>0:+
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 198/846
Nr% Fëi~lr Pqa}anl Qrç`y
>'?1
>%=% Ile ramaraeik` `c zrlocag` >%1# ¼Iqàc ay c` zrlo`okckn`n na pqa# qe agzca`nl
yacaiikle`nl# `ca`~lrk`gae~a# ~aenrà iq`enl gaely 0 `iiknae~a & V ≡ 0+ ae ac
ykbqkae~a `ðl<>%2% _amkrkëenlya `c zrlocag` >%18
&`+ ¼Iq`c ay c` zrlo`okckn`n na pqa qe agzca`nl ~aenrà# av`i~`gae~a# qe
`iiknae~a< &:%??2:+
&o+ Ay~kg`r c` zrlo`okckn`n na pqa qe agzca`nl ~aenrà ` cl gày qe `iiknae~a%
>%6% @yqgkr pqa ac eøgarl na `q~ly pqa `rrko`e ` c` i`ya~` na ikar~` `q~lzky~` ykbqa
` qe` nky~rkoqikùe na Zlkyyle% Yk ac eøgarl zrlgankl na `q~ly pqa `rrko`e ae 0
flr` ay na =# ¼iqàc ay c` zrlo`okckn`n na pqa ae 0 flr` n`n`# el ccabqa ekebøe `q~l<
>%;% _amkrkëenlya `c zrlocag` `e~arklr# ¼Iqàc ay c` zrlo`okckn`n na pqa#
av`i~`gae~a# 1 i`rrly ccabqae ae 0 flr`< &:%02=:+
>%0:% _amkrkëenlya `c zrlocag` `e~arklr# ¼Iq`c ay c` zrlo`okckn`n na pqa gày na 1
i`rrly & V 5 1+ `rrkoae ae 0 flr`<
>%00% Ae qe ay~qnkl na ile~`gke`ikùe nac `kra# ae c`y ~argke`cay i`gkleay na i`rb`#
ya y`oa pqa ac eøgarl zrlgankl na i`gkleay pqa ccab`e nk`rk`gae~a# ` qe`
~argke`c na i`gkleay na i`rb` ay na 9% Z`r` pqa cly ilgzcahly f`ok~`ikle`cay el ya
ile~`gkeae ile cly fqgly na cly i`gkleay# ya ray~rkeba ac eøgarl na i`gkleay
pqa `rrko`e { ya ay~`ocaia qe cçgk~a na el gày na 6 zlr nç`% F`iar qe` bràmki` ile
cly rayqc~`nly { i`ciqc`r cl ykbqkae~a8
&`+ C` zrlo`okckn`n na pqa ya cay ekabqa c` ae~r`n` ` cly i`gkleay iq`enl ac
eøgarl avian` 6% &:%9;+&o+ C` zrlo`okckn`n na pqa `rrkoae ae~ra ? { > i`gkleay# keicqyl% &%::0?+
&i+ C` zrlo`okckn`n na pqa el `rrkoa ekebøe i`gkùe% &:%:1:+
&n+ Z& V 7 :# 0# ?# 9#%%%%%# ∕+ &0%:::+
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 199/846
Nr% Fëi~lr Pqa}anl Qrç`y
>'?=
&a+ @yqgkaenl pqa ay~a ay~qnkl ya fkikar` ae c` ëzli` i`cqrly`# ¼iqàc yarç` c`
nkmaraeik` ae cly rayqc~`nly# yk ac ay~qnkl ya fkikar` ae ke}karel8 `qgae~`rç` l
nkygkeqkrç` c` zrlo`okckn`n<>%0?% Yqzlekaenl pqa c` zrlo`okckn`n na pqa ikar~l ~kzl na yagkcc` el bargkea ya`
na %:>% Yk ya zc`e~`e ?1 yagkcc`y# ¼Iqàc ay c` zrlo`okckn`n na pqa 1 l gaely
yagkcc`y el bargkeae<
>%09% @yqgkr pqa ac eøgarl na `q~loqyay pqa ccab`e ` qe` ~argke`c ykb` ` qe
zrliayl Zlkyyle% Yk ac zrlgankl na `q~loqyay pqa ccab`e nqr`e~a qe` flr` ay na 1#
i`ciqc`r cly ykbqkae~ay aeqeik`nly8
&`+ C` zrlo`okckn`n na pqa ae 0 flr` n`n` el ccabqa ekebøe `q~loøy% &:%::2+
&o+ C` zrlo`okckn`n na pqa av`i~`gae~a 1 ccab`ràe ae 0 flr`% &:%02=+
&i+ C` zrlo`okckn`n na pqë gày na 1 `q~loqyay ccab`ràe ae qe` flr`% &:%96>+
>%0>% Ac eøgarl zrlgankl na i`rrly ~`epqa pqa `rrko`e i`n` nç` ` ikar~l zqar~l
g`rç~kgl ay na ;% C`y m`ikckn`nay zlr~q`rk`y zqanae g`eah`r ` cl gày 01 i`rrly
~`epqay & V ≮ 01+ zlr nç`% ¼Iqàc ay c` zrlo`okckn`n na pqa ae qe nç` n`nl ~aeb`e
pqa yar rabray`nl cly i`rrly ~`epqay iq`enl ac eøgarl avian` 01<
>%01% Ae c` g`eqm`i~qr` na qe `c`gora na ilora# yqzùeb`ya pqa ac eøgarl na m`cc`y
ykbqa ` qe` nky~rkoqikùe Zlkyyle# ile qe zrlgankl na ?%9 m`cc`y zlr gkcçga~rl%
I`ciqc`r c` zrlo`okckn`n na8
&`+ ^aear av`i~`gae~a ? m`cc`y ae qe gkcçga~rl nac c`gora na ilora% &:%?=1:+
&o+ ^aear 0: m`cc`y ae ikeil gkcçga~rly na `c`gora% &:%009+
&i+ C` zrlo`okckn`n na ~aear `c gaely qe` m`cc` ae nly gkcçga~rly% &%6;;+>%0=% Ae qe ay~qnkl na fkbkaea kenqy~rk`c { yabqrkn`n ay~`~`c# ac eøgarl na o`ifay
ae qe` yaiikùe na qe` i`rra~ar` ke~aray~`~`c pqa rapqkarae na raz`r`ikùe qrbae~a#
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 200/846
Nr% Fëi~lr Pqa}anl Qrç`y
>'?2
z`r` a}k~`r `iiknae~ay# zqana glnac`rya ile qe` nky~rkoqikùe Zlkyyle% Yk c`
i`rra~ar` ~kaea qe zrlgankl na ? o`ifay zlr gkcc`# ae~leiay8
&`+ ¼Iqàc ay c` zrlo`okckn`n na pqa el f`{` o`ifay pqa raz`r`r ae qe ~r`gl na 1gkcc`y<
&o+ ¼Iqàc ay c` zrlo`okckn`n na pqa ya` eaiay`rkl raz`r`r `c gaely 0 o`ifa ae qe
~r`gl na gank` gkcc`<
>%02% C` zrlo`okckn`n na pqa qe z`ikae~a ya raiqzara na qe` r`r` aemargan`n ay na
:%>% Yk ya y`oa pqa 01 zaryle`y f`e ile~r`çnl c` aemargan`n { `yqgkaenl qe`
nky~rkoqikùe okelgk`c# ¼Iqàc ay c` zrlo`okckn`n na pqa<8
&`+ Iq`enl gaely 0: z`ikae~ay ylora}k}`e Z& V ≡ 0:+% &:%996:+
&o+ Na 9 ` 6 z`ikae~ay ylora}k}`e Z&9 ≮ V ≮ 6+
&i+ Av`i~`gae~a 1 z`ikae~ay ylora}k}`e Z& V 7 1+ &:%061;+
>%06% Yk ac eøgarl na cc`g`n`y ~acamùeki`y pqa qe lzar`nlr raikoa ae qe ke~ar}`cl
na 0: gkeq~ly ykbqa ` qe` nky~rkoqikùe Zlkyyle# ileδ 7 ± 7 0 &qe zrlgankl na na
qe` cc`g`n` i`n` 0: gkeq~ly+% &Zm`mmaeoarbar# a~ `c% 0;62+% F`iar cl ykbqkae~a8
&`+ ¼Iqàc ay c` zrlo`okckn`n na pqa c` lzar`nlr` el raikokrà ekebqe` cc`g`n` ae ac
ke~ar}`cl na 0: gkeq~ly<
&o+ ¼Iqàc ay c` zrlo`okckn`n na pqa raikokrà gaely na > cc`g`n`y<
&i+ ¼Iqàc ay ac eøgarl gày zrlo`oca na cc`g`n`y pqa c` lzar`nlr` raikokrà<
>%0;% Yqzlekaenl pqa ~aeagly qe zrliayl na zrlnqiikùe na apqkzl na ile~rlc na
ikicleay pqa naoae na ~aear qe` amkikaeik` na railcaiikùe nac 21 { y`oagly pqa
c` zrlo`okckn`n na pqa el iqgzc`e ile ay~` amkikaeik` ay na z 7 :%:0% Qe` gqay~r``ca`~lrk` na 0:: ikicleay ya yacaiikle`% ¼Iqàc ay c` zrlo`okckn`n na pqa f`{` e
ikicleay pqa el iqgzc`e ile ac 21 na amkikaeik` ae ay~` gqay~r`< Qy`r c`
nky~rkoqikùe okelgk`c { c` nky~rkoqikùe na Zlkyyle ilgl qe` `zrlvkg`ikùe% F`iar
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 201/846
Nr% Fëi~lr Pqa}anl Qrç`y
>'?6
qe` ~`oc` ile cly rayqc~`nly na c`y nly nky~rkoqikleay { cqabl f`iar qe` br`mki` {
ilgz`r`r cly rayqc~`nly%
^`oc` gly~r`enl cly rayqc~`nly qy`enl c` nky~rkoqikùe okelgk`c { c` nky~rkoqikùe naZlkyyle% &Ac`olr`ikùe zrlzk`+
Nky~rkoqikùe okelgk`c Nky~rkoqikùe na Zlkyyle
e o& v40::#:%:0+ z& v40+
: :%9== :%:9? :%9=2 :%62;
0 :%9=; :%29: :%9=2 :%62;
? :%06> :%6=1 :%069 :%;>:
9 :%:=: :%;;; :%:=0 :%909
> :%:0> :%;>? :%:01 :%9?61 :%::? :%6;6 :%::9 :%:==
= :%::: :%>=9 :%::: :%100
2 :%::: :%:=9 :%::: :%:29
6 :%::: :%::2 :%::: :%::;
; :%::: :%::0 :%::: :%::0
>%?:% Ileyknëraya c` zrlnqiikùe na f`iar ilhkea~ay na rln`gkae~l &acagae~ly pqa
ykr}ae z`r` ylzlr~`r { bqk`r qe aha l àrolc na ~r`eygkykùe nac }afkiqcl+# yk c`
zrlo`okckn`n na qel na ëy~ly# na yar namai~qlyl# ay na :%:0% Yk ~aeagly qe`
gqay~r` na 0::: ilhkea~ay# i`ciqc`r c` zrlo`okckn`n pqa f`{`8 &^`rl [`g`ea# 0;=0+
&`+ v ilhkea~ay namai~qlyly
&o+ > ilhkea~ay namai~qlyly
&i+ iq`enl gaely 9 ilhkea~ay namai~qlyly%
>%?0% Ac zarkùnkil QY@ ^ln`{ &el}kagora 01# 0;;9+# razlr~ù pqa Z`rdmkacn#
I`ckmlrek`# ay~à ileyknar`n` ilgl c` i`zk~`c nac gqenl# ae iq`e~l ` ~agoclray na~karr` naoknl ` pqa ay~à yk~q`n` aeikg` na c` m`cc` na Y`e @enrëy% Nayna 0612#
Z`rdmkacn f` ~aeknl ~agoclray na ~karr` ile qe zrlgankl na qel i`n` ?? `ðly%
&Or`ya# a~ `c% 0;;1+%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 202/846
Nr% Fëi~lr Pqa}anl Qrç`y
>'?;
&`+ Avzcki`r zlrpqa c` nky~rkoqikùe na Zlkyyle yarç` qe` oqae` yacaiikùe z`r` r 7
eøgarl na ~arragl~ly ae qe ke~ar}`cl na ~kagzl n`nl
&o+ I`ciqc`r c` zrlo`okckn`n na pqa iq`enl gaely qe ~arragl~l# na br`enayg`bek~qnay# liqrr` ae cly ykbqkae~ay ?? `ðly% _anlena`r δ ` c`y iae~ëykg`y% &%=9+
&i+ I`ciqc`r c` zrlo`okckn`n na pqa el f`orà qe ~arragl~l na br`enay g`bek~qnay
ae cly ykbqkae~ay ?? `ðly% _anlena`r δ ` c`y iae~ëykg`y% &:%92+
&n+ I`ciqc`r c` zrlo`okckn`n na pqa liqrr` iq`enl gaely qe br`e ~arragl~l ae cly
ykbqkae~ay 1: `ðly% &0%:+
&a+ I`ciqc`r c` zrlo`okckn`n na pqa el liqrr` qe br`e ~arragl~l nae~rl cly
ykbqkae~ay 1: `ðly% &:%;6:+
>%??% Ae qe ay~qnkl na fkbkaea kenqy~rk`c { yabqrkn`n ya y`oa pqa ac eøgarl na
`iiknae~ay pqa z`y`e ae c` cçea` na aey`goc`ha ~kaea qe zrlgankl yag`e`c na 9%
Aeile~r`r cl ykbqkae~a8
&`+ C` zrlo`okckn`n na pqa qe` yag`e`# c` cçea` na aey`goc`ha el ~aenrà ekebøe
`iiknae~a%
&o+ C` zrlo`okckn`n na pqa# iq`enl gaely 9 `iiknae~ay liqrrkràe ae qe` yag`e`%
&i+ C` zrlo`okckn`n na pqa liqrr`e ae~ra ? { = `iiknae~ay avicqyl%
>%?9% Ae qe` ke}ay~kb`ikùe rac`ikle`n` ile ac `flrrl na ilgoqy~koca# ae ac >: na
cly ilifay el `garki`ely na > ikckenrly# ac ileyqgl na b`ylcke` ya ranqia
ileyknar`ocagae~a &ile rac`ikùe ` cly ilifay `garki`ely na = q 6 ikckenrly+% Yk ya
y`i` qe` gqay~r` `ca`~lrk` na 01 ilifay na > ikckenrly# i`ciqc`r c` zrlo`okckn`n pqa
> na ay~ly `q~ly na > ikckenrly ya`e amkikae~ay ae ac `flrrl na b`ylcke`% F`iar ay~a zrlocag` qy`enl c` nky~rkoqikùe okelgk`c { c` nky~rkoqikùe elrg`c ilgl
`zrlvkg`ikùe% &O&>401#:%:>+ 7 :%0?=6+4 Z&X 3 '%2++
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 203/846
Nr% Fëi~lr Pqa}anl Qrç`y
>'9:
>%?>% Ae qe ay~qnkl na fkbkaea kenqy~rk`c { yabqrkn`n# ay naikr# ae qe` màorki` na
aey`goc`gkae~ly na i`roqr`nlray ae c` m`orki`ikùe na `q~ly# ac eøgarl na
`iiknae~ay ae ay~` zc`e~` na aey`gocay ~kaea qe zrlgankl na 1%: `iiknae~ay zlr gay% Ykaenl `yç# ay~kg`r cly ykbqkae~ay aeqeik`nly8
&`+ ¼Iqàc ay c` zrlo`okckn`n na pqa liqrr`e 0: `iiknae~ay ae qe gay n`nl<
&o+ ¼Iqàc ay c` zrlo`okckn`n na pqa liqrr`e iq`enl gaely ? `iiknae~ay<
&i+ ¼Iqàc ay c` zrlo`okckn`n na pqa el f`oràe gày na ?1 `iiknae~ay ae qe `ðl<
>%?1% Qy`enl cly n`~ly na c` ~`oc` na Glcke` na `o`hl# z`r` }`rkly }`clray na V {
na δ gly~r`r pqë# ` gankn` pqa ac }`clr na δ &l ±+ `qgae~`# c`y nky~rkoqikleay na
Zlkyyle ya `zrlvkg`e ` c` nky~rkoqikùe elrg`c% Z`r` ay~l# f`iar qe` bràmki`
ylorazqay~` qy`enl cly }`clray na c` ~`oc` na Glcke` n`n` `o`hl%
^`oc` na Glcke` gly~r`enl cly n`~ly na ay~a zrlocag`% \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
]`rk`oca `ca`~lrk` V ]`clray na C`gon` &δ + \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
:%6 0%: ?%: 9%: 2%:
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
: :%>1 :%92 :%0> :%:1 :%::
0 :%9= :%92 :%?2 :%01 :%:0? :%0> :%06 :%?2 :%?? :%:?
9 :%:> :%:= :%06 :%?? :%:1
> :%:0 :%:? :%:; :%02 :%0:
1 :%:> :%0: :%09
= :%:0 :%:1 :%01
2 :%:? :%01
6 :%:0 :%01
; :%0:
0: :%:2
00 :%:1
0? :%:9
09 :%:00> :%:0
01 :%:: \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
Mqae~a8 ^`rl [`g`ea% Y~`~ky~kiy# @e Ke~rlnqi~lr{ @e`c{yky &0;=>+
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 204/846
Nr% Fëi~lr Pqa}anl Qrç`y
1'0
I@ZK^QCL 1
Nky~rkoqikleay na zrlo`okckn`n ile~keq`%' Mqeikùe na naeykn`n na zrlo`okckn`n
na c` }`rk`oca `ca`~lrk` ile~keq` V %' Mùrgqc` mqen`gae~`c nac iàciqcl%'
Nky~rkoqikùe elrg`c { yqy i`r`i~arçy~ki`y%' _ac`ikùe ae~ra c` iqr}` elrg`c { c`
okelgk`c%' Àra`y o`hl c` iqr}` elrg`c%' Nky~rkoqikùe avzleaeik`c%' Nky~rkoqikùe
B`gg`%' Nky~rkoqikùe Rakoqcc%' Ke~ar}`cly na ilemk`ex` z`r` ±%' Ay~`nçy~ki` na
kemaraeik`8 ~alrç` na naikykùe ay~`nçy~ki` { zrqao`y na fkzù~ayky%' Zrqao`y na
fkzù~ayky ay~`nçy~ki`y% Fkzù~ayky eqc` &Fl8+ a fkzù~ayky `c~are`~k}`y &F08# F?8# F98+%'
^kzly na arrlray K &`cm`+ { KK &oa~`+%' Zrqao`y na fkzù~ayky el ~r`nkikle`cay
qy`enl ac }`clr na c` zrlo`okckn`n z%' Zrqao`y na fkzù~ayky z`r` qel { nly
zrlgankly zloc`ikle`cay &±0# { ±?+%' Zrqao`y na fkzù~ayky z`r` c`y nkmaraeik`y na
nly zrlgankly zloc`ikle`cay &±0 ’ ±?+# z`r` gqay~r`y br`enay &e 9:+ qy`enl c`
nky~rkoqikùe elrg`c# ile }`rk`ex`y ilelikn`y a kbq`cay &ω?0 7 ω
??+%' Ke~ar}`cly na
ilemk`ex` z`r` nly zrlgankly zloc`ikle`cay%' Zrqao`y na fkzù~ayky a ke~ar}`cly
na ilemk`ex` z`r` zrlzlrikleay%
Namkekikùe na }`rk`oca ca`~lrk` ile~keq`
Qe` }`rk`oca `ca`~lrk` V ya nkia pqa ay ile~keq` yk# yq ilehqe~l na }`clray zlykocay ay
qe ke~ar}`cl ilgzca~l na eøgarly# ay~l ay# yk zlr ` 3 o iq`cpqkar eøgarl V ae~ra ` { o
ay zlykoca% Ae ~ërgkely ykgzcay# c` }`rk`oca `ca`~lrk` V ya namkea# ilgl c` }`rk`oca
`ca`~lrk` pqa ile~kaea qe ke~ar}`cl mkek~l l kemkek~l na eøgarly ra`cay% Na ay~` g`ear`#
qe` }`rk`oca pqa# ~aùrki`gae~a# zqan` `yqgkr iq`caypqkar }`clr ae~ra nly }`clray
n`nly# ay ile~keq`4 na l~r` g`ear` ay nkyira~`% Ae baear`c# gankikleay n`e cqb`r `
n`~ly ile~keqly# gkae~r`y pqa aeqgar`ikleay l ile~al n` cqb`r ` n`~ly nkyira~ly% Zlr
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 205/846
Nr% Fëi~lr Pqa}anl Qrç`y
1'?
ahagzcl# ac eøgarl na ekðly ae qe` m`gkck` zqanae ~aear }`clray na :# 0# 9# >#… zarl
el zqanae ~aear }`clray na ?%1# l 9%6>?% Zlr l~rl c`nl# ahagzcly na }`rk`ocay
ile~keq`y yle c`y `c~qr`y na qe brqzl na zaryle`y pqa ya zqanae avzray`r ilgl =?#=9%6 zqcb`n`y# =1%69>0? zqcb`n`y# a~i% Zlr ahagzcl# naikr iq`cay na cly ykbqkae~ay
~ërgkely razrayae~`e n`~ly nkyira~ly l ile~keqly%
&`+ Eøgarl na `iikleay oqryà~kcay }aenknly i`n` nç` ae ac gari`nl oqryà~kc%
Ylcqikùe8 @pqç c` }`rk`oca ay `ca`~lrk` nkyira~`%
&o+ C`y ~agzar`~qr`y rabky~r`n`y i`n` gank` flr` ae c` lmkike` na ga~alrlclbç`%
Ylcqikùe8 @pqç c` }`rk`oca ay `ca`~lrk` nkyira~`%
&i+ C`y clebk~qnay na 0::: ~lrekccly zrlnqiknly zlr qe` màorki`%
Ylcqikùe8 @pqç c` }`rk`oca ay ile~keq`%
Mqeikùe na naeykn`n na zrlo`okckn`n na c` }`rk`oca `ca`~lrk` ile~keq` V
Qe` mqeikùe m& v+ ay qe` mqeikùe na naeykn`n na zrlo`okckn`n na c` }`rk`oca `ca`~lrk`
ile~keq` V z`r` ac ilehqe~l na zlykocay }`clray na V ay~àe ae iq`cpqkar ke~ar}`cl na
eøgarly ra`cay T v0 # v?W% Ay~` mqeikùe ccae` cl ykbqkae~a8 &Gle~blgar{ a~ `c% 0;;=+
0% m v & v+ ≡ : &1'0+∕
?% ∤ '∕ m& v +nv 7 0 &1'?+v?
9% Z & v0 ≮ V ≮ v?+ 7 ∤ v0 m v&q+ nq &1'9+
Namkekikùe8 Nahagly pqa V ya` qe` }`rk`oca `ca`~lrk` ile~keq`# ae~leiay c`
nky~rkoqikùe na zrlo`okckn`n l mqeikùe na naeykn`n na zrlo`okckn`n na V ay qe`
mqeikùe m& v+ na ~`c g`ear` pqa# z`r` iq`caypqkar nly eøgarly ` { o ile ` ≮ o#
ae~leiay8
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 206/846
Nr% Fëi~lr Pqa}anl Qrç`y
1'9
o
Z&` ≮ V ≮ o+ 7 ∤ ` m& v+ nv &1'>+
Z`r` pqa c` mqeikùe na naeykn`n na zrlo`okckn`n ya` }àckn` naoa na y`~kym`iar c`y
ykbqkae~ay nly ilenkikleay8
0% m & v+ ≡ : z`r` ~ln`y c`y v‛y &1'1+
∕
?% ∤ '∕ m & v+ nv 7 àra` o`hl c` bràmki` ilgzca~` na m & v+ 7 0 &1'=+
Mùrgqc` mqen`gae~`c nac iàciqcl
Nahagly pqa m ya` qe` mqeikùe pqa ya namkea ae ac ke~ar}`cl iarr`nl na T`# oW#
o
ae~leiay# ac ke~abr`c namkeknl na m na ` ` o# nael~`nl zlr ∤ ` m&v+ nv ya n` zlr8 o
∤ ` m & v+ nv 7 ckg Ωm &k+ ∍vk &1'2+
ssZss↟: k
Iq`cpqkar `e~knark}`n` na m& v+ zqana yar qy`n` z`r` a}`cq`r ac ke~abr`c &1'2+%
Ae~leiay# yk M ay qe` e~knark}`n` na m# qy`gly c` mùrgqc` na o`hl8 o
∤ ` m & v+ nv 7 M&o+ ’ M&`+ &1'6+
Z`r` ilgzq~`r c` mqeikùe na rrko` M&o+ ’ M&`+ ke~rlnqikgly c` rac`ikùe8
o o
∤ ` m & v+ nv 7 M & v+ ┏ ` 7 M &o+ ’ M &`+ &1';+
Ahagzcl )0% Yk qe` }`rk`oca `ca`~lrk` ~kaea c` naeykn`n na zrlo`okckn`n na8
M & v+ ta'?v z`r` v 5 :# l z`r` v ≮ :w
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 207/846
Nr% Fëi~lr Pqa}anl Qrç`y
1'>
Na~argke`r c`y zrlo`okckn`nay na pqa c` }`rk`oca `ca`~lrk` V `nlz~a qe }`clr na8
&`+ Ae~ra 0 { 9
&o+ G`{lr pqa :%1&i+ @ cl gày 9
Ylcqikùe8 o
Qy`enl Z&` ≮ V ≮ o+ 7 ∤ ` m& v+ nv Ke~ar}`cl T0# 9W
9
&`+ ∤ 0 a'?v nv 7 &':%1+ Ta'? ’ a'=W 7 :%:=2
∕
&o+ ∤ :%1 a'?v nv Ke~ar}`cl T:%1# ∕W
9
&i+ ∤ : a'?v nv 7 &':%1+ Ta'= ’ 0W 7 :%1 Ke~ar}`cl T:# 9W
Ahagzcl )?% Ykgolckx`r ile V c` i`e~kn`n na ~kagzl na keiqo`ikùe na o`i~ark`y ae qe
zc`~l na zrqao` nqr`e~a ? flr`y% Yqzùeb`ya pqa c` }`rk oca `ca`~lrk` V ~kaea mqeikùe
na naeykn`n na m &v+ 7 :%1v# z`r` ac ilehqe~l zlykoca na }`clray na V ae ac ke~ar}`cl &:
≮ V ≮ ?+% Ykaenl `yç# i`ciqc`r c`y ykbqkae~ay zrlo`okckn`nay8
&`+ Z & V ≮ 0+
&o Z &%1 ≮ V ≮ 0%1+
&i+ Z &0%1 3 V +
Ylcqikùe8
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 208/846
Nr% Fëi~lr Pqa}anl Qrç`y
1'1
0 0
&`+ ∤ : :%1v nv 7 :%1 &v?,?+┏: 7 :%1 &:%1 ’ :,?+ 7 :%?1
0%1 0%1
&o+ ∤ :%1 :%1 v nv 7 :%1 &v?,?+┏:%1 7 &:%1+&0%0?1 ' %0?1+ 7 :%1
?%: ?%:
&i+ ∤ 0%1 :%1 v nv 7 :%1&v?,?+┏0%1 7 :%1&?%: ’ 0%0?1+ 7 :%>>
Ahagzcl )9% Yqzùeb`ya pqa ac arrlr ae c` ra`iikùe na ~agzar`~qr`# ae lI# na qe`
keiqo`nlr` na qe c`olr`~lrkl na o`i~arklclbç`# z`r` c` keiqo`ikùe na qe zc`~l na `b`r#ay qe` }`rk`oca V ile~keq` pqa ~kaea qe` naeykn`n na zrlo`okckn`n na m&v+ 7 v?,9#
nlena V zqana `yqgkr }`clray na ae~ra &'0 3 V ≮ ?+% Aeile~r`r c` zrlo`okckn`n na
naeykn`n na pqa c` ~agzar`~qr` ay~ë ae~ra : lI { 0 lI%
Ylcqikùe8
@pqç pqaragly aeile~r`r Z&: 3 V ≮ 0+ ae ac ke~ar}`cl T:#0W% Ae~leiay…0 0
Z&: 3 V ≮ 0+ 7 ∤ : v?,9 nv 7 v9,; ┏ : 7 0,;
Ahagzcl )>% C` zrlzlrikùe na kenqy~rk`y pqa rayzlenae ` ikar~l iqay~kle`rkl
ailcùbkil &}lcqe~`rkl# zarl pqa# `i~q`cgae~a# }` ` yar lockb`~lrkl+ ay qe` }`rk`oca
`ca`~lrk` ile~keq` V iq{` mqeikùe na naeykn`n ay m&v+ 7 ?&v $ ?+,1% Ay~` mqeikùe ~kaea
qe` }`rk`oca `ca`~lrk` V zqana `yqgkr }`clray na : 3 V 3 0% F`iar cl ykbqkae~a8
&`+ Gly~r`r pqa Z&: 3 V 3 0+ 7 0
&o+ Aeile~r`r c` zrlo`okckn`n na pqa gày na ?1# zarl gaely pqa 1: na c`y
kenqy~rk`y ile~`i~`n`y rayzlenaràe }lcqe~`rk`gae~a ` ay~` ylckik~`ikùe%
Ylcqikùe8
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 209/846
Nr% Fëi~lr Pqa}anl Qrç`y
1'=
&`+ Qy`enl c` rac`ikùe g`~agà~ki` na m&v+ { b&v+ iq{`y mqeikleay yle ile~keq`y {
~kaeae qe` `e~knark}`n` ae ac ke~ar}`cl T`# oW# ykaenl `yç# ae~leiay# qy`gly c` mqeikùe
&1'0:+ na `o`hl { yqy~k~q{aenl n`8
o o o
∤ ` Tm&v+ $ b&v+W nv 7 ∤ ` m&v+ nv $ ∤ ` b&v+ nv &1'0:+
0 0 0
∤ : &?v,1 $ >,1+ nv 7 ∤ : ?v,1 nv $ ∤ : >,1 nv 7 ?v?,&?+&1+ $ >v,10 0
7 v?,1┏:
$ >v,1┏:
7 T0,1 ’ :W $ T>,1 ’ :W7 0
&o+ @pqç ac ke~ar}`cl ay T:%?1 3 V 3 %1:W% Ay~l nkia pqa# ` 7 :%?1 { o 7 :%1:%1: %1: %1:
Zlr cl ~`e~l8 ∤ %?1 &?v,1 $ >,1+ nv 7 v?,1┏ %?1 $ >v,1 ┏ %?1
7 T&:%1+? ,1 ’ &:%?1+?,1W $ T>&:%1+,1 ’ >&:%?1+,1W7 0;,6:
C` nky~rkoqikùe elrg`c
C` nky~rkoqikùe elrg`c ay ac ahagzcl gày kgzlr~`e~a na qe` nky~rkoqikùe na
zrlo`okckn`n ile~keq % @or`f`g Na Glk}ra &0==2'021>+ c` kekikù ae 0299%
Nay`mlr~qe`n`gae~a# yq ~r`o`hl ya zarnkù {# i`yk 0:: `ðly nayzqëy# D`rc B`qyy &0222'
0611+ { Zkarra Ykgle# G`rpqay na C`zc`ia nay`rrlcc`rle# na g`ear` kenazaenkae~a# c`
nky~rkoqikùe elrg`c% Zlr ay~` r`xùe# ` c` nky~rkoqikùe elrg`c ~`gokëe ya ca cc`g`
nky~rkoqikùe B`qyyk`e`%I`r`i~arçy~ki y na c` nky~rkoqikùe elrg`c
0% Ay ykgë~rki` `crananlr na yq zrlgankl ± { ae mlrg` na i`gz`e`%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 210/846
Nr% Fëi~lr Pqa}anl Qrç`y
1'2
?% Ac zrlgankl# c` gank`e` { c` gln` yle kbq`cay%
9% Ac àra` ~l~`c o`hl c` iqr}` ay kbq`c ` qel% Ac 1: na c`y loyar}`ikleay ay~àe ` c`
naraif` nac zrlgankl { ac l~rl 1: na c`y loyar}`ikleay ay~àe ` c` kxpqkarn` nac zrlgankl%
>% C` nky~rkoqikùe elrg`c ya na~argke` ilgzca~`gae~a zlr yqy z`ràga~rly ± { ω%
Iq`enl ± 7 : { ω 7 0 c` nky~rkoqikùe elrg`c ay~à ae yq mlrg` ay~`en`rkx`n`%
C` nky~rkoqikùe elrg`c ay ra`cgae~a qe` m`gkck` na nky~rkoqikleay nky~kebqkn`
qe` na c` l~r` zlr cly }`clray na ± { na ω% Yke ago`rbl# ac gkagorl gày kgzlr~`e~a na
ay~` m`gkck` na nky~rkoqikleay ay c` pqa ~kaea qe zrlgankl na : { qe` nay}k`ikùe
ay~àen`r na 0% C` aiq`ikùe na c` nky~rkoqikùe elrg`c ay~àen`r ya ayirkoa ilgl8
m &x+ 7 0,∘ ?ϊ avz ':%1 x ? ' ∕ 3 x 3 $ ∕ &1'00+
Qyq`cgae~a# ya qy` c` ca~r` gkeøyiqc` x zlr c` }`rk`oca `ca`~lrk` pqa rayqc~`%
Àra`y o`hl c` iqr}` elrg`c
C` iqr}` na iq`cpqkar nky~rkoqikùe ile~keq` na zrlo`okckn`n l mqeikùe na naeykn`n ya
iley~k~q{a na ~`c glnl pqa# ac àra` ay~à ckgk~`n` zlr cly nly zqe~ly v 7 vk { v 7 v? { ay
kbq`c ` c` zrlo`okckn`n na pqa c` }`rk`oca `ca`~lrk` V `yqg` qe }`clr ae~ra v 7 vk { v 7
v?% Ae~leiay# ac àra` z`r` c` iqr}` elrg`c ya n` zlr c` mqeikùe &R`czlca a~ `c% 0;;?+8
V?
Z&v0 3 V 3 v?+ 7 ∤ v0 e &v4±#ω+ nv &1'0?+v?
7 0,∘ ?ϊ ω ∤ v0 avz'&:%1+T&v'±+,ωW? nv &1'0?`+
Yke ago`rbl# ay nkmçikc raylc}ar c`y ke~abr`cay na c`y mqeikleay na naeykn`n elrg`c#
naoknl ` pqa el ya zqanae ke~abr`r ae mlrg` iarr`n`# ae~ra i`n` z`r na cçgk~ay na ` {
o% Naoknl ` ay~` yk~q`ikùe ya f`ia eaiay`rkl c` ~`oqc`ikùe na c`y àra`y na c` iqr}`
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 211/846
Nr% Fëi~lr Pqa}anl Qrç`y
1'6
elrg`c% Na iq`cpqkar g`ear`# c` ~`oc` na c` nky~rkoqikùe elrg`c ay~àen`r ile ± 7 : { ω
7 0 { yqy ae~r`n`y yle cly }`clray na8
x
M&x+ 7 0,∘ ?ϊ ∤ ∕ avz T':%1 ~?W n~ &1'09+
@nagày# yarç` gq{ nkmçikc f`iar qe` ~`oc` zlr yaz`r`nl z`r` i`n` }`clr na ± {
ω% @mlr~qe`n`gae~a# ay zlykoca ~r`eymlrg`r ~ln`y c`y loyar}`ikleay na iq`cpqkar
}`rk`oca `ca`~lrk` elrg`c V ` eqa}l ilehqe~l na loyar}`ikleay na qe` }`rk`oca
`ca`~lrk` elrg`c x ile zrlgankl na : { }`rk`ex` na 0% C`y ~r`eymlrg`ikleay ya f`iae
qy`enl c` mùrgqc` na c` }`rk`oca `ca`~lrk` elrg`c ay~`en`rkx`n` x pqa ya qy` z`r`~r`eymlrg`r iq`cpqkar }`rk`oca `ca`~lrk` elrg`c V ile zrlgankl ± { nay}k`ikùe
ay~àen`r ω ` c` nky~rkoqikùe elrg`c ay~`en`rkx`n`% Ay~` mùrgqc` z`r` i`ciqc`r c`y
zrlo`okckn`nay na iq`cpqkar nky~rkoqikùe elrg`c &el ay~`en`rkx`n + ya n` ilgl8
V ’ ±X 7 „„„„ &1'0>+
ω
Yq ay~kg`nlr na c` gqay~r` ay8
x 7 &V ’ V + , y &1'01+
Ilgl ya nkhl `rrko`# f`{ ~`oc`y pqa n`e cly rayqc~`nly na c` ke~abr`ikùe# zlr cl
~`e~l# el ~aeagly pqa f`iar c` ke~abr`ikùe z`r` i`ciqc`r c`y zrlo`okckn`nay% Ae iq`e~l
` c`y zrlzkan`nay na c` nky~rkoqikùe elrg`c# ëy~`y ya n`e ae c` ^@OC@ 1%: na `o`hl%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 212/846
Nr% Fëi~lr Pqa}anl Qrç`y
1';
^@OC@ 1%:% ^`oc` gly~r`enl c`y zrlzkan`nay na c` iqr}` elrg`c l B`qyyk`e`% \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ Zrlgankl `rk~gë~kil ±
]`rk`ex` ω?
Nay}k`ikùe ay~àen`r ω Glgae~l nac ilamkikae~a nac yaybl ΰ9 7 :Glgae~l nac ilamkikae~a na dqr~lyky ΰ> 7 9 Nay}k`ikùe nac zrlgankl `rk~gë~kil ω∘ ?,ϊ 7 :%2;2; ω
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ &Mqae~a8 Yzkabac# 0;=0+
C` bràmki` na c` iqr}` elrg`c ya gqay~r` ae c` mkbqr` na `o`hl% Na c` bràmki` zqana
}arya pqa# c` iqr}` ay ae mlrg` na i`gz`e`# ay ykgë~rki` ile rayzai~l ` c` cçea`nkoqh`n` zarzaenkiqc`rgae~a# `c aha flrkxle~`c ae ac zrlgankl ±% C` bràmki` }` ae
mqeikùe na c` mraiqaeik` rac`~k}` { c`y nay}k`ikleay ay~àen`ray% C` nay}k`ikùe ay~àen`r
na~argke` ac ayz`rikgkae~l na c` iqr}`% @ gankn` pqa f`{ gày }`rk`ikùe ae qe`
gqay~r` l ae qe` zloc`ikùe# c` iqr}` ya f`ia gày `if`~`n`% Ac àra` ~l~`c o`hl c` iqr}`
ay na 0::% C`y bràmki`y na `o`hl gqay~r`e c` nky~rkoqikùe elrg`c ae nkmarae~ay
mlrg`y%
Mkbqr` 1%0% Aypqag` gly~r`enl c`y àra`y o`hl c` iqr}` elrg`c% &Or`ya a~ `c% 0;;1+
Na c` Mkbqr` 1%0# ya }a pqa# `zrlvkg`n`gae~a# ac =6%?2 na cly }`clray na cly n`~ly
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 213/846
Nr% Fëi~lr Pqa}anl Qrç`y
1'0:
ay~àe nae~rl na qe` nay}k`ikùe ay~àen`r ` i`n` c`nl nac zrlgankl4 `zrlvkg`n`gae~a#
ac ;1%>1 na cly }`clray ay~àe nae~rl na nly nay}k`ikleay ay~àen`r ae i`n` c`nl nac
zrlgankl4 { `zrlvkg`n`gae~a# ac ;;%29 na c`y loyar}`ikleay ay~àe nae~rl na ~raynay}k`ikleay ay~àen`r nac zrlgankl%
Mkbqr` 1%? % Ay~` mkbqr` gqay~r` c`y àra`y ilrrayzlenkae~ay ` }`clray na x 7 °0# x 7 °?
{ x 7 °9# ilrrayzlenkae~ay ` c`y zrlo`okckn`nay na =6%?2# ;1%>1 { ;;%29#
rayzai~k}`gae~a% &Ac`olr`ikùe zrlzk`+%
Na c` Mkbqr` 1%?# eù~aya pqa# ae ~ërgkely na mraiqaeik` rac`~k}`# c` zrlo`okckn`n ae~ra x 7 : { x 7 $0#
ay kbq`c ` %9>094 c` zrlo`okckn`n ae~ra x 7 $0 { x 7 ? ay kbq`c ` %091; {# c` zrlo`okckn`n ae~ra x 7 $?
{ x 7 $9 ay kbq`c ` %:??6% Zlr ykga~rç` ac àra` ~l~`c ay ≆ 0%
Nky~rkoqikùe elrg`c ay~àen`r
Ilgl ya nkhl `e~arklrgae~a# c` nky~rkoqikùe na qe` }`rk`oca `ca`~lrk` elrg`c# ile
zrlgankl kbq`c ` : { }`rk ex` kbq`c ` 0# ya cc`g` nky~rkoqikùe elrg`c ay~àen`r { ya
nael~` ile c` }`rk`oca `ca`~lrk` elrg`c ay~`en`rkx`n` X 7 & V ’ ±+,ω% Yke ago`rbl#
iq`enl ya ilelia ac zrlgankl { c` nay}k`ikùe ay~àen`r# ya cc`g` nky~rkoqikùe elrg`cel ay~`en`rkx`n`% @pqç# ay kgzlr~`e~a nky~kebqkr c`y nkmaraeik`y ae~ra c` iqr}` elrg`c
ay~àen`r { c` el ay~àen`r%
Ahagzcl )1% N`n` qe` nky~rkoqikùe elrg`c ay~àen`r# aeile~r`r ac àra` o`hl c` iqr}`
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 214/846
Nr% Fëi~lr Pqa}anl Qrç`y
1'00
pqa ay~ë ae~ra x 7 '0%;2 { x 7 :%6=%
Ylcqikùe8
Ac àra` ae~ra x 7 '0%;2 { x 7 :%6= ya nael~` ilgl8
Z&'0%;2 3 x 3 :%6=+ 7 Z&x 3 :%6=+ ’ Z&x 3 '0%;2+7 :%6:10 ’ :%:?>> 7 :%26:2 &na c` ~`oc` na c` nky~rkoqikùe elrg`c+
Cly }`clray na c` zrlo`okckn`n pqa ilrrayzlenae ` c` }`rk`oca ay~`en`rkx`n` x yle na x
7 :%6= { x 7 '0%;2 { ya aeiqae~r`e ac c` ~`oc` na c` nky~rkoqikùe elrg`c%
Ahagzcl )=% N`n` qe` nky~rkoqikùe elrg`c# ile ± 7 1: { ω 7 0:% Aeile~r`r c`
zrlo`okckn`n na pqa V ay~ë ae~ra >1 { =?%
Ylcqikùe8
Zrkgar`gae~a# `pqç ay~`gly f`oc`enl na ~lnl ac ilehqe~l# el na qe` gqay~r`% Z`r`
ay~l ~r`eymlrg`gly cly }`clray na V ` }`clray na X qy`enl c` mùrgqc` na c` }`rk`oca
`ca`~lrk` ay~`en`rkx`n` &1'0>+# X 7 & V ’ ±+ , ω% Cly }`clray na X ilrrayzlenkae~ay ` V
7 >1 { V 7 =? ya f`iae qy`enl ay~` mùrgqc`% Yqy~k~q{aenl cly }`clray n`8
X>1 7 &>1 ’ 1:+,0: 7 ':%1[ X=? 7 &=? ’ 1:+,0: 7 0%?
Zlr cl ~`e~l8 Z&>1 3 V 3 =?+ 7 Z&':%1 3 X 3 0%?+
7 Z&X 3 0%?+ ' Z&X 3 ':%1+
7 :%66>; ’ :%9:61
7 :%12=>
Ahagzcl )2% Ya y`i` qe` gqay~r` `ca`~lrk` na qe` nky~rkoqikùe elrg`c zloc`ikle`c {
ya i`ciqc` qe zrlgankl na V 7 9:: { qe` nay}k`ikùe ay~àen`r na 1:% Aeile~r`r c`
zrlo`okckn`n na pqa V `yqg` qe }`clr g`{lr pqa 9=?%
Ylcqikùe8
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 215/846
Nr% Fëi~lr Pqa}anl Qrç`y
1'0?
Zrkgar`gae~a# `pqç ya el~` pqë# {` el ay~`gly f`oc`enl na qe` zloc`ikùe qy`enl ±#
ilgl ac zrlgankl { ω ilgl c` nay}k`ikùe ay~àen`r% @flr` qy`gly c` }`rk`oca elrg`c
ay~`en`rkx`n` na c` gqay~r` na x 7 & V ’ V +,y# c` iq`c ay ac ay~kg`nlr na c` }`rk`oca X 7& V ’ ±+,ω% Ykaenl `yç# zrkgarl ~r`eymlrg`gly V 7 9=? `c }`clr ilrrayzlenkae~a na x
qy`enl c` mùrgqc` `e~arklr% &Eù~aya pqa ae ay~a i`yl qy`gly c` mùrgqc` na c` }`rk`oca
ay~`en`rkx`n` x ilgl ac ay~kg`nlr na c` }`rk`oca X zloc`ikle`c# ay~l ay# x 7 & V ’
V +,y+%
x9=? 7 &9=? ’ 9::+,1: 7 0%?>
Zlr cl ~`e~l# Z& V 5 9=?+ 7 Z&x 5 0%?>+ 7 0 ’ Z&x 3 0%?>+7 0 ’ :%6;?1 7 :%0:21
C` nky~rkoqikùe elrg`c ay qe` nky~rkoqikùe na zrlo`okckn`n ile~keq` &ae
ile~r`y~a ile c` Zlkyyle# okelgk`c# fkzarbalgë~rki`# balgë~rki`# a~i%# pqa yle
nky~rkoqikleay na zrlo`okckn`n nkyira~`y+% Ay~l pqkara naikr pqa# cly rayqc~`nly na qe
avzarkgae~l na zrlo`okckn`n ileyky~ae na qe keeqgar`oca a kemkek~l ile~al na
}`clray% @yç# qe` nky~rkoqikùe na zrlo`okckn`n ile~keq` ely zargk~a gankr eqay~r`}`rk`oca ` iq`cpqkar br`nl na zraikykùe rapqarkn` { ay~à `ylik`n` ile }`rk`ocay
`ca`~lrk`y ile~keq`y% Ae ile~r`y~a# c`y nky~rkoqikleay na zrlo`okckn`n nkyira~` yle
ilgl c` nky~rkoqikùe okelgk`c l na Oarelqcck { c` Zlkyyle# c`y iq`cay ay~àe `ylik`n`y
ile }`rk`ocay ca`~lrk`y nkyira~`y%
C`y }`rk`ocay nkyira~`y yle gankikleay zraiky`y% Ahagzcly na }`rk`ocay
nkyira~`y yle ac ~`g`ðl na qe` m`gkck` l ac eøgarl na `q~ly pqa ya ~kaeae# l ac
eøgarl na ay~qnk`e~ay na qe` ic`ya% ^ln`y ay~`y yle }`rk`ocay nkyira~`y% Ay~l pqkara
naikr pqa# iq`enl `cbl ya zqana gankr ile zraikykùe# ae~leiay# ay qe` }`rk`oca
nkyira~`% Ae ile~r`y~a# c`y }`rk`ocay ile~keq`y el ya zqanae gankr# zraiky`gae~a#
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 216/846
Nr% Fëi~lr Pqa}anl Qrç`y
1'09
iq`enl keiragae~`gly c` zraikykùe na c` gankikùe y`i`gly qe yky~ag` na ile~al gày
mkel% Qe` }`rk`oca ile~keq` el }kaea ae z`pqa~ay na qekn`n# ykel pqa gkna l
razrayae~` qe br`nl na zraikykùe `rok~r`rkl# ay naikr# ranlena`nl% Ahagzcly na}`rk`ocay ile~keq`y yle cly zayly# c` ~agzar`~qr`# c` `c~qr`# c`y naeykn`nay# a~i%
_ac`ikùe ae~ra c` iqr}` elrg`c { c` okelgk`c
Yk e ay br`ena {# yk ek z ek ~`gzlil p ay~àe iari`e`y ` iarl# c` nky~rkoqikùe okelgk`c
zqana `zrlvkg rya ` c` elrg`c# ile c` }`rk oca ay~`en`rkx`n` n`n` zlr c` mqeikùe
ay~`nçy~ki` x 7 &v ’ ez+, ∘ ezp%
C` `zrlvkg`ikùe elrg`c ` c` nky~rkoqikùe okelgk`c ay oqae` yk e ay o`y~`e~a
br`ena ile rayzai~l ` z% Ae z`r~kiqc`r# ay~l ay ikar~l iq`enl ez 5 0: { e&0 ' Z+ 5 0:%
Z`r` f`iar c`y `zrlvkg`ikleay okelgk`cay qy`enl c` nky~rkoqikùe elrg`c# ya qy` c`
}`rk`oca `ca`~lrk` ay~`en`rkx`n` x% &¼Iqàc }arykùe na c` }`rk`oca `ca`~lrk` na x ya
qy`rç`8 c` }`rk`oca `ca`~lrk` ay~`en`rkx`n` na x zloc`ikle`c l c` }`rk`oca `ca`~lrk`
ay~`en`rkx`n` na x gqay~r`c<+%
Zlr l~r` z`r~a# C`zke &0;6?+ rai`zk~qc` ac faifl na pqa c` bràmki` na c`
nky~rkoqikùe okelgk`c ~kaena ` c` nky~rkoqikùe elrg`c ` gankn` pqa e `qgae~`% Ay~l
yqbkara pqë# z`r` gqay~r`y na ~`g`ðly br`enay# c` nky~rkoqikùe okelgk`c ya `zrlvkg`
` c` elrg`c% C`zke n` c`y bqç`y `iaz~`n`y z`r` qy`r c`y `zrlvkg`ikleay elrg`cay# na
`iqarnl ` c` rabc` zlzqc`r na pqa c` `zrlvkg`ikùe elrg`c ` c` nky~rkoqikùe okelgk`c
ay `naiq`n`# ykagzra { iq`enl# ez ≡ 1 { e&0 ’ z+ ≡ 1% C` ^@OC@ 1%0 gqay~r` c`y
bqç`y ilgøegae~a `iaz~`n`y z`r` qy`r c` `zrlvkg`ikùe elrg`c ` c` nky~rkoqikùe
okelgk`c% Yke ago rbl# ya `rbqgae~ pqa `cbqely ay~`nçy~kily keyky~ae na pqa~`g`ðly na gqay~r`y gày br`enay pqa cly n`nly ae c` ^@OC@ 1%0 naoae yar qy`nly
`e~ay na pqa c` `zrlvkg`ikùe ya` `iaz~`oca% Ay~l ya naoa ` pqa# ac yaybl na c`
nky~rkoqikùe okelgk`c ay ~`e zrleqeik`nl z`r` ~`g`ðly na z br`enay l zapqaðly na ~`c
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 217/846
Nr% Fëi~lr Pqa}anl Qrç`y
1'0>
g`ear` pqa# c` mlrg` na i`gz`e` ya `yqga zlr c` nky~rkoqikùe okelgk`c# ylc`gae~a#
z`r` qe ~ g`ðl na e gq{ br ena%
^@OC@ 1%0% ^`oc` gly~r`enl c`y bqç`y gày ilgøegae~a `iaz~`n`y z`r` qy`r c``zrlvkg`ikùe elrg`c ` c` nky~rkoqikùe okelgk`c%
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ Ykagzra pqa z kbq`ca `8 Qy`r c` `zrlvkg`ikùe elrg`c# ylc`gae~a#
yk e el ay gày zapqað` pqa8 \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
%1 0:%>: l %=: 09%9: l %2: 02%?: l %6: ?1%0: l %;: 1:%:1 l %;1 0::%:0 l %;; 1::%::1 l %;;1 0#:::%::0 l %;;; 1#:::
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ Mqae~a8 Y~`~ky~kiy mlr Glnare Oqykeayy Naikykle% C`raia C% C`zke &0;60+%
Cly ykbqkae~ay ahagzcly i`ciqc`e c`y zrlo`okckn`nay z`r` c` nky~rkoqikùe elrg`cay~`en`rkx`n`
Ahagzcl )6% ¼Iqàc ay ac àra`# c` zrlo`okckn`n# zrlzlrikùe l ac zlriae~`ha na aeile~r`r
qe }`clr na x o`hl c` iqr}` l nky~rkoqikùe elrg`c ae~ra cly }`clray na x 7 '0%29 { x 7
$?%>1< Nkoqh`r c` bràmki`%
Ylcqikùe8
Nackea`r ac ke~ar}`cl na c` }`rk`oca `ca`~lrk` x# ay~l ay# &'0%29 ≮ x ≮ ?%>1+ r`xle`enl
na c` ykbqkae~a g`ear`8
Z&'0%29 ≮ x ≮ ?%>1+ 7 Z&x ≮ ?%>1+ ’ Z&x ≮ '0%29+
Oqyi`enl x 7 ?%>1 ae c` ~`oc` na c` nky~rkoqikùe elrg`c n` qe` zrlo`okckn`n na %;;?;%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 218/846
Nr% Fëi~lr Pqa}anl Qrç`y
1'01
Aeyabqkn`# ya f`ia cl gkygl ile ac }`clr na x 7 '0%29 { n` qe` zrlo`okckn`n na %:>06%
Zlr cl ~`e~l8
Z&'0%29 ≮ x ≮ ?%>1+ 7 Z&x ≮ ?%>1+ ’ Z&x ≮ '0%29+7 %;;?; ' %:>06
7 %;100
Mkbqr` 1%9% Bràmki` na c` iqr}` elrg`c z`r` ac Ahagzcl )6% &Ac`olr`ikùe zrlzk`+
Ahagzcl );% ¼Iqàc ay c` zrlo`okckn`n# ae c` iqr}` elrg`c ae~ra qe }`clr na x 7 '0%1> {
qe }`clr na x 7 '%2=<
Ylcqikùe8
Z&'0%1> ≮ x ≮ '%2=+ 7 Z&x ≮ '%2=+ ’ Z&x ≮ '0%1>+
7 %??9= ' %:=06 7 %0=06 &na c` ~`oc` na x+
Ahagzcl )0:% ¼Iqàc ay ac àra` o`hl c` iqr}` elrg`c ` c` kxpqkarn` na x 7 '0%;=<
Ylcqikùe8
Ae c` ~`oc` na c` nky~rkoqikùe elrg`c ya oqyi` ac }`clr na c` }`rk`oca `ca`~lrk` x 7 '0%;=
{ n` %:?1 ay naikr#
Z&x 3 '0%;=+ 7 %:?1:
Ahagzcl )00% ¼Iqàc ay ac àra` o`hl c` iqr}` elrg`c ` c` kxpqkarn` na qe }`clr na x 7
0%>?<
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 219/846
Nr% Fëi~lr Pqa}anl Qrç`y
1'0=
Ylcqikùe8
Ya oqyi` ac }`clr na x 7 0%>? ae c` ~`oc` na x { n` %;???% Ay~l ay cl gkygl pqa# àra`
rapqarkn` na %1::: $ %>??? 7 %;???%Ahagzcl )0?% Aeile~r`r c` zrlo`okckn`n na pqa c` }`rk`oca X ay~ë ae~ra ':1 { 0%?1%
Ylcqikùe8
Z&':%1 3 X 3 0%?1+ 7 0%: ’ :%9:61 ’ :%0:1= 7 :%161;
C` Mkbqr` 1%> gqay~r` aypqagà~ki`gae~a ay~` yk~q`ikùe%
Mkbqr` 1%>% Mkbqr` gly~r`enl c` zrlo`okckn`n na Z&':1 3 X 3 0%?1+% &Ac`olr`ikùe zrlzk`+
Cly ykbqkae~ay zrlocag`y i`ciqc`e c`y zrlo`okckn`nay z`r` iq`cpqkar }`rk`oca
elrg`c nky~rkoqkn` qy`enl c` }`rk`oca ay~`en`rkx`n` X 7 & V ' ±+,ω {,l yq
ay~kg`nlr ay~`nçy~kil ilrrayzlenkae~a# x 7 & V ' V +,y%
Ahagzcl )09% Yk V ay qe` }`rk`oca elrg`cgae~a nky~rkoqkn`# ile qe zrlgankl
`rk~gë~kil na V 7 ?> { qe` nay}k`ikùe ay~àen`r na 9# ¼iqàc ay ac }`clr na c` }`rk`oca
elrg`c ay~`en`rkx`n` &~kzkmki`n`+# x pqa ilrrayzlena ` qe }`clr na V 7 0;<
Ylcqikùe8
Zrkgarl ~r`eymlrg`gly &ay~`en`rkx`gly+ ac }`clr na V 7 0; ` }`clray na x# ay naikr8
x0; 7 & V ' V +,y 7 &0; ' ?>+,9 7 '0%=2
@pqç# ya el~` pqa# ac }`clr na V 7 0; ay~à 0%=2 nay}k`ikleay ay~àen`r `o`hl nac
zrlgankl na ?>%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 220/846
Nr% Fëi~lr Pqa}anl Qrç`y
1'02
Ahagzcl )0>% Yk V ay qe` }`rk`oca elrg`cgae~a nky~rkoqkn`# ile qe zrlgankl
`rk~gë~kil na 01: { qe` nay}k`ikùe ay~àen`r na ?># ¼iqàc ay ac }`clr na x
ilrrayzlenkae~a ` qe }`clr na V 7 06?<Ylcqikùe8
x06? 7 &06? ' 01:+,?>
7 0%99
Ay~a }`clr na 06? ay~à ` 0%99 nay}k`ikleay ay~àen`r `rrko` nac zrlgankl na V 7 01:%
Ahagzcl )01% Yk V ay qe` }`rk`oca elrg`cgae~a nky~rkoqkn`# ile qe zrlgankl na 0::
{ qe` nay}k`ikùe ay~àen`r na 01# i`ciqc`r c` zrlo`okckn`n na8 Z&2: 3 V 3 09:+%Ylcqikùe8
Zrkgarl ~r`eymlrg`gly &ay~`en`rkx`gly+ cly }`clray na V 7 2: { V 7 09: ` }`clray na
c` }`rk`oca `ca`~lrk` x% Ay~l ay8
x2: 7 &2: ' 0::+,01 7 '?%::
x09: 7 &09: ' 0::+,01 7 ?%::
Ac }`clr na x ilrrayzlenkae~a `c ke~ar}`cl &2: 3 V 3 09:+ ay na &'?%:: 3 x 3 $?%::+ {
c` zrlo`okckn`n ay8
Z&2: ≮ V ≮ 09:+ 7 Z&'?%: ≮ x ≮ ?%:+
7 Z&x ≮ ?%:+ ’ Z&x ≮ '?%:+
7 %;22? ' %:??6
7 :%;1>>
@pqç# ya zqana }ar pqë# yke ileyqc~`r c` ~`oc` na c` x# {` y`oagly pqa# ` ?%: qekn`nay
na x `rrko` nac zrlgankl ay~àe ilgzraenkn`y ac ;2%2? na c`y loyar}`ikleay%Ykgkc`rgae~a# ` '?%: qekn`nay `o`hl nac zrlgankl ay~àe ilgzraenkn`y ac ?%?6 na c`y
loyar}`ikleay4 zlr cl ~`e~l# %;22? ' %:??6 7 :%;1>>% C` Mkbqr` 1%1 gqay~r` ay~l%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 221/846
Nr% Fëi~lr Pqa}anl Qrç`y
1'06
Mkbqr` 1%1% Bràmki` gly~r`enl c` iqr}` elrg`c z`r` ay~a zrlocag`% &Ac`olr`ikùe
zrlzk`+%
Ahagzcl )0=% Ae qe` ke}ay~kb`ikùe na fkbkaea kenqy~rk`c { yabqrkn`n# rac`ikle`n` ile
qe zrliayl kenqy~rk`c# ya rapqkara qe` `z~k~qn gae~`c gq{ `c~`% Z`r` ay~l# cly
~r`o`h`nlray ya ylga~karle ` qe` zrqao` nac ilamkikae~a na ke~acai~l &KP+% Yk ya y`i`
qe` gqay~r` `c `x`r pqa n` V 7 0?: zqe~ly { y 7 ?: zqe~ly# ¼Iqàc ay c` zrlo`okckn`n
na pqa qe ~r`o`h`nlr yacaiikle`nl ~aenrà qe }`clr na ilamkikae~a na ke~acai~l pqa ay~ë
ae~ra 6: { 0>: zqe~ly<
Ylcqikùe8
@pqç ay~`gly oqyi`enl c` zrlo`okckn`n na Z&6: 3 V 3 0>:+ 7 Z&'?%:: 3 x 3 $0%::+%
Zlr cl ~`e~l# ac àra` ~l~`c l c` zrlo`okckn`n rapqarkn` ay kbq`c ` :%6061% Ay~l nkia pqa#
iari` nac 6? na c` zloc`ikùe ~kaea qe KP na ay~` zrqao` nac ke~acai~l pqa ay~à ae~ra
6: { 0>: zqe~ly%
Ahagzcl )02% Yk qe` gqay~r` `ca`~lrk` na qe` zloc`ikùe elrg`c na ke~aeykn`nay na
}kae~l# ae g,yabqenl# ~kaea qe zrlgankl na 0: g,yab { qe` }`rk`ex` na >8
&`+ ¼Pqë zlriae~`ha {,l zrlo`okckn`n na c`y ke~aeykn`nay nac }kae~l i`ae ae~ra ; { 0>g,yab%<
&o+ ¼Ae~ra 09 { 01<
Ylcqikùe8
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 222/846
Nr% Fëi~lr Pqa}anl Qrç`y
1'0;
&`+ 7 &Àra` na '∕ ` ?+ ' &àra` na '∕ ` '%1+ 7 %;22? ' %9:61
L ya`8 Z&; ≮ V ≮ 0>+ 7 Z&':%1 ≮ x ≮ ?+ 7 :%;22? ' :%9:61
7 :%==62 7 ==%62@pqç# ya el~` pqa# yk ac }`clr na y ay kbq`c ` ?# zlr cl ~`e~l# f`{ ? qekn`nay na
nay}k`ikùe ay~àen`r z`r` cly }`clray na V 7 0> { V 7 ;# ay naikr# ` c` naraif` a
kxpqkarn` nac zrlgankl%
&o+ @pqç ay~`gly nkikaenl pqa 09 ay~à ` 0%1 qekn`nay `o`hl nac zrlgankl { 01 ay~à `
?%1 qekn`nay `rrko` nac zrlgankl% Ay~l ay8
Z&09 ≮ V ≮ 01+ 7 Z&0%1 ≮ x ≮ ?%1+ 7 %;;96 ’ %;99? 7 :%:= 7 =Aeile~r`enl cly }`clray na x n`enl c`y zrlo`okckn`nay
Ahagzcl )06% Qe àra` na %>92: ay~à o`hl c` iqr}` elrg`c ae~ra ac zrlgankl { qe }`clr
zlyk~k}l na x% ¼Iqàc ay ac } clr na x<
Ylcqikùe8
Oqyi`enl ac }`clr na :%>92: ae c` ~`oc` na c` x }agly pqa ilrrayzlena ` x 7 $0%19%
Ahagzcl )0;% Qe àra` na %>6:6 ay~à o`hl c` iqr}` elrg`c ae~ra ac zrlgankl { qe }`clr
na x eab`~k}l% ¼Iqàc ay ac }`clr na x<
Ylcqikùe8
Oqyi`gly ac }`clr na %>6:6 ae c` ~`oc` { n` x 7 '?%:2%
Ahagzcl )?:% Ac ;: na c` nky~rkoqikùe na z`r~çiqc`y `~glymërki`y na qe` iqr}`
elrg`c ay~à ` c` kxpqkarn` na qe }`clr na x ae z`r~kiqc`r% ¼Iqàc ay ac }`clr na x<
Ylcqikùe8
Ac }`clr na x naoa na ay~`r ` c` naraif` nac zrlgankl# zlrpqa ac 1: na c`nky~rkoqikùe ay~à ` c` kxpqkarn` nac zrlgankl% Ayl nah` :%>:: na c` iqr}` ae~ra ac
zrlgankl { ac }`clr rapqarknl na x% Na g`ear` pqa# `flr` ~aeagly pqa aeile~r r ac
}`clr na x pqa ilrrayzlena ` qe` àra` na %>:: ae c` ~`oc` na c` iqr}` elrg`c &%;:: '
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 223/846
Nr% Fëi~lr Pqa}anl Qrç`y
1'?:
%1::+% Yke ago`rbl# `pqç }agly pqa el f`{ ae~r`n` na :%>:::# el loy~`e~a# cl gày
iari`el ay %9;;2 pqa ilrrayzlena `c }`clr na x 7 0%?6%
Aeile~r`enl cly }`clray x nac zqe~l na avz`eykùe z`r` }`rk`ocay elrg`cgae~anky~rkoqkn`y
Ahagzcl )?0% I`ciqc`r nae~rl na pqa r`ebl ay~`ràe ilgzraenkn`y ac ;1 na c`y
loyar}`ikleay iae~r`cay l na ae gankl# yk ac zrlgankl ay na 0: { c` nay}k`ikùe
ay~àen`r ay na ?% F`iar qe` bràmki`%
Ylcqikùe8
@pqç# }`gly ` qy`r c` rac`ikùe8 V 7 V ° x &y+# ile V 7 0:%: { y 7 ?%:# ay naikr80: ° 0%;= &?+ 7 0: ° 9%;? z`r` n`r &=%:6 ≮ V ≮ 09%;?+% C` mkbqr` na `o`hl gqay~r`
ay~` yk~q`ikùe%
Mkbqr` 1%=% Br`mki` gly~r`enl cly rayqc~`nly na ay~a zrlocag`% &Ac`olr`ikùe zrlzk`+
Ahagzcl )??% Yk V 7 0: { y 7 ?# ¼Nae~rl na pqa r`ebl ay~àe ilgzraenkn`y ac ;; na
c`y loyar}`ikleay na ae gankl na c` iqr}` elrg`c<
&o+ ¼Ac ;:<
Ylcqikùe8
Qy`enl c` rac`ikùe V 7 V ° x%:0 &y+ { yqy~k~q{aenl n`8
&`+ V 7 V ° x%:0&y+ 7 0: ° ?%12&?+7 &>%61 ≮ V ≮ 01%01+
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 224/846
Nr% Fëi~lr Pqa}anl Qrç`y
1'?0
Mkbqr` 1%2% Bràmki` gly~r`enl cly rayqc~`nly nac ahagzcl )??%
Ahagzcl )?9% Yk V 7 ?: { y? 7 ;# nae~rl na pqa r`ebl ay~àe ilgzraenkn`y8&`+ ¼Ac ;; na c`y loyar}`ikleay na ae gankl na c` iqr}` elrg`c<
&o+ ¼Ac ;:<
&i+ ¼Ac 6:<
Ylcqikùe8
Qy`enl c` rac`ikùe V 7 V ° x%:0 &y+ { yqy~k~q{aenl n`8&`+ V 7 V ° x%:0&y+ 7 ?: ° ?%12&9+
7 ?: ° 2%>0
7 &0?%?; ≮ V ≮ ?2%90+
&o+ Qy`enl c` rac`ikùe V 7 V ° x:%0&y+ { yqy~k~q{aenl n`8
V 7 V ° x:%0&y+ 7 ?: ° 0%=>1&9+
7 ?: ° >%;917 &?0%;= ≮ V ≮ ?>%;>+
&i+ Qy`enl c` rac`ikùe V 7 V ° x:%:1 &y+ { yqy~k~q{aenl n`8
V 7 V ° x:%?: &y+ 7 ?: ° 0%?6 &9+
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 225/846
Nr% Fëi~lr Pqa}anl Qrç`y
1'??
7 ?: ° 9%6>
7 &0=%0= ≮ V ≮ ?9%6>+
Zrlocag`y kcqy~r`enl c`y zrlvkg`ikleay elrg`cay c` nky~rkoqikùe okelgk`c
Ahagzcl )?>% Qe` gàpqke` zrlnqia ~lrekccly na cly iq`cay 0: yle namai~qlyly%
Aeile~r`r c` zrlo`okckn`n na pqa# ae qe` gqay~r` `ca`~lrk` na >:: ~lrekccly
zrlnqiknly zlr ay~` gàpqke`8
&`+ @ cl gày 9: ~lrekccly ay~`ràe namai~qlyly
&o+ Ae~ra 9: { 1: ay~`ràe namai~qlyly
&i+ Ae~ra 91 { >1 ay~`ràe namai~qlyly
&a+ 11 l gày ~lrekccly ay~`ràe namai~qlyly
Ylcqikùe8
Zrkgarl ya i`ciqc` ac zrlgankl { c` nay}k`ikùe ay~àen`r8
± 7 ez 7 &>::+&:%0+ 7 >: { ω 7 ezp 7 T&>::+&:%0+&%;:+W:%1 7 =%:
Aeyabqkn`# ya i`ciqc` ac }`clr na c` }`rk`oca `ca`~lrk` X qy`enl c` rac`ikùe8
X 7 &V ’ ±+ , ω%
&`+ Z&V ≮ 9:+% Z`r` i`ciqc`r ay~l# zrkgarl ya ~r`eymlrg` ac }`clr na 9:# ` }`clray na X
qy`enl c` mqeikùe na `rrko`# ay naikr#
X9: 7 &9: ’ >:+,=%: 7 '0%=2Zlr cl ~`e~l8
Z&V ≮ 9:+ 7 Z&X ≮ '0%1;+ 7 :%:11;
&o+ Z&9: ≮ V ≮ 1:+% Z`r` i`ciqc`r ay~l# zrkgarl ~r`eymlrg`gly cly }`clray na 9: {
1: ` }`clray na X# ay naikr#
X9: 7 &9: ’ >:+ , =%: 7 '0%=24 X1: 7 &1: ’ >:+ , =%: 7 0%=2
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 226/846
Nr% Fëi~lr Pqa}anl Qrç`y
1'?9
Zlr cl ~`e~l#
Z&9: ≮ V ≮ 1:+ 7 Z&V ≮ 1:+ ’ Z&V ≮ 9:+
7 Z&X ≮ 0%1;+ ' &X ≮ '0%1;+7 %;>>0 ' :%:11;
7 %666?
&i+ C` zrlo`okckn`n na pqë# ae~ra 91 { >1 ~lrekccly ay~ëe namai~qlyly# ay# Z&91 ≮ V ≮
>1+% Z`r` ay~l# zrkgarl ~r`eymlrg`gly cly }`clray na V ` }`clray na X%
X91 7 &91 ’ >:+,=%: 7 ':%69 { X>1 7 &>1 ’ >:+,=%: 7 :%69%
Zlr cl ~`e~l# Z&91 ≮ V ≮ >:+ 7 Z&'0%1; ≮ X ≮ :%2;+7 %261? ' %:11;
7 :%2?;9
&a+ Zrkgarl ay~`en`rkx`gly ac }`clr na V 7 11 ` }`clray na X%
x11 7 &11 ’ >:+ , =%: 7 ?%1: pqa ilrrayzlena ` qe` zrlo`okckn`n na %;;96% Zlr cl ~`e~l8
Z&V ≡ 11+ 7 0 ’ Z&V ≮ 11+ 7 0 ' %;;96 7 %::=?
Ahagzcl )?1% C` zrlo`okckn`n na pqa V `yqg` qe }`clr av`i~l na > ya n` `o`hl# ay~l
ay# qy`enl c` nky~rkoqikùe okelgk`c%
Z&V 7 >+ 7 O&>401#:%>+ 7 :%0?=6
Ykaenl `yç qy`r c` nky~rkoqikùe elrg`c ilgl qe` `zrlvkg`ikùe8
Ylcqikùe8
Zrkgarl ya i`ciqc` ac }`clr nac zrlgankl# ± 7 ez { n` ± 7 &01+&:%>+ 7 =% C` nay}k`ikùe
ay~àen`r ay ω 7 ezp 7 :%=+&01+&:%>+& 7 0%6;2% @nagày# V 7 > zqana `yqgkr }`clray
na 9%1 { >%1% Aeyabqkn`# ~r`eymlrg`enl cly }`clray qy`enl c` }`rk`oca `ca`~lrk` elrg`c
X n`8
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 227/846
Nr% Fëi~lr Pqa}anl Qrç`y
1'?>
X9%1 7 &9%1 ’ =+ , 0%; 7 '0%9? { X>%1 7 &>%1 ’ =+ , 0%; 7 ':%2;
Yk V ay qe` }`rk`oca `ca`~lrk` okelgk`c { X ay qe` }`rk`oca elrg`c# ae~leiay8
Z&V 7 >+ 7 O&>401#:%>+ ≆ Z&'0%9? 3 X 3 ':%2;+7 Z&X 3 ':%2;+ ’ Z&X 3 '0%9?+
7 :%?0>6 ’ :%:;9> 7 :%0?0>
Ay~a }`clr y`i`nl qy`enl c` nky~rkoqikùe elrg`c ilgl qe` `zrlvkg`ikùe ` c`
nky~rkoqikùe okelgk`c ay~à gq{ iari`el `c na :%0?=6 i`ciqc`nl zlr c` nky~rkoqikùe
okelgk`c%
Nky~rkoqikùe avzleaeik`c
C` nky~rkoqikùe avzleaeik`c ay qe` nky~rkoqikùe ile~keq` na zrlo`okckn`n z`r`
nayirkokr ac ~kagzl pqa ya ~`rn` ae ra`ckx`r qe` `i~k}kn`n% Ay~` nky~rkoqikùe ay qe i`yl
ayzaik`c na c` nky~rkoqikùe b`gg`% Ay~` mqeikùe ya qy` z`r` glnac`r c`y }kn`y na c`y
o`~arç`y# na ~r`eyky~lray# na }`carly# a~i% ^`gokëe ya qy` z`r` glnac`r c` nky~ eik
ae~ra cly zrkeikz`cay namai~ly ae qe` i`rra~ar`# a~i% @ zay`r na pqa c` nky~rkoqikùe
avzleaeik`c ay ile~keq`# ay~` nky~rkoqikùe ay~à iari`e`gae~a rac`ikle`n` ile c`
nky~rkoqikùe na Zlkyyle# pqa ay nkyira~`% Ay~l liqrra ae ac yae~knl pqa# qe` }`rk`oca
`ca`~lrk` Zlkyyle iqae~` ac eøgarl na liqrraeik`y na qe a}ae~l nqr`e~a qe ke~ar}`cl
na ~kagzl n`nl% Ae ile~r`y~a c` }`rk`oca `ca`~lrk` avzleaeik`c V pqa zqana yar qy`n`
z`r` gankr ac ~kagzl pqa ~r`eyiqrra `e~ay na c` zrkgar` liqrraeik` na qe a}ae~l#
nlena c`y liqrraeik`y nac a}ae~l ykbqae ` qe` nky~rkoqikùe Zlkyyle%
Apqk}`cae~agae~a# qe` }`rk`oca `ca`~lrk` avzleaeik`c zqana yar qy`n` z`r` gankr ac
~kagzl pqa ~r`eyiqrra ae~ra c`y liqrraeik`y na qe a}ae~lC`y `zcki`ikleay na c` nky~rkoqikùe avzleaeik`c# ` c` kebaekarç` `gokae~`c yle
}`rk`y% Zlr ahagzcl# ya zqana qy`r z`r` glnac`r ac ~kagzl pqa ya ~`rn`e cly zay~kikn`y
ae nabr`n`rya ae c` ~karr` l z`r` gankr ac ~kagzl ae pqa ya ~lg` ae nabr`n`rya qe`
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 228/846
Nr% Fëi~lr Pqa}anl Qrç`y
1'?1
yqy~`eik` r`nk`i~k}`% Kbq`cgae~a# ay ø~kc z`r` gankr c` ikeë~ki` na c` nag`en`
oklpqçgki` na lvçbael &NOL1+% @eàclb`gae~a# ya zqana qy`r z`r` gankr ac ~kagzl
pqa ~`rn`e c`y z`r~çiqc`y ~glymërki`y ae i`ar c` yqzarmkika na c` ~karr`%Qe` }`rk`oca `ca`~lrk` ile~keq` V ya nkia pqa ay~à avzleaeik`cgae~a
nky~rkoqkn` yk yq mqeikùe na naeykn`n ay8
m &v+ 7 δ a'δv z`r` V ≡ :# δ ≡ : &1'0=+
Nlena8 δ ay qe z`ràga~rl na c` nky~rkoqikùe# { a qe` iley~`e~a kbq`c ` ?%206?6
V { y? na c` }`rk`oca `ca`~lrk` avzleaeik`c V yle A&V+ 7 0,δ { ]&V+ 7 0,δ?#
rayzai~k}`gae~a% Ya zqana nagly~r`r pqa ac zrlgankl { c` nay}k`ikùe ay~àen`r na qe`
nky~rkoqikùe avzleaeik`c yle kbq`cay ac qel `c l~rl# ay~l ay8 ± 7 ω 7 0,δ%
Zlr l~rl c`nl# Daccar a~ `c% &0;;:+ `mkrg` pqa# ae ac i`yl na qe` }`rk`oca
`ca`~lrk` avzleaeik`c V# ya zqana nagly~r`rya pqa c` zrlo`okckn`n na pqa V zqan`
~lg`r qe }`clr gày br`ena pqa qe eøgarl ayzaikmki`nl el eab`~k}l `# ay a'δ`% Ay~l ya
zqana avzray r qy`enl iàciqcl ke~abr c# ay naikr8
∕ ∕
Z&V ≡ `+ 7 ∤ ` δ a'δv nv 7 'a'δv s ` 7 a'δ` &1'0=`+
Ac iàciqcl na c`y mqeikleay avzleaeik`cay ke}lcqir` c` a}`cq`ikùe na ke~abr`cay
na zrlo`okckn`n ae~ra cly cçgk~ay na ` { o% Z`r` ay~l# ya n` qe` ~`oc` na zrlo`okckn`nay
avzleaeik`cay% C`y ykbqkae~ay mùrgqc`y ya qy`e ile ay` ~`oc`%
Z&` ≮ V ≮ o+ 7 a'δ` ’ a'δ o &1'02`+
Z&V ≮ `+ 7 0 ’ a'δ`
&1'02o+Z&V ≡ `+ 7 a'δ` &1'02i+
C` nky~rkoqikùe avzleaeik`c ay qe` m`gkck` na nky~rkoqikleay glnkmki`n`y zlr
qe ylcl z`ràga~rl# δ# c`y iq`cay ya gqay~r`e ae c` Mkbqr` 1%6%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 229/846
Nr% Fëi~lr Pqa}anl Qrç`y
1'?=
Mkbqr` 1%6% Bràmki`y na ~ray nky~rkoqikleay avzleaeik`cay% Mqae~a8 Na}lra &?:::+
Ahagzcl )?=% Yqzleb`gly pqa ac ~kagzl zrlgankl pqa ya ~`rn` qe` yqy~`eik`
r`nk`i~k}` ae nabr`n`rya ay na ± 7 01 { yq mqeikùe na naeykn`n ay m&v+ 7 0,01 a'v,01% Yk
cly }`clray na c` }`rk`oca `ca`~lrk` v yle 1# 01# ?1# 91# { >1# i`ciqc`r c`y ykbqkae~ay zrlo`okckn`nay8
&`+ @ cl gày = `ðly
&o+ Ae~ra = { 06 `ðly
Ylcqikùe8
&`+ Qy`enl &1'02o+ ile ± 7 0,δ# ay naikr# δ 7 0,01 { ` 7 v 7 = { yqy~k~q{aenl ae c`mqeikùe Z&V ≮ `+ 7 0 ’ a'δ`# n`8
Z&V ≮ =+ 7 0 ’ a'&=,01+ 7 :%9?;2
&o+ Qy`enl c` mqeikùe &1'02`+ { yqy~k~q{aenl n`8
Z&^kagzl na i`çn` = ≮ V ≮ 06 `ðly+ 7 %=;66 ' %9?;2 7 %9=;0
Ahagzcl )?2% _amkrkëenlya `c Ahagzcl )?=# ¼Iqàc ay c` zrlo`okckn`n na pqa c`
nabr`n`ikùe na c` yqy~`eik` r`nk`i~k}` nqra iq`enl gaely 0: `ðly%
Ylcqikùe8
Qy`enl c` mqeikùe &1'02i+# ay naikr8 Z&V ≡ `+ 7 a'0:,01 7 :%10
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 230/846
Nr% Fëi~lr Pqa}anl Qrç`y
1'?2
Mkbqr` 1%;% Bràmki`y gly~r`enl cly rayqc~`nly z`r` &`+ { &o+# nac ahagzcl )?=%
Ahagzcl )?6% Ac ~kagzl rapqarknl z`r` pqa liqrr` qe` ra`iikùe pqçgki` ay~àavzleaeik`cgae~a nky~rkoqkn` ile qe ~kagzl ayzar`nl na 1 gkeq~ly% &`+ ¼Pqë
zrlzlrikùe na c` yqy~`eik` ya mlrg rà nae~rl na 0 gkeq~l< &o+ ¼Ae 1 gkeq~ly< &i+
¼Ae~ra > { 6 gkeq~ly< &n+ Yk c` i`e~kn`n na c` yqy~`eik` pqçgki` ay na 1%:: br`gly#
¼iqàe~l ay cl pqa ya }` mlrg`enl ae i`n` qel na cly ke~ar}`cly< &Ac cai~lr cl f`rà+
Ylcqikùe8
Qy`ragly ke~ar}`cly na 0 gkeq~l z`r` i`ciqc`r c` zrlo`okckn`n% Zlr cl ~`e~l# naoknl
` pqa c` ra`iikùe ya f`ia ae 1 gkeq~ly ae zrlgankl &zaey`gly na ay~l
zrlnqikëenlya ae 1 ke~ar}`cly ile~keqly na qe gkeq~l+ ac eøgarl ayzar`nl na
zrlnqiikùe ae qe gkeq~l ay 0,1 7 %?: 7 δ &l ya` ac eøgarl ayzar`nl na liqrraeik`y
ae 0 gkeq~l+% C` }`rk`oca `ca`~lrk` V ya namkea ilgl ac ~kagzl# ae gkeq~ly#
rapqarknl z`r` ilgzca~`r c` ra`iikùe% Zlr cl ~`e~l8
&`+ ZT V ≮ 0W 7 0 ’ a'&:%?:&0+ 7 :%6062 7 60%62 &Qy`enl c` mùrgqc` 1'02o+
&o+ ZTV ≮ 1W 7 0 ’ a'&:%?:+&1+ 7 :%9=2; 7 9=%62 &qy`enl c` mùrgqc` 1'02o+&i+ ZT> 3 V 3 6W 7 a'&:%?:+&>+ ’ a'&:%?:+&6+ 7 :%?>2> &qy`enl c` mùrgqc` 1'02`+
&n+ >%:; b%# 0%6> { 0%?9 b# rayzai~k}`gae~a%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 231/846
Nr% Fëi~lr Pqa}anl Qrç`y
1'?6
Nky~rkoqikùe b`gg` ile~keq`
@ zay`r na pqa c` nky~rkoqikùe elrg`c zqana raylc}ar gqifly zrlocag`y ae
kebaekarç`# f`{ l~r`y yk~q`ikleay pqa rapqkarae na nkmarae~ay ~kzly na mqeikleay nanaeykn`n% Mqeikleay ilgl ëy~`y yle c` avzleaeik`c# c` b`gg`# c` Rakoqcc# c` oa~`#
a~i% F`{ gqif`y yk~q`ikleay ae pqa c` }`rk`oca na ke~arëy# z`r` ac avzarkgae~`nlr#
zqan` ~aear qe` nky~rkoqikùe lockiq`% Ykaenl `yç# ae~leiay# qe` m`gkck` na
mqeikleay na zrlo`okckn`n na naeykn`n &znm+ pqa n`e qe` `gzck` }`rkan`n na
nky~rkoqikleay yayb`n`y ay c` m`gkck` na nky~rkoqikleay b`gg`%
Ilgl ya nkhl `e~ay# c` nky~rkoqikùe b`gg` ay qe i`yl ayzaik`c na c`
nky~rkoqikùe avzleaeik`c% C`y mqeikleay avzleaeik`cay { c` mqeikùe b`gg` hqab`e
qe z`zac gq{ kgzlr~`e~a ae c` ~alrç` na mkc`y pqa ayzar`e ac lrnae na yq ccab`n`%
C` nky~rkoqikùe b`gg` zqana yar }ky~` ilgl qe` nky~rkoqikùe b`gg`
ay~`en`rkx` l ilgl qe` nky~rkoqikùe b`gg` el ay~`en`rkx`n`%
Yk qe` }`rk`oca `ca`~lrk` ile~keq` v ~kaea qe` nky~rkoqikùe b`gg`# ile
z`ràga~rly ΰ { ο# ae~leiay# z`r` iq`cpqkar v 5 : c` nky~rkoqikùe `iqgqc`n` na
mraiqaeik` &inm+ na v ay~à n`n` zlr8
Z&V ≮ v+ 7 M&v4ΰ#ο+ 7 M&v,ο4ΰ+ &1'06+
Nlena8
ΰ ay ac zrkgar z`ràga~rl na mlrg` pqa namkea c` nky~rkoqikùe b`gg`
ο ay ac z`ràga~rl na ayi`c` pqa namkea c` nky~rkoqikùe b`gg` &zlrpqa }`clray
g`{lray pqa c` ilgzrkgae l ay~kr`e c` mqeikùe na zrlo`okckn`n na naeykn`n &znm+
ae c` nkraiikùe na v+4 M&v,ο4ΰ+ ay qe` mqeikùe na b`gg` keilgzca~`%Ae c` m`gkck` na nky~rkoqikleay b`gg` qe` }`rk`oca `ca`~lrk` ile~keq` V ya
nkia pqa ~kaea qe` nky~rkoqikùe b`gg` el ay~`en`rkx`n` yk c` znm na V ay8
m&v4ΰ#ο+ 7 t0,οΰ Й&ΰ+ vΰ'0 a'v,ο v ≡ : &1'0;+
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 232/846
Nr% Fëi~lr Pqa}anl Qrç`y
1'?;
l na l~r` g`ear`
Nlena cly z`ràga~rly ΰ { ο y`~kym`iae ΰ 5 : { ο 5 :
Yk ya zlea ο 7 0 c` avzraykùe &1'0;+ ya ranqia ` c` mlrg` na na c` nky~rkoqikùeb`gg` ay~àen`r nayirk~` `o`hl%
v
m &v4ΰ+ 7 ∤ : vΰ'0 a'v , Й&ΰ+ nv v 5 : &1'?:+
C` mqeikùe &1'?:+ ya cc`g` mqeikùe na b`gg` keilgzca~`# iq`enl el ~kaea ac
naelgke`nlr ile Й&ΰ+ ae ac ke~abr`nlr%
Iq`enl ya qy`e c`y mqeikleay &1'0;+ { &1'?:+ c` ~`ra` ya m`ikck~` qy`enl c`~`oc` na c` nky~rkoqikùe b`gg`# ile }`clray na ΰ 7 0# ?# 9#…#0: { na v 7 0# ?#…#01%
Ac zrlgankl { c` }`rk`ex` na c` nky~rkoqikùe b`gg` yle# rayzai~k}`gae~a8
A&V+ 7 ± 7 ΰο &1'?0+
{ ]&V+ 7 ω? 7 ΰο? &1'?0`+
Mkbqr` 1%0:% Bràmki`y ile nky~rkoqikleay b`gg` na naeykn`n ile nkmarae~ay }`clray
na ΰ { ο { iqr}`y na naeykn`n b`gg` ay~àen`r% Eù~aya pqa iq`enl ο 7 0# ay c`iqr}` avzleaeik`c% &Na}lra ?:::+%
Ahagzcl )?;% Yqzùeb`ya pqa ya ~kaea qe` nky~rkoqikùe b`gg` ay~àen`r ile
z`ràga~rl ΰ 7 9# i`ciqc`r8
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 233/846
Nr% Fëi~lr Pqa}anl Qrç`y
1'9:
&`+ C` zrlo`okckn`n na pqa V ay~ë ae~ra > { 1%
&o+ C` zrlo`okckn`n na pqa V ya` g`{lr pqa >
Ylcqikùe8Naoknl ` pqa Z&` ≮ V ≮ o+ 7 M&o+ ’ M&`+ iq`enl V ay ile~keq`# zlr cl ~`e~l8
&`+ Z&> ≮ V ≮ 1+ 7 M&149+ ’ M&>49+
7 :%621 ’ :%2=? 7 :%009
&o+ Z&V 5 >+ 7 0 ’ Z&V ≮ >+ 7 0 ’ M&>#9+
7 0 ' %2=?
7 :%?96
Ahagzcl )9:% Ay~a zrlocag` ke}lcqir` qe avzarkgae~l ile ileahkccly na Kenk`
yacaiikle`nly `c `x`r% Ay~a ay qe ay~qnkl rac`ikle`nl ile ac ~kagzl V na
yqzar}k}aeik`# ae yag`e`y% Cly `ekg`cay mqarle avzqay~ly ` qe` r`nk`ikùe na >::
r`ny &nlyky na r`nk`ikùe `oylrokn`+# ay naikr# na r`nk`ikùe b`gg` &aearbç`
r`nk`e~a+% Ya `yqga pqa ay~` yk~q`ikùe ykbqa ` qe` nky~rkoqikùe b`gg` ile
z`ràga~rly na ayi`c` na ΰ 7 0: { ο 7 ?:% Ykaenl `yç# f`iar cly ykbqkae~ay iàciqcly8
&`+ I`ciqc`r c` gank` na yqzar}k}aeik` { c` }`rk`ex`%
&o+ I`ciqc`r c` zrlo`okckn`n na pqa qe ileahkccl ylora}k}` ae~ra 6: { 0?: nç`y%
&i+ C` zrlo`okckn`n na pqa qe `ekg`c ylora}k}`# iq`enl gaely ?: nç`y%
Ylcqikùe8
@pqç qy`gly c` nky~rkoqikùe b`gg` el ay~`en`rkx`n`%
&`+ Ac zrlgankl ay8 A&V+ 7 ± 7 ΰο 7 &0:+&?:+ 7 ?:: nç`y%
C` }`rk`ex` ay8 ]&V+ 7 ω?
7 ΰο?
7 &0:+&?:+?
7 >::: nç`y%&o+ Z&6: ≮ V ≮ 0?:+ 7 M&0?:,?:40:+ ’ M&6:,?:40:+
7 M&=40:+ ’ M&>40:+
7 :%:6> ' :%::6 &na c` ~`oc` na c` nky~rkoqikùe na b`gg`+
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 234/846
Nr% Fëi~lr Pqa}anl Qrç`y
1'90
7 :%:2=
Ay~l nkia pqa ac }`clr na :%:2= ay c` zrlo`okckn`n na pqa qe ileahkccl ylora}k}`
ae~ra 6: { 0?: nç`y%&i+ Z&V ≡ ?:+ 7 0 ' Z&V 3 ?:+
7 0 ' M&?:,?:40:+
7 :%::: &na c` ~`oc` na c` nky~rkoqikùe b`gg`+
Nky~rkoqikùe Rakoqcc
C` nky~rkoqikùe Rakoqcc mqa ke~rlnqikn` zlr ac mçykil yqail R`clnnk Rakoqcc ae
0;9;% Ae mlrg` `eàclb` ` c`y nky~rkoqikleay b`gg` { avzleaeik`c c` nky~rkoqikùe
na Rakoqcc ~kaea `zcki`ikleay rac`ikle`n`y ile ~kagzl na m`cc` l clebk~qn na }kn`%
Ay naikr# z`r` gankr c` ilemk`okckn`n na qe ilgzleae~a l zrlnqi~l# ilgl c`
zrlo`okckn`n na pqa yk mqeikle`rà `zrlzk`n`gae~a# zlr iq`enl gaely qe ~kagzl
ayzaikmki`nl o`hl ilenkikleay avzarkgae~`cay ayzaikmki`n`y% Ay~` mqeikùe#
kbq`cgae~a# ya qy` ae ac nkyaðl na yky~ag`y ilgzcki`nly# iq{` lzar`ikùe l
yabqrkn`n nazaena na cly }`rkly ilgzleae~ay ke}lcqir`nly ae ac yky~ag`% Zlr
ahagzcl# qe` ilcqge` na `iarl zqana }aeiarya% L~r` `zcki`ikùe ay ac glnac`nl na
`cbøe `z`r`~l yaeykoca `c i`clr pqa zqan` m`cc`r% L~r` `zcki`ikùe yarç` ac ay~qnkl na
ilgzleae~ay knëe~kily yqha~ly ` ilenkikleay `gokae~`cay knëe~ki`y# pqa zqan`e
m`cc`r ` ~kagzly nkmarae~ay a kgzranaikocay%
C` mqeikùe na zrlo`okckn`n na naeykn`n &znm+ na c` nky~rkoqikùe Rakoqcc ay8
m &v+ 7 ΰ vΰ'0 avz'&v,ο+? , οΰ # v 5 : &1'??+
Nlena ΰ { ο yle cly z`ràga~rly ilenkikle`nly ` ΰ 5 : { ο 5 :
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 235/846
Nr% Fëi~lr Pqa}anl Qrç`y
1'9?
Mkbqr` 1%00% Bràmki` gly~r`enl c` iqr}` na naeykn`n na Rakoqcc% Eù~aya pqa iq`enl
ΰ7 0 {
ο7 0# c` iqr}` ya ~lre` avzleaeik`c% &Na}lra# ?:::+
Zrlzlykikùe8 C` mqeikùe na nky~rkoqikùe `iqgqc`n` &inm+ na qe` }`rk`oca `ca`~lrk`
pqa ~kaea z`ràga~rly ΰ { ο ay8
M&v4ΰ#ο+ 7 t0 ’ avz'&v,ο+ΰ v ≡ : &1'??`+
Ahagzcl )90% Yqzùeb`ya pqa V ~kaea qe` nky~rkoqikùe na Rakoqcc ile z`ràga~rly ΰ 7
?: { ο 7 0:: &Na}lra# ?:::+% Ae~leiay# i`ciqc`r8
&`+ Z&V ≮ 0:1+&o+ Z&;6 ≮ V ≮ 0:?+
Ylcqikùe8
&`+ Z&V ≮ 0:1+ 7 M&0:14?:#0::+ 7 0 ’ avz'&0:1,0::+?:
7 0 ' %:2: 7 %;9:
&o+ Z&;6 ≮ V ≮ 0:?+ 7 M&0:?4?:#0::+ ’ M&;64?:#0::+
7 avz'&%;6+?: ’ avz'&0%:?+?:
7 %109 ' %??= 7 %?62
Ke~ar}`cly na ilemk`ex` z`r` ± ile ω? ilelikn`
Ya y`oa pqa c` ay~`nçy~ki` X 7 & V ' ±+ , ω,∘ e ykbqa ` c` nky~rkoqikùe elrg`c# ile ± 7 :
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 236/846
Nr% Fëi~lr Pqa}anl Qrç`y
1'99
{ ω 7 0% Yk ~ln`y c`y gqay~r`y zlykocay na qe ~`g`ðl e yle y`i`n`y na qe` zloc`ikùe {
ac }`clr na X ya i`ciqc` z`r` i`n` gqay~r`# ac ;1 l ;; na cly }`clray na X i`aràe
ae~ra ’xΰ,? { xΰ,?% Y`oagly pqa c` zrlo`okckn`n na pqa x ay~ë ae~ra ’xΰ,? { xΰ,? ay 0 ’ ΰ%Ay~l ya zqana avzray`r ilgl8
Z&'xΰ,? 3 X 3 xΰ,?+ 7 0 ’ ΰ% &1'?9+
Z`r` ay~l ya yqy~k~q{a ac }`clr na X { ya gqc~kzcki` i`n` acagae~l na &1'?9+ zlr ω,∘ e
{ cqabl ya ca ray~` V na i`n` ~ërgkel% Nayzqëy na ay~l# ya gqc~kzcki` zlr '0
&ra}ary`enl ac yae~knl na c`y naykbq`cn`nay+ { ely n` c` mqeikùe na `o`hl8
Z& V ' xΰ,? ω, e 3 ± 3 V $ xΰ,? ω, e + 7 0 ’ ΰ &1'?>+
Nlena8
C` zrlo`okckn`n 0 ’ ΰ ya cc`g` ac ek}ac na ilemk`ex`
V ’ xΰ,? ω, e ya cc`g` ac cçgk~a na ilemk`ex` kemarklr
V $ xΰ,? ω, e ya cc`g` ac cçgk~a na ilemk`ex` yqzarklr
^@OC@ 1%?% ^`oc` gly~r`enl cly ek}acay na ilemk`ex` gày ilgqeay &Ac`olr`ikùe zrlzk`+ \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ Ek}ac na ilemk`ex` 0 ’ ΰ ΰ ΰ,? xΰ,?
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ %;1 %:1 %:?1 0%;=%;; %:0 %::1 ?%16%;: %0: %:1 0%=>1
Ahagzcl )9?% Ae qe` gqay~r` `ca`~lrk` na 0:: loyar}`ikleay na ileiae~r`ikleay na
ùvknly na ek~rùbael &EL+ `~glymërkil y`i`n` na qe` zloc`ikùe elrg`c ~kaea ω 7 ?1 {
V 7 ?:# ile qe ~`g`ðl na gqay~r` na e 7 0::% Aeile~r`r ac ke~ar}`cl na ilemk`ex`
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 237/846
Nr% Fëi~lr Pqa}anl Qrç`y
1'9>
ay~kg`nl nac ;1 z`r` ac zrlgankl zloc`ikle`c ±%
Ylcqikùe8
Qy`enl c` aiq`ikùe &1'?>+# ile ω 7 ?1# V 7 ?: { e 7 0::# ΰ 7 :%:1 { ile rabkleayirç~ki`y na °0%;= n`8
Z&?: ’ 0%;= &?1+, 0:: 3 ± 3 ?: $ 0%;= &?1+, 0:: + 7 0 ' ΰ
01%0 3 ± 3 ?>%;
Ay~`nçy~ki` kemaraeik c% alrç` na naikykùe ay~`nçy~ki`% Zrqao`y na fkzù~ayky
Ae c` zrài~ki` ay eaiay`rkl f`iar naikykleay `iari` na zrlocag`y o`yàenlya aegqay~r`y ay~`nçy~ki`y% Yagah`e~ay naikykleay ya cc`g`e naikykleay ay~`nçy~ki`y% C`y
zrqao`y na fkzù~ayky ya zqanae f`iar ile c` nky~rkoqikùe elrg c z`r` ay~kg`r cly
z`ràga~rly na zloc`ikùe ±# ω?# ω# π# ο# a~i%# yk ac ~`g`ðl na c` gqay~r` ay e ≡ 9:
loyar}`ikleay% Yke ago`rbl# yk ac ~`g`ðl na c` gqay~r` ay e 3 9: i`yly# ae~leiay# ya
zqana qy`r cl pqa ya cc`g` ~alrç na gqay~ral zapqaðl# qy enl c` nky~rkoqikùe na ~ na
Ay~qnk`e~a%
Ae ay~a raebcùe# z`r` f`iar zrqao`y na fkzù~ayky a ke~ar}`cly na ilemk`ex`#
~`gokëe ya zqana qy`r c` nky~rkoqikùe na c` HK iq`nr`n` &ς ?+# z`r` ay~kg`r cly
z`ràga~rly zloc ikle`cay ω? { ω% @nagày# ya zqana qy`r c` nky~rkoqikùe M# ay naikr#
f`ikaenl ~`oc`y na `eàckyky na }`rk`ex`y# a~i%
Zrqao`y na Fkzù~ayky ay~`nçy~ki`y% Fkzù~ayky eqc` &Fl8+ a fkzù~ayky c~are`~k}`y
Z`r` zrlocag`y na zrqao`y na fkzù~ayky icàyki`y &ya nkia pqa yle icàyki`y# zlrpqa ya
nkyað`rle ac ykbcl `e~az`y`nl+ ya ile~r`y~`e ile ac eqa}l aemlpqa glnarel naciàciqcl na c` zrlo`okckn`n# z na zrlbr`g`y na ilgzq~`nlr` na c` ar` ikoareë~ki`%
@c ~r`~`r na `ci`ex`r naikykleay ay~`nçy~ki`y# ay eaiay`rkl `yqgkr yk~q`ikleay
`iari` na c`y zloc`ikleay ke}lcqir`n`y ae ~r`o`hly na ke}ay~kb`ikùe% Yagah`e~ay
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 238/846
Nr% Fëi~lr Pqa}anl Qrç`y
1'91
yqzlykikleay# pqa zqanae l el yar }arn`nar`y# ya cc`g`e fkzù~ayky eqc`y% Ae gqif`y
li`ykleay mlrgqc`gly qe` fkzù~ayky ay~`nçy~ki`# ile ac zrlzùyk~l na raif`x`rc` l#
iq`enl gaely# na el `iaz~`rc` ae o`ya na c` a}knaeik` lo~aekn`%Ile c` ke}ay~kb`ikùe ikae~çmki` c` kna` na~rày na f`iar zrqao`y na fkzù~ayky ay c`
na ~r`~`r na zrlnqikr a}knaeik` z`r` raif`x`r c` fkzù~ayky% Ay~l ya naoa ` pqa# ac
raif`xl na qe` fkzù~ayky# ae ~r`o`hly na ke}ay~kb`ikùe nael~` nkyaðly avzarkgae~`cay
mqar~ay# zraikyly { ileikyly% @nagày# ile c` kebaekarç` na g`eqm`i~qr`# ac zrlzùyk~l
na f`iar zrqao`y na fkzù~ayky ay ile ac loha~l na }arkmki`r ac ile~rlc na i`ckn`n na cly
zrlnqi~ly zrlnqiknly zlr c` kenqy~rk g`eqm`i~qrar`% El loy~ e~a# yk c` fkzù~ayky el
ya zqana raif`x`r zqan` naoarya ` pqa c` a}knaeik` pqa zqnkar` raif`x`r c` fkzù~ayky#
el ya zqana zrlnqikr% Ay~l zqana rayqc~`r na qe` gqay~r` zapqað` l na qe arrlr
avzarkgae~`c aviayk}l &nlena f`{ gqif` }`rk`ikùe+%
C` g`ear` na zrlnqikr a}knaeik` z`r` raif`x`r c` fkzù~ayky ay `e`ckx`enl ac
arrlr ay~àen`r nac zrlgankl# ac iq`c zrqao` pqa `goly arrlray# K &`cm`+ { KK &oa~`+
zqanae yar ranqiknly `qgae~`enl ac ~`g`ðl na c` gqay~r` l nkygkeq{aenl c`
nay}k`ikùe ay~àen`r% Ay~l cl zlnagly nagly~r`r `e`ckx`enl ac arrlr ay~àen`r na c`
gqay~r`% Ay~l ay8
ω V 7 ω , E &1'?1+l yq ay~kg`nlr y 7 y , e &1'?1`+
@pqç# yke ago`rbl# i`oa el~`r pqa avky~ae }`rkly arrlray ay~àen`ray na c`y
nky~rkoqikleay ay~`nçy~ki`y% Ay~` kemlrg`ikùe ya n` ae c` ~`oc` na `o`hl%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 239/846
Nr% Fëi~lr Pqa}anl Qrç`y
1'9=
^@OC@ 1%9% Arrlray ay~àen`ray na nky~rkoqikleay ay~`nçy~ki`y% &Ac`olr`ikùe zrlzk`+ \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ Ay~`nçy~ki` Arrlr ay~àen`r Loyar}`ikleay ayzaik`cay
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ Zrlgankly ωv 7 ω , e Ay~l ay }arn`narl z`r` gqay~r`y zapqað`y l
br`enay% C` nky~rkoqikùe na c` gqay~r` ayelrg`c z`r` e ≡9:# `qe yk c` zloc`ikùe el ayelrg`c%
Nay}k`ikleay ωy 7 ω , E ? &0+ay~àen`ray ωy 7 ±>' ±?,>E±? &?+
Z`r` E ≡ 0:: i`yly# c` nky~rkoqikùe na y ayelrg`c% ωy ya n` ae &0+# ylc`gae~a yk f`{elrg`ckn`n% El loy~`e~a# yk c` zloc`ikùe elay elrg`c# c` aiq`ikùe &?+ ya zqana qy`r%
Eù~aya pqa &?+ ya ranqia ` &0+ iq`enl ±?7ω? { ±> 7 9ω># z`r` zloc`ikleay elrg`cay%Z`r` e ≡ 9:y 7 ω
]`rk`ex`y ω?y 7 ω? E ,? &9+
ωy? 7 ±> ' ±?? , E &>+
C`y loyar}`ikleay faif`y z`r` c` nay}k`ikùe ay~àen`r `zcki`e `pqç ~`gokëe% Eù~ayapqa &?+ n` &0+ ae ac i`yl na qe` zloc`ikùe elrg`c%
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 240/846
Nr% Fëi~lr Pqa}anl Qrç`y
1'92
^kzly na arrlray K &`cm`+ { KK &oa~`+
Iq`caypqkar` na c`y fkzù~ayky pqa ya` ilrrai~`# ykagzra f`{ c` zlykokckn`n na pqa qe
arrlr na gqay~ral ely keickea ` ilga~ar cl pqa ya cc`g`e arrlray K l KK% @yç# zlnagly
raif`x`r qe` fkzù~ayky eqc` Fl8 pqa ay }arn`nar` l zlnagly `iaz~`r qe` fkzù~ayky
eqc` pqa ay m`cy`% Yk ya raif`x` qe` fkzù~ayky# iq`enl naokù yar `iaz~`n`# ya nkia pqa
ya ilga~kù ac arrlr K% Ae ile~r`y~a# yk ya `iaz~` qe` fkzù~ayky m`cy`# iq`enl naokù yar
raif`x`n`# ya nkia pqa ya ilga~kù ac arrlr KK% Ilgl ya }arà# ay~ly nly arrlray ya
zqanae a}k~ r `qgae~`enl ac ~`g`ðl na c` gqay~r` ay~`nçy~ki` {,l ranqikaenl c`
nay}k`ikùe ay~àen`r &ay~l ya zqana zrlo`r ` ~r`}ëy nac arrlr ay~àen`r nac zrlgankl#pqa ay kbq`c ` y, e +%
Na iq`cpqkar g`ear`# ilgl ya `yae~ù `e~ay# c` kna` na qe` zrqao` na fkzù~ayky
ay ~r`~`r na zrlnqikr a}knaeik` z`r` raif`x`r c` fkzù~ayky eqc`# Fl8 Yk el ya zqana
raif`x`r c` fkzù~ayky eqc`# ay~` m`c~` na a}knaeik` zqana rayqc~`r# {` ya` ` ~r`}ëy na qe`
gqay~r` keyqmkikae~agae~a br`ena l ` ~r`}ëy na qe arrlr na c`olr`~lrkl aviayk}`gae~a
br`ena &pqa ya ramcah` ae c` nay}k`ikùe ay~àen`r# ω+% ^`gokëe# c` `iaz~`ikùe na qe`fkzù~ayky m`cy` zqana naoarya ` qe` }`rk`ikùe kefarae~a na c` zloc`ikùe pqa ay~`gly
gqay~ra`enl &ilgl ae ac i`yl na c`y ~agzar`~qr`y ` ek}ac gqenk`c iq{ly rabky~rly ya
ay~àe rlgzkaenl i`n` `ðl# zlr ac i`cae~`gkae~l bclo`c naoknl ` c`y agkykleay na
okùvknl na i rolel+% Na iq cpqkar g`ear` na ay~`y ~ray }`rk ocay# ω ay c` gày yaeykoca%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 241/846
Nr% Fëi~lr Pqa}anl Qrç`y
1'96
Mkbqr`% 1%0?% Nky~rkoqikùe na cly zrlgankly na nly gqay~r`y na c`y iqr}`y @ { Okcqy~r`enl ac ~kzl na arrlr KK l oa~` ile ± 7 1: &ae iqr}` @+ { ω 7 0:# ile qe ek}ac na
ykbekmki`eik` na ΰ 7 :%:1 { ile qe ~`g`ðl na gqay~r` na e 7 0=% &Ck 0;=>+
Mkbqr` 1%09% Bràmki`y gly~r`enl ilgl ya ranqia c` zrlo`okckn`n na ilga~ar cly
arrlray K { KK# `c `qgae~`r ac ~`g`ðl na e% &Ck 0;=>+%
Ek}acay na ykbekmki`eik`
Ae c` zrqao` na qe` fkzù~ayky# c` gàvkg` zrlo`okckn`n ile c` iq`c zqnkër`gly
`rrkayb`r ac ~kzl na arrlr K ya cc`g` ac ek}ac na ykbekmki`eik` na c` zrqao`% Ay~a ek}ac ya
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 242/846
Nr% Fëi~lr Pqa}anl Qrç`y
1'9;
ayzaikmki` `e~ay na pqa ya y`pqae c`y gqay~r`y { ya f`b` c` zrqao` na fkzù~ayky# z`r`
pqa el f`{` kemcqaeik` ylora cly rayqc~`nly lo~aeknly%
C` zrqao` na ykbekmki`eik` ay iq`enl ya ayzaikmki` c` zrlo`okckn`n ile c` iq`cay~`gly nkyzqay~ly ` `rrkayb`r ac raif`xl na c` fkzù~ayky# `iari` nac zrlgankl
zloc ikle`c# `qe iq`enl ay }arn`narl% Cly ek}acay na ykbekmki`eik` gày qy`nly ae c`y
zrqao`y na fkzù~ayky yle ac na ΰ 7 :%:1 { ΰ 7 :%:0% Ay~ly }`clray ilrrayzlenae `
ek}acay irç~kily na 0%;= { ?%16# iq`enl ya qy` c` nky~rkoqikùe elrg`c x% Zlr ahagzcl#
ae zrqao`y na fkzù~ayky okc`~ar`cay# ile ΰ 7 :%:1# yk c` x i`ciqc`n` ay x 3 '0%;= l x 5
0%;=# ya raif`x` c` fkzù~ayky% Kbq`cgae~a liqrra yk ac ek}ac na ykbekmki`eik` ayΰ
7 :%:0#
ay naikr# iq`enl x 3 '?%16 { x 5 ?%16# ae~leiay# ya raif`x` c` fkzù~ayky% Na l~r`
g`ear` ya ra~kaea l ya nkia pqa el fqol yqmkikae~a a}knaeik` z`r` raif`x`r Fl8 Ay~`
zrqao` na ykbekmki`eik ely `{qn` ` naiknkr yk c` nkmaraeik` ae~ra ac zrlgankl na c`
gqay~r` ay~`nçy~ki` { ac zrlgankl zloc`ikle`c `yqgknl# ya `~rkoq{a ` c` i`yq`ckn`n l
yk ay ay~`nçy~ki`gae~a ykbekmki`e~a# ay~l ay# yk ay gq{ br`ena z`r` yar `~rkoqknl ` c`
i`yq`ckn`n% C` ^@OC@ 1%> n` cly }`clray irç~kily gày ilgqeay%
^@OC@ 1%>% ^`oc` gly~r`enl c`y rabkleay irç~ki`y pqa ya namkeae na `iqarnl `c }`clr nac ek}ac na ykbekmki`eik` qy`nl# ay naikr# yk c` zrqao` na fkzù~ayky ay okc`~ar`c#qekc`~ar`c naraif` l qekc`~ar`c kxpqkarn`% &Ac`olr`ikùe zrlzk`+
Ek}ac na ykbekmki`eik ΰ :%0: :%:1 :%:0 :%::1 :%::?
]`clray irç~kily na x
z`r` zrqao`y qekc ~ar cay °0%?6 °0%=>1 °?%99 °?%16 °?%66
&naraif` l kxpqkarn`+]`clray irç~kily na x z`r` °0%=>1 °0%;= °?%16 °?%60 °9%:6
zrqao`y okc ~ar`cay
Zlr ahagzcl yk qy`gly qe ek}ac na ilemk`ex` na ;1# ay naikr# qe ek}ac na
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 243/846
Nr% Fëi~lr Pqa}anl Qrç`y
1'>:
ykbekmki`eik` na ΰ 7 :%:1# z`r` qe` zrqao` na fkzù~ayky qekc`~ar`c kxpqkarn`# ae~leiay#
o`hl ay~ y ilenkikleay# ac }`clr irç~kil na x ay '0%?6% Ykgkc`rgae~a# yk ya qy` ac ek}ac
na ykbekmki`eik` na ΰ 7 :%0:# z`r` qe` zrqao` na fkzù~ayky qekc`~ar`c naraif`#ae~leiay# ac }`clr irç~kil na x ay na $0%?6%
I`oa el~`r pqë# z`r` c`y zrqao`y na fkzù~ayky# cly }`clray na cly ek}acay na
ykbekmki`eik` gày ilgqeay yle cly na ΰ 7 :%:1 { na ΰ 7 :%:0% Zlr ahagzcl# z`r` qe`
zrqao` okc`~ar`c ile ΰ 7 :%:1# cly }`clray irç~kily na x yle na °0%;=% El loy~`e~a# z`r`
qe` zrqao` qekc`~ar`c kxpqkarn` ile ΰ 7 :%:1# ac }`clr irç~kil na x yarç` na na '0%=>1 {
`yç yqiayk}`gae~a%
¼Iqàc ay c` nkmaraeik` ae c` naikykùe na `iaz~`r l na raif`x`r qe` fkzù~ayky eqc`<
Z`r` }ar ay~` yk~q`ikùe# yqzleb`gly pqa ac }`clr na c` fkzù~ayky eqc` ay kbq`c ` qe
}`clr ayzar`nl na ±l 7 0:# ay~l ay# Fl8± 7 0:% @nagày# yqzleb`gly pqa V 7 0?# ω 7
>%1 { e 7 ?1 {# yk nayzqëy na yqy~k~qkr cly }`clray ae c` }`rk`oca `ca`~lrk` elrg`c
i`ciqc`n` zlr xi`ci% 7 & V ' ±l+ , ω,∘ e# ile ΰ 7 :%:1 ile yqy }`clray irç~kily na °0%;=#
ae~leiay# xi`ci% 7 ?%??# {# zlr cl ~`e~l# ?%?? 5 0%;= { ya raif`x` Fl8 @pqç# c`
ilemk`okckn`n ay nki~`n` zlr ac }`clr na c` zrlo`okckn`n z# ay~l ay# z 7 0 ' %;6=0 7
%:09;% Ay~l nkia pqa# c` zrlo`okckn`n na f`oar faifl qe` naikykùe apqk}li`n` ae
raif`x`r qe` fkzù~ayky }arn`nar` ay na# `zrlvkg`n`gae~a# 0 ae 0::%
@flr`# yqzleb`gly pqa xi`ci% 7 0%?# ile ω 7 6%999 { ile c`y nagày }`rk`ocay
iley~`e~ay% O`hl ay~`y ilenkikleay# 0%? 3 0%;= {# ya `iaz~` Fl8 ile qe }`clr na
zrlo`okckn`n na z 7 0 ' %66>; 7 %0?% @pqç# ac }`clr na z nkia pqa# c` zrlo`okckn`n na
f`oar faifl qe` naikykùe arrùea`# ae f`oar `iaz~`nl qe` fkzù~ayky m`cy` ay na 0 zlykokckn`n ae 0:% Ae~leiay# na `iqarnl `c r`xle`gkae~l avzqay~l `e~arklrgae~a# ¼ae
iqàc na c`y nly yk~q`ikleay f`{ gày ilemk`okckn`n# ay naikr# gày iar~ax` ae eqay~r`y
naikykleay<
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 244/846
Nr% Fëi~lr Pqa}anl Qrç`y
1'>0
Ile c` kebaekarç` na g`eqm`i~qr`# qe raif`xl na fkzù~ayky &ayzaik`cgae~a# yk ac
}`clr na c` zrlo`okckn`n z ay gqifl gq{ ykbekmki`e~a# a%b%# z 7 %::0+ yqbkara pqa c`
cçea` na zrlnqiikùe na c` màorki` l na c` g`eqm`i~qr` kenqy~rk`c na farr`gkae~`y#`r~çiqcly# ram`iikleay# a~i%# ay~à ~r`o`h`enl ae mlrg` ùz~kg`% Ae mlrg` `eàclb`# ile c`
ke}ay~kb`ikùe ikae~çmki`# qe raif`xl na fkzù~ayky# na qe` gqay~r` na cly rayqc~`nly
lo~aeknly na c`olr`~lrkl# kenki` qe nkyaðl avzarkgae~`c ileikyl { zraikyl% El
loy~`e~a# qe` `iaz~`ikùe na fkzù~ayky yqbkara pqa naoae na ramke`rya c`y ~ëieki`y na
c`olr`~lrkl l na c` zrlnqiikùe% ^`gokëe ya zqana f`iar yacaiikle`enl ~`g`ðly na
gqay~r`y gày br`enay &`qepqa ay~l ay gày ily~lyl+%
Ilgzleae~ay na c` zrqao` na fkzù~ayky mlrg`c
0% Zrqao`y na fkzù~ayky icàyki`y% Ay~`y zrqao`y ~r`nkikle`cay ya nkyað`rle ac ykbcl 0;%
Ay~`y zrqao`y na fkzù~ayky eqc`y ya nael~`e zlr Fl8 { ay qe` `mkrg`ikùe `iari` nac
}`clr nac z`ràga~rl na zloc`ikùe# ±% Ay~` zrqao` na fkzù~ayky eqc` &Fl8+ ya nael~`
qy`enl naykbq`cn`nay `cbaor`ki`y# c`y iq`cay ya nayirkoae ile cly yçgolcly 7# ≮ # ≡ %
Ay~l pqkara naikr pqa# c` zrqao` na fkzù~ayky eqc` {# c`y fkzù~ayky `c~are`~k}`y ~kaeae
~ray mlrg`y zlykocay8
&`+ Fl8 ± 7 ±l &1'?=+
Ay~` rac`ikùe pqkara naikr pqa ay /kbq`c/ `c }`clr ayzar`nl na ±l%
&o+ Fl8 ± ≡ ±l &1'?=`+
Ay~` rac`ikùe ile c` naykbq`cn`n ≡ pqkara naikr /iq`enl gaely/ pqa l /kbq`c l g`{lr
pqa/ ac }`clr ayzar`nl na ±l+
&i+ Fl8 ± ≮ ±l &1'?=o+Ae ay~` rac`ikùe c` naykbq`cn`n ≮ pqkara naikr /` cl gaely/# /elgày pqa/ l /kbq`c l
gaelr pqa/ ac }`clr ayzar`nl na ±l
?% Zrqao`y `c~are`~k}`y% Ay~`y zrqao`y ya nael~`e zlr cly yçgolcly F08# F?8 l F98%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 245/846
Nr% Fëi~lr Pqa}anl Qrç`y
1'>?
Ay~`y zrqao`y `c~are`~k}`y el naoae na ile~aear kbq`cn`nay# ilgl ae ac i`yl na qy`r
cly yçgolcly 7# ≡ # l ≮ # pqa nael~`e c`y fkzù~ayky eqc`y# ykel pqa naoae na ~aear
naykbq`cn`nay ilgl 5 l 3% Na g`ear` pqa# z`r` nael~`r c`y fkzù~ayky `c~are`~k}`y#baear`cgae~a# avky~ae ~ray g`ear`y na avzray`rc`y# ay~l ay8
F08 ± ≪ ±l# yk Fl8± 7 ±l &1'?=i+
F08 ± 3 ±l# yk Fl8± ≡ ±l &1'?=n+
F08 ± 5 ±l# yk Fl8± ≮ ±l &1'?=a+
Zlr ahagzcl# yk ya zrqao` c` fkzù~ayky eqc` na pqa ac }`clr ayzar`nl zloc`ikle`c ay ± l
7 1:%:# ae~leiay# c` zrqao` na fkzù~ayky eqc` ay Fl8± 7 1:%:# { c`y fkzù~ayky
`c~are`~k}`y yle Fl8± ≪ 1:%:# F08± 5 1: { F?8± 3 1:% @nagày# yk ay~q}kër`gly
zrlo`enl c`y fkzù~ayky eqc`y na Fl8± ≡ 1:%:# ae~leiay# c` fkzù~ayky `c~are`~k}` ay F08±
3 1:% Na kbq`c g`ear`# yk ay~q}kër`gly zrlo`enl c` fkzù~ayky eqc` na pqa Fl8± ≮ 1:#
ae~leiay c` fkzù~ayky `c~are`~k}` naoa yar F08± 5 1:%:
El~` 0% Yk ay~`gly f`ikaenl eqay~r`y zrlzk`y zrqao`y# naoarç`gly `rrabc`r c`y
fkzù~ayky eqc`y { c`y `c~are`~k}`y na ~`c g`ear` pqa# ac arrlr gày yarkl mqar` ac raif`xl
na qe` zrqao` na fkzù~ayky }arn`nar` &arrlr K+% @pqç# ae ay~a ~av~l# ay~`gly yqgkaenl
pqa ay~`gly f`ikaenl c`y zrqao`y faif`y zlr `cbqkae gày% Kna`cgae~a# naoarç`gly
f`iar ~ln`y c`y zrqao`y { raic`gly na ~`c g`ear` pqa ~ln`y mqar`e fkzù~ayky eqc`y%
Ay~`y cçea`y mqarle ayirk~`y ile ac ae~aenkgkae~l na pqa# el ~lnly cly raic`gly yle
ilgl naoarç`e na yar# zlrpqa `cbqely aharikikly ke}lcqir`e raic`gly pqa yle zrqao`y
na fkzù~ayky eqc`y { l~rly pqa ke}lcqir`e fkzù~ayky `c~are`~k}`y% Zlr ahagzcl# yk
pqaragly naiknkr yk qe zrliankgkae~l ay gahlr pqa l~rl# mlrgqc`gly c` fkzù~ayky napqa el f`{ nkmaraeik`y ae~ra cly zrliankgkae~ly% Yagah`e~ay fkzù~ayky ya naelgke`e
fkzù~ayky eqc`y { ya nael~`e zlr Fl% ^`gokëe zlnagly agzax`r ile fkzù~ayky
`c~are`~k}`y &l fkzù~ayky na ke}ay~kb`ikùe+ pqa nkmkarae na qe` fkzù~ayky eqc`
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 246/846
Nr% Fëi~lr Pqa}anl Qrç`y
1'>9
yqy~ae~`n`% Ae ac ay~`ocaikgkae~l na c`y fkzù~ayky# ay~l# yke ago`rbl# naoa ay~`r
o`y`nl ae faifly# zarl el ae zrahqkikly% Na iq cpqkar g`ear`# yk ya raif`x` c`
fkzù~ayky eqc`# &Fl8+ ely keicke`gly zlr c` fkzù~ayky `c~are`~k}` &F08+%Ac irk~arkl z`r` raif`x`r l `iaz~`r Fl8 ykbqkaenl ac gë~lnl icàykil# ay na pqa yk
ac }`clr na c` x i`ciqc`n`# ay naikr# qy`enl c` ay~`nçy~ki` x 7 & V ' ±+ , ω, e # ay g`{lr
pqa c` x ~`oqc`n` &xΰ+# ile yq ilrrayzlenkae~a }`clr irç~kil na ΰ# ae~leiay ya raif`x`
Fl8 { ya keicke` zlr c` fkzù~ayky `c~are`~k}` F08% Na l~r` g`ear`# el ya raif`x` Fl8 l ya
zlyzlea c` naikykùe%
9% Zrqao`y na fkzù~ayky el ~r`nkikle`cay% Ay~` zrqao`y ke}lcqir`e cly iàciqcly na c` zrlo`okckn`n# z% Ay~`y zrqao`y yle mlrg`y el icàyki`y na f`iar zrqao`y na fkzù~ayky
eqc`y# Fl8 Ay~`y zrqao`y }kaeae ae ~lnly cly zrlbr`g`y na ilgzq~`nlr` { ya zqanae
f`iar ile c` nky~rkoqikùe na x# ile c` nky~rkoqikùe na ~ na Ay~qnk`e~a# ile c` HK
iq`nr`n` l c` nky~rkoqikùe M%
Knae~kmki`enl c`y zrqao`y na ay~`nçy~ki` na kemaraeik` okc`~ar`cay na &ile c` ilc`
naraif` l ile c` kxpqkarn`+ { zrqao`y okc`~ar`cay &ile nly ilc`y na c`y
nky~rkoqikleay zrlo`n`y+
Iq`enl ay~`gly f`ikaenl zrqao`y na fkzù~ayky# `cbqe`y }aiay ay eaiay`rkl f`iar ay~`y
zrqao`y ae mlrg` okc`~ar`c l qekc ~ar c &qekc`~ar`c naraif` l kxpqkarn`+% Ay~l ya f`ia
qy`enl c` nky~rkoqikùe elrg`c# c` ~ na Ay~qnk`e~a# c` HK iq`nr`n`# c` nky~rkoqikùe na
Mkyfar# a~i%
Iq`enl f`iagly eqay~r`y zrlzk`y zrqao`y na fkzù~ayky# { y`oagly zlr
avzarkaeik` pqa cly }`clray ayzar`nly na ±l }`e ` yar g`{lray na ikar~ly }`clray &liq`enl naikgly pqa F08± 5 ±l+# ae~leiay qy`gly ac av~ragl naraifl na c`
nky~rkoqikùe { zleagly ac ek}ac na ykbekmki`eik` na ΰ 7 :%:1 l :%:0# ile Fl8 7 ` ikar~l
}`clr% Ae mlrg` `eàclb`# yk cly }`clray ayzar`nly }`e ` yar gaelray na ikar~ly }`clray
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 247/846
Nr% Fëi~lr Pqa}anl Qrç`y
1'>>
&l F08± 3 pqa ikar~l }`clr+# qy`gly ac av~ragl kxpqkarnl na c` nky~rkoqikùe { zleagly
ac ek}ac na ykbekmki`eik` na ΰ 7 :%:1 l :%:0% Mke`cgae~a# yk ayzar`gly pqa cly }`clray
}`{`e ` yar gaelray l g`{lray na ikar~`y i`e~kn`nay &Fl8± 7 ` qe na~argke`nl }`clr+#ae~leiay qy`gly cly nly av~ragly na c` nky~rkoqikùe { nk}knkgly ΰ# kbq`cgae~a# ae~ra
nly# z`r` pqa ely në# ΰ 7 :%:1,? l ΰ 7 :%:0,?# a~i%
Yk f`iagly c`y zrqao`y na l~rly# zlr gankl nac av`gae na c` fkzù~ayky eqc`#
Fl8 zlnagly nanqikr yk c` zrqao` ay na nly ilc`y l na qe` ilc` &naraif` l kxpqkarn`+%
Zlr ahagzcl# yk Fl8± 7 ;6%=# ae~leiay F08± ≪ ;6%= { ya nkia pqa c`y zrqao`y
`c~are`~k}`y yle na F08± 5 ;6%= { F
08± 3 ;6%=% El loy~`e~a# yk c` zrqao` na fkzù~ayky
eqc` ay na Fl8± ≡ ;6%=# ae~leiay# c` ilc` na c` fkzù~ayky `c~are`~k}` &pqa ay cl
ile~r`rkl na c` fkzù~ayky eqc` Fl8+ `zqe~` ` c` kxpqkarn` &ilgl ± 3 ;6%=+# { c` zrqao`
ay na c` ilc` kxpqkarn` &qekc`~ar`c kxpqkarn`+% Yke ago`rbl# yk Fl8± ≮ ;6%=# ae~leiay c`
zrqao` ay na pqa ± 5 ;6%=# { c` zrqao` ay na c` ilc` naraif` &qekc`~ar`c naraif`+%
_ayqgae ae ac ay~`ocaikgkae~l na c`y zrqao`y na fkzù~ayky okc`~ar`cay &nly ilc`y+
l qekc`~ar`cay &na c` ilc` naraif` l na c` kxpqkarn`+8
Yk c` zrqao` na fkzù~ayky eqc` ay Fl8± 7 ±l# ae~leiay# c` zrqao` ay okc`~ar`c { c`y
fkzù~ayky `c~are`~k}`y yle8 F08± ≪ ±l% F?8± 5 ±l { F98± 3 ±l# nlena ±l ay ac }`clr
ayzar`nl%
Yk c` zrqao` na fkzù~ayky eqc` ay Fl8± ≡ ±l# ae~leiay# c` zrqao` ay qekc`~ar`c kxpqkarn`
{ c` fkzù~ayky `c~are`~k}` ay F08± 3 ±l%
Yk c` zrqao` na fkzù~ayky eqc` ay Fl8± ≡ ±l# ae~leiay# c` zrqao` ay qekc`~ar`c naraif` {
c` fkzù~ayky `c~are`~k}` ay F08± 5 ±l%Namkekaenl cly z`yly icàykily ae ac zrliankgkae~l z`r` f`iar zrqao`y na
fkzù~ayky
0% Ay~`ocaiar c` zrqao` na fkzù~ayky eqc` &Fl8+ { ac zrlgankl ayzar`nl ±l { c`y zrqao`y
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 248/846
Nr% Fëi~lr Pqa}anl Qrç`y
1'>1
na fkzù~ayky `c~are`~k}`y &F0# F?8# F98+% Ae ay~a raebcùe# yk ya qy` c` nky~rkoqikùe
elrg`c# ~`gokëe ay eaiay`rkl f`iar ikar~`y yqzlykikleay na pqa c` gqay~r` ay `ca`~lrk`#
na pqa c` zloc`ikùe gqay~ra`n` ay elrg`c {# na pqa c` nay}k`ikùe ay~àen`r zloc ikle`c# ω ay ilelikn`%
?% Yacaiikle`r ac ek}ac na ykbekmki`eik` na ΰ naya`nl &:%:1# :%:0# a~i%# a% b% nac ;1#
nac ;;# a~i%+% @pqç# z`r` c` zrqao` na nly ilc`y# ay xΰ,? { z`r` c`y ilc`y naraif` l
kxpqkarn`# ykgzcagae~a ΰ%
9% Na~argke`r c` zrqao` ay~`nçy~ki` pqa ya }` ` qy`r z`r` ac zrlgankl# c` }`rk`ex`# c`y
zrlzlrikleay# a~i%# pqa ya }`e ` zrlo`r# ay naikr# qy`enl c`y nky~rkoqikleay x# ~#ς
?# M#
a~i% Na ay~` g`ear`# yk e ≡ 9: i`yly ya qy` c` nky~rkoqikùe na x z`r` ac zrlgankl% Na
l~r` g`ear`# yk e 3 9: ya qy` cl pqa ya cc`g` ~alrç` na gqay~ral zapqaðl# ilgl c` ~ na
ay~qnk`e~a# c` HK iq`nr`n`# a~i%
>% Namkekr c`y rabkleay irç~ki`y# ay naikr# na qe` ilc` &kxpqkarn` l naraif`+ l na nly
ilc`y% &]ar rayqgae na zrqao`y z`r` nly ilc`y# z`r` c` naraif` l z`r` c` kxpqkarn`+%
Zlr ahagzcl# yk F08± 5 qe }`clr# ya qy` c` ilc` naraif`% Yk ± 3 pqa qe }`clr# ya qy` c`
ilc` kxpqkarn`# zarl yk ± ay naykbq`c ` qe }`clr n`nl ya qy`e nly ilc`y% @pqç# yke
ago`rbl# ay na el~`rya pqa ay~`y ikriqey~`eik`y nazaenae nac nkyaðl avzarkgae~`c pqa
ya pqkar` f`iar%
1% Namkekr c` rabc` na naikykùe# ay naikr# na raif`x`r l na ra~aear l `iaz~`r c` fkzù~ayky
eqc`# Fl8 {,l na keicke`rya zlr c`y fkzù~ayky `c~are`~k}`y# F08# F?8# a~i%
=% F`iar cly iàciqcly eaiay`rkly na cly n`~ly na c` gqay~r` { i`ciqc`r ac }`clr na c`
mqeikùe ay~`nçy~ki` na c`y nky~rkoqikleay na x# na ~# na ς ?
# a~i%# pqa ya }`{`e ` qy`r% Zlr ahagzcl# yk qy`gly c` nky~rkoqikùe elrg`c na x l c` na ~ na ay~qnk`e~a z`r` ac zrlgankl
`rk~gë~kil# qy`gly8
x 7 & V ' ±+ , ω, e l okae ~ 7 & V ' ±+ , y, e
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 249/846
Nr% Fëi~lr Pqa}anl Qrç`y
1'>=
Eù~aya pqa c øeki` nkmaraeik` ae~ra c` x { c` ~ ay na pqa ae c` x ya qy` ω { ae c` ~ ya qy`
y z`r` c` nay}k`ikùe ay~àen`r%
2% Ilgz`r`r ac }`clr na c` mqeikùe qy`n` ile c` rabc` na naikykùe ay~`ocaikn`# { f`iar c` naikykùe ay~`nçy~ki` icàyki` l ~r`nkikle`c &pqa ya nkyaðù ae ac ykbcl `e~az`y`nl+
`iari` na c` fkzù~ayky eqc`% @pqç ~`gokëe ya zqana f`iar c` zrqao` na c` zrlo`okckn`n
na z# pqa ay qe` zrqao` el ~r`nkikle`c l glnare` na c` ar` ikoareë~ki`% @yç# yk ac }`clr
na c` ay~`nçy~ki` i`ciqc`n` ay g`{lr pqa c` xi l ~ ~`oqc`n`y# ya raif`x` c` fkzù~ayky
eqc`# Fl8 { ya keicke` zlr c` fkzù~ayky `c~are`~k}`% Na l~r` g`ear` el ya raif`x` Fl8 l
el ya f`ia ekebqe` naikykùe% Ay~` zrqao` na zrlo`okckn`n z# ya f`ia z`r` }ar# ile pqë
~`e~` mknackn`n &ae ~ërgkely na zrlo`okckn`n+ zqnkër`gly ay~`r `iar~`nly l
apqk}li`nly ae f`oar raif`x`enl c` fkzù~ayky eqc`% @pqç# zlr ahagzcl# yk ac }`clr na z
ay gaelr pqa ac }`clr nac ek}ac na ykbekmki`eik` na ΰ# ya raif`x` c` fkzù~ayky { ya nkia
yk ay ykbekmki`e~a l gq{ ykbekmki`e~a# a~i%
_abc`y na naikykùe o`hl }`rk`y ilenkikleay ile c`y nky~rkoqikleay x { ~
Z`r` c` nky~rkoqikùe elrg`c8
Iq`enl e ≡ 9: i`yly# ω ilelikn` {# ` y`okaen`y pqa c` nky~rkoqikùe ay elrg`c%
Z`r` zrqao`y okc`~ar`cay &nly ilc`y+8 raif`x`r Fl8 { `iaz~`r F08# yk ac }`clr na c`
ay~`nçy~ki` x ay g`{lr pqa c` xi ~`oqc`n`4 na l~r` g`ear` `iaz~`r Fl8 l zly~arb`r c`
naikykùe%
Z`r` zrqao`y qekc`~ar`cay &qe` ylc` ilc`+# nkb`gly# c` kxpqkarn`8 raif`x`r Fl8 { `iaz~`r
F08 yk ac }`clr na c` ay~`nçy~ki` x ay gaelr pqa c` xi ~`oqc`n`% Na l~r` g`ear`# `iaz~`r
l ra~aear Fl8Z`r` zrqao`y qekc`~ar`cay &qe` ylc` ilc`+# nkb`gly c` naraif`8 raif`x`r Fl8 { `iaz~`r
F08 yk ac }`clr na c` ay~`nçy~ki` x ay g`{lr pqa c` xi ~`oqc`n`4 na l~r` g`ear`# `iaz~`r
c` Fl8
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 250/846
Nr% Fëi~lr Pqa}anl Qrç`y
1'>2
Z`r` c` nky~rkoqikùe na ~ na Ay~qnk`e~a8
Iq`enl e ≡ 9: i`yly# ω nayilelikn` { y`okaenl pqa c` zloc`ikùe gqay~ra`n` ay
elrg`c%Z`r` zrqao`y okc`~ar`cay &nly ilc`y+8 raif`x`r Fl8 { `iaz~`r F08 yk ac }`clr na c`
ay~`nçy~ki` ~ 5 $~ΰ,?% F`iarcl na kbq`c g`ear`# yk ac }`clr na c` ay~`nçy~ki` ~ 3 '~ΰ,?4 na
l~r` g`ear`# ra~aear Fl8
Z`r` qe` ylc` ilc`# nkb`gly c` kxpqkarn`# raif`x`r Fl8 { `iaz~`r F08 yk ac }`clr na c`
ay~`nçy~ki` ~ 3 '~ΰ4 na l~r` g`ear` `iaz~`r Fl l el f`iar ekebqe` naikykùe
Z`r` qe` ylc` ilc`# nkb`gly c` naraif`8 raif`x`r Fl8 { `iaz~`r F
08 yk ac }`clr na c`
ay~`nçy~ki` ~ 5 $~ΰ% Na l~r` g`ear` `iaz~`r Fl8
C` kna` na~rày na f`iar zrqao`y na fkzù~ayky
Ilgl ya nkhl `e~ay# c` kna` na f`iar zrqao`y na fkzù~ayky ay c` na `iqgqc`r a}knaeik`
z`r` raif`x`r c` fkzù~ayky eqc % Ae ac i`gzl na c` ke}ay~kb`ikùe ikae~çmki`# ~lnly cly
ke}ay~kb`nlray ykagzra ay~àe ayzar`ex`nly ae raif`x`r c`y fkzù~ayky eqc`y na yqy
~r`o`hly na ke}ay~kb`ikùe% Ily` ykgkc`r liqrra ile c` kebaekarç` kenqy~rk`c { na
g`eqm`i~qr`% Cly kebaekarly kenqy~rk`cay ykagzra ~kaeae pqa f`iar zrqao`y na
fkzù~ayky zarkùnki`y na cly zrlnqi~ly g`eqm`i~qr`nly l na cly `r~çiqcly zrlnqiknly
zlr c` kenqy~rk na g`eqm`i~qr % Ay~l ya f`ia ile ac loha~l na ra}ky`r c` amkikaeik` na
c` cçea` na zrlnqiikùe na c` màorki`% Ay~l ya naoa ` pqa# `c raif`x`r qe` fkzù~ayky eqc`#
ay~l nael~` qe nkyaðl avzarkgae~`c mqar~a { ilemk`oca% Ae c` kenqy~rk` na g`eqm`i~qr`
cly raif`xly na fkzù~ayky kenki`e pqa c` cçea` na zrlnqiikùe ay~à lzar`enl
elrg`cgae~a%Ae c`y zrqao`y na fkzù~ayky# ac }`clr na y l na ω ay gq{ kgzlr~`e~a# zlrpqa `fç
ya ramcah` c`y ~ëieki`y nac c`olr`~lrkl ramke`n`y l namai~qly`y% Qe }`clr o`hl na y
ramcah` ~ëieki`y na c`olr`~lrkl gq{ ylmky~ki`n`y l ramke`n`y# gkae~r`y pqa qe }`clr `c~l
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 251/846
Nr% Fëi~lr Pqa}anl Qrç`y
1'>6
na y# ramcah` cl ile~r`rkl% ^lnl ay~l ya avzcki` { ya zrqao` ` ~r`}ëy nac zlnar na c`
zrqao` { na cly arrlray ay~àen r nac zrlgankl# na c` nay}k ikùe ay~àen`r# a~i%
Ac }`clr na z ae c` ~lg` na naikykleayAe c`y zrqao`y na fkzù~ayky f`{ l~r` mlrg` `c~are`~k}` glnare` ilgzq~`rkx`n` na
zrlo`r c` gkyg` yk~q`ikùe &pqa ya f`ia ile c` zrqao` icàyki` na fkzù~ayky pqa ya
nkyaðù ae ac ykbcl `e~az`y`nl+# ay naikr# ac aemlpqa glnarel% Ae }arn`n# ac }`clr na z
ay c` zrlo`okckn`n# o`hl c` fkzù~ayky eqc` &l c` zrlo`okckn`n# yk c` fkzù~ayky eqc` ay
}arn`nar`+# na lo~aear qe }`clr ~`e keqyq`c l gày keqyq`c pqa `pqëc pqa lo~q}kgly na
c` gqay~r`# iq`enl c` fkzù~ayky eqc` ay }arn`nar` &qe` yk~q`ikùe keqyk~`n`+%
Ay~` zrqao` el ~r`nkikle`c ya f`ia qy`enl ac }`clr na c` zrlo`okckn`n z% Zlr
ahagzcl# iq`enl raif`x`gly l `iaz~`gly qe` fkzù~ayky eqc` Fl8 { ely keicke`gly zlr
c` fkzù~ayky `c~are`~k}`# F08# ile qe ek}ac na ykbekmki`eik` na ΰ 7 :%:1 l kbq`c ` :%:0#
a~i%# pqaragly y`oar# pqë ~`e~` ilemk`okckn`n zlnagly zlear `c f`iar eqay~r`y
naikykleay ay~`nçy~ki`y% Ay~a br`nl na ilemk`okckn`n ya n` zlr c` zrlo`okckn`n# z%
Ae }arn`n# ac ileiaz~l mkclyùmkil nac }`clr na z ay pqa ay~a }`clr razrayae~` qe
nairagae~l ae ac br`nl na ilemk`okckn`n ae qe rayqc~`nl% Ay~a aemlpqa ay~à nkyað`nl
z`r` n`rely c` `c~are`~k}` &ae ~ërgkely na zrlo`okckn`n+# na raif`x`r l el raif`x`r c`
fkzù~ayky yqy~ae~`n`% @yç# ae~ra gày o`hl ya` ac }`clr na z# gaely zlnagly iraar ae c`
fkzù~ayky eqc`% Ayzaiçmki`gae~a f`oc`enl# ac ek}ac na z razrayae~` c` zrlo`okckn`n na
arrlr ae `iaz~`r cly rayqc~`nly loyar}`nly ilgl }àcknly%
Zlr ahagzcl# ile qe }`clr na z 7 %:1 ay~l ykbekmki` 0,?:# ay naikr pqa
zqnkër`gly ay~`r apqk}li`nly ile qe` zrlo`okckn`n na 0 ae ?: ae c` naikykùe naraif`x`r c` fkzù~ayky eqc`# Fl8 yqy~ae~`n`% @nagày# yk z 7 %:0# ay~l ay# 0,0::# kenki`
pqa zqnkër`gly ay~`r apqk}li`nly ae eqay~r` naikykùe na raif`x`r c` fkzù~ayky ile
qe` zrlo`okckn`n na 0 ae 0::% &@pqç# ae ay~ly i`yly# e`nka }` ` `rbqgae~`r pqa }`gly
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 252/846
Nr% Fëi~lr Pqa}anl Qrç`y
1'>;
` apqk}li`rely ae eqay~r` naikykùe# ile ay~` zrlo`okckn`n ~`e o`h`+%
Ae ~ërgkely baear`cay# }`clray br`enay na z# nkb`gly 5 :%0 `zl{`e ac el
raif`xl na c` fkzù~ayky &ay naikr ya `iaz~` l ya rayar}` qe` naikykùe+% Zlr l~rl c`nl#}`clray zapqaðly na z `zl{`e ac raif`xl na c` fkzù~ayky%
Cly ~kzly na gai`ekygly pqa ya ykbqae z`r` ay~`ocaiar c`y zrqao`y na fkzù~ayky
0% C` fkzù~ayky eqc` ya zqana f`iar ilgl8 Fl8± 7 ±l% O`hl ay~`y ilenkikleay na
kbq`cn`n# c`y fkzù~ayky `c~are`~k}`y yle8
F08± ≪ ±l# F?8± 3 ±l { F98± 5 ±l# nlena ±l ay ac zrlgankl zloc`ikle`c pqa ya pqkara
zrlo`r% @pqç# i`oa el~`r pqa ae ay~a i`yl# c` zrqao` na fkzù~ayky ay okc ~ar c l na nly
ilc`y%
?% ^`gokëe c` fkzù~ayky eqc` ya zqana f`iar ilgl8 Fl8± ≡ ±l% Ae ay~a i`yl# c`
fkzù~ayky `c~are`~k}` ay Fl8± 3 ±l% @pqç# c` zrqao` na fkzù~ayky ay qekc`~ar`c kxpqkarn`%
9% Kbq`cgae~a# c` fkzù~ayky eqc` ya zqana f`iar ilgl8 Fl8± ≮ ±l% Ae ay~a i`yl c`
fkzù~ayky `c~are`~k}` ay F08± 5 ±l% @pqç# c` zrqao` na fkzù~ayky ay qekc`~ar`c naraif`%
>% Yacaiikle`r qe ek}ac na ykbekmki`eik` na ~`g`ðl ΰ# ay~l ay# ΰ 7 %:1 l ΰ 7 %:0 ile yqy
rayzai~k}ly ek}acay na ilemk`ex` na ;1 { ;;% ^`gokëe# ya zqanae qy`r l~rly
ek}acay na ykbekmki`eik`# ilgl ac %0:# %?:# a~i%# zarl cly gày ilgqeay yle cly na :%:1
{ %:0%
1% Yacaiikle`r c` ay~`nçy~ki` `zrlzk`n` &zlr ahagzcl# yk e 5 9: i`yly ya qy` c`
nky~rkoqikùe x% Yk c` gqay~r` ay e 3 9: i`yly { c` zloc`ikùe gqay~ra`n` el ay elrg`c
ya qy` c` nky~rkoqikùe na ~# a~i%
=% Ya ay~`ocaiae c`y rabkleay irç~ki`y qy`enl ek}acay na ilemk`ex` nac ;1# ;;# ;:#6: a~i% &;1 { ;; cly gày ilgqeay+
2% Ya ay~kg` ac }`clr na c` zrqao` na ay~`nçy~ki` na c` gqay~r` { ya ilgz`r` ile ac
}`clr na c` ay~`nçy~ki` i`ciqc`n`# ay naikr# xi`ci% l ~i`ci% &na c`y rabkleay irç~ki`y+ { ya
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 253/846
Nr% Fëi~lr Pqa}anl Qrç`y
1'1:
ilgz`r`e ile x~`o% l ~~`o% Yk c` ay~`nçy~ki` i`ciqc`n` ay g`{lr pqa c` ay~`nçy~ki` ~`oqc`n`
&na c`y rabkleay irç~ki`y+ ya raif`x` c` fkzù~ayky eqc`% Na l~r` g`ear`# ya `iaz~` c`
fkzù~ayky l el ya f`ia ekebqe` naikykùe% Na ay~` g`ear`# yk ac }`clr na c` ay~`nçy~ki`i`ciqc`n` ya ga~a ae c`y rabkleay irç~ki`y ya raif`x` c` fkzù~ayky eqc` &l ~`gokëe yk ac
}`clr na z ay gaelr l kbq`c c ek}ac na ykbekmki`eik # ΰ naya`nl+%
El~ 8 @pqç ay kgzlr~`e~a railrn`r pqa# c` zrqao` na fkzù~ayky eqc` ay~`nçy~ki` ya
nkyaðù ac ykbcl `e~az`y`nl% Ae ~kagzly glnarely na c` ar` ikoareë~ki`# avky~a c`
zrqao` el ~r`nkikle`c rac`ikle`n` ile ac }`clr na c` zrlo`okckn`n z% ^`gokëe ay
kgzlr~`e~a el~`r pqa gqifly zrlbr`g`y na ilgzq~`nlr` n`e øeki`gae~a ac }`clr na
z { ac ke}ay~kb`nlr l cai~lr ~kaea pqa ke~arzra~ rcl ilrnagae~a%
Gai`ekygly z`r` i`ciqc`r cly }`clray na c` zrlo`okckn`n z &z`r` c` nky~rkoqikùe
elrg`c+ iq`enl ya f`iae c`y zrqao`y na fkzù~ayky el ~r`nkikle`cay &i`ciqc`enl ac
}`clr na z+
0% Z`r` i`ciqc`r ac }`clr na c` zrlo`okckn`n z# ya oqyi` ac }`clr na c` x i`ciqc`n` ae c`
~`oc` na c` nky~rkoqikùe elrg`c# ile ac }`clr nac ek}ac na ykbekmki`eik` qy`nl% Cly
irk~arkly pqa ya ykbqae ya f`iae ilgz`r`enl ac }`clr na c` z ile ac }`clr na ΰ%
?% Cly irk~arkly pqa ya ykbqae z`r` ke~arzra~`r ac }`clr na z yle8
Z ≮ %:1 C` zrqao` ay~à ae ac qgor`c na c` ykbekmki`eik`% @pqç i`yk ykagzra ya
`iaz~` c` fkzù~ayky eqc`% Ay qe `rbqgae~l nëokc { el ile}keiae~a ae c` zrqao`y na
fkzù~ayky% Ely nah` ae qe` yk~q`ikùe na keiar~knqgora% Ely nkia pqa# –~`c }ax `yç
ya`‗%
Z ≮ %:0 C` zrqao` ay `c~`gae~a ykbekmki`~k}`% Ya ileyknar` qe `rbqgae~lay~`nçy~kil gq{ mqar~a ae ile~r` na c` `iaz~`ikùe na c` fkzù~ayky eqc`% C`
zrlo`okckn`n na %:0 nkia pqa zqnkër`gly f`oarely apqk}li`nl ae c`
naikykùe na raif`x`r c` fkzù~ayky eqc`# ile qe` zrlo`okckn`n na 0 ae 0:: na f`oar
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 254/846
Nr% Fëi~lr Pqa}anl Qrç`y
1'10
raif`x`nl qe` fkzù~ayky }arn`nar`# iq`enl naokù yar `iaz~`n`%
Z ≮ %::0 C` zrqao` ay gqifl gq{ ykbekmki`~k}`% Ya ileyknar` qe `rbqgae~l
ay~`nçy~kil gqifl gq{ mqar~a# ileikyl { zraikyl% @pqç# c` zrlo`okckn`n ile c` iq`c zqnkër`gly f`oarely apqk}li`nl ae f`oar faifl qe` naikykùe arrùea` ae ac
raif`xl na c` fkzù~ayky eqc` ay na qe` gkcëykg`# ay naikr# na 0 ae 0:::%
Ke~arzra~`ikùe g`~agà~ki` na cly }`clray na c` zrlo`okckn`n z &Zm`mmaeoarb a~ `c%
0;62+
Mlrg` K% ]`clr na z 7 ?ZTV 5 vW# yk Fl8 ± 7 ±l ile F08 ± 5 ±l
]`clr na z 7 ?ZTV 3 vW# yk F?8 ± 3 ±
l
Mlrg` KK% ]`clr na z 7 TV 3 vW# yk Fl8 ± ≡ ±l# ile F08 ± 3 ±l
Mlrg` KKK% ]`clr na z 7 ZTV 5 vW# yk Fl8 ± ≮ ±l ile F08 ± 5 ±l
Nlena8 V ay ≮ # ≡ # 7 # pqa# ac zrlgankl gqay~r`c V
Ahagzcl )99% @o`hl ya n`e cly }`clray na c` x i`ciqc`n`% I`ciqc`r ac }`clr na c`
zrlo`okckn`n z# yk8
&`+ Ac }`clr na x 7 9%?# ile Fl8± 7 ±l%
&o+ Ac }`clr na x 7 9%:# ile Fl8± ≮ ±l
&i+ Ac }`clr na x 7 '9%?# ile Fl8± ≡ ±l%
Ylcqikùe8
&`+ Oqyi`gly ac }`clr na x 7 9%? ae c` ~`oc` na c` nky~rkoqikùe elrg`c { n` qe }`clr na
%;;;9% Ae~leiay# z`r` i`ciqc`r ac }`clr na c` zrlo`okckn`n z zrlianagly ilgl ykbqa8
z 7 0 ' %;;;9 7 %:::2%
Yke ago`rbl# naoknl ` pqa c` zrqao` ay okc`~ar`c# ay~a }`clr na z ya gqc~kzcki` zlr ? z`r` n`r z 7 %::0>%
&o+ Oqyi`gly ac }`clr na x 7 9%: ae c` ~`oc` na c` nky~rkoqikùe elrg`c { ely n` %;;62%
Ae~leiay# z`r` i`ciqc`r ac }`clr na c` zrlo`okckn`n# z zrlianagly ilgl8
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 255/846
Nr% Fëi~lr Pqa}anl Qrç`y
1'1?
z 7 0 ' %;;62 7 %::09
Ilgl c` zrqao` ay qekc`~ar`c# `yç ya pqan`%
&i+ Z`r` x 7 '9%? ile Fl8 ± ≡ ±l% Ay~` ay qe` zrqao` qekc`~ar`c kxpqkarn` &zlrpqa ac}`clr na x ay eab`~k}l+% Oqyi`gly ay~a }`clr ae c` ~`oc` na c` nky~rkoqikùe elrg`c { n`
%:::2%
Ga~lnlclbç` z`r` i`ciqc`r cly }`clray na c` zrlo`okckn`n z nazaenkaenl na c`
ay~`nçy~ki` qy`n`
Z`r` c`y zrqao`y na fkzù~ayky el ~r`nkikle`cay# ay naikr# qy`enl ac }`clr na c`
zrlo`okckn`n z# ay eaiay`rkl f`iar ke~arzlc ikleay na cly }`clray lo~aeknly% Yke
ago`rbl# ae ac i`yl na c` nky~rkoqikùe elrg`c# z`r` ay~kg`r ac }`clr na c` zrlo`okckn`n
z# el ay eaiay`rkl f`iar ke~arzlc`ikleay# zlrpqa ya zqana caar nkrai~ gae~a ae c` ~ oc`
na c` nky~rkoqikùe elrg`c ac }`clr na c` ay~`nçy~ki` x i`ciqc`n`% El loy~`e~a# z`r` c`
nky~rkoqikùe na ~ na ay~qnk`e~a# z`r` c` nky~rkoqikùe Mkyfar# z`r` c` nky~rkoqikùe na c` HK
iq`nr`n`# a~i%# yk ay eaiay`rkl f`iar ke~arzlc`ikleay% Ay~l ya f`ia oqyi`enl ac }`clr na
c` ay~`nçy~ki` i`ciqc`n` ae c` ~`oc` na c` nky~rkoqikùe pqa ya ay~à qy`enl ile yq
ilrrayzlenkae~a }`clr na br`nly na ckoar~`n { nac }`clr zlriae~q`c naya`nl%
Mùrgqc` agzçrki` z`r` f`iar ke~arzlc`ikleay { i`ciqc`r ac }`clr na c`
zrlo`okckn`n z
@pqç }`gly ` n`r qe gë~lnl z`r` f`iar ke~arzlc`ikleay qy`enl qe` mùrgqc` agzçrki`
nkyað`n` zlr ac `q~lr na ay~a ckorl# ac Nr% Fëi~lr Pqa}anl Qrç`y &`q~lr na ay~a ckorl+ {
zlr c` Nr`% Ylilrrl @r~a`b`% Ay~` mùrgqc` ya n` ilgl8&δ? ’ δ0+,&^_ ? ’ ^_ 0+ 7 &δ? ’ V+,&^_ ? ’ ^_ i`ci%+ &1'?2+
Nlena8
δ? 7 Ac ek}ac na ilemk`ex` gày `c~l na c` ~`oc` na c` nky~rkoqikùe qy`n`%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 256/846
Nr% Fëi~lr Pqa}anl Qrç`y
1'19
δ0 7 Ac ek}ac na ilemk`ex` gày o`hl na c` ~`oc` qy`n`%
^_ ? 7 zrlo`okckn`n na c` ay~`nçy~ki` qy`n` ilrrayzlenkae~a ` δ?%
^_ 0 7 zrlo`okckn`n na c` ay~`nçy~ki` qy`n` ilrrayzlenkae~a ` δ0%V 7 }`clr pqa ya pqkara ke~arzlc`r% @pqç# iq`enl c` zrqao` ay okc`~ar`c ay~a }`clr ya
gqc~kzcki` zlr ?%
^_ i`ci% 7 }`clr na c` ay~`nçy~ki` i`ciqc`n`%
Mùrgqc`y z`r` i`ciqc`r ac }`clr na z zlr gankl na ke~arzlc`ikleay z`r`
nkmarae~ay nky~rkoqikleay
Z`r` c` nky~rkoqikùe na ~ na Ay~qnk`e~a &c` iq`c ya ra~lg`rà ae ac i`zç~qcl =+8
&δ? ’ δ0+ , &~? ’ ~0+ 7 &δ? ’ V+ , &~? ’ ~i`ci%+ &1'?6+
Z`r` c` nky~rkoqikùe na c` HK iq`nr`n`8
&δ? ’ δ0+ , &ς ?? ’ ς ?0+ 7 &δ? ’ V+ , &ς ?? ’ ς ?i`ci%+ &1'?;+
Z`r` i`ciqc`r c` nky~rkoqikùe M8
&δ? ’ δ0+ , &M? ’ M0+ 7 &δ? ’ V+ , &M? ’ Mi`ci%+ &1'9:+
Ahagzcly gly~r`enl c` g`ear` na i`ciqc`r ac }`clr na c` zrlo`okckn`n z
Ahagzcl )9>% Yqzùeb`ya pqë# yk ac }`clr i`ciqc`nl na c` ay~`nçy~ki` na c` nky~rkoqikùe
na x mqar`# nkb`gly na x 7 ' 9%> ile qe` zrqao` na fkzù~ayky okc`~ar`c# ae~leiay#
oqyi`gly ay~a }`clr ae c ~`oc` na c` nky~rkoqikùe elrg`c { ely n` %:::9% Ay~a }`clr ay
zraiky`gae~a ac }`clr na c` zrlo`okckn`n z% Zarl ilgl c` zrqao` ay okc ~ar c# ya
gqc~kzcki` zlr nly { n` z 7 %:::=%
Ahagzcl )91% Yqzùeb`ya `flr` pqa ac }`clr na c` ay~`nçy~ki` x mqar` nkb`gly x 7 9%>
ile qe` zrqao` okc`~ar`c% Ae~leiay oqyi`gly ay~a }`clr ae c` ~`oc` na c` nky~rkoqikùeelrg`c { }agly pqa ay~à `c av~ragl naraifl ile qe }`clr na %;;;6% @flr` ca
ray~`gly 0 { ely n` z 7 0 ' %;;;6 7 %:::?% Eqa}`gae~a# ilgl c` zrqao` ay okc`~ar`c#
ac }`clr cl gqc~kzcki`gly zlr nly { n` z 7 %:::>%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 257/846
Nr% Fëi~lr Pqa}anl Qrç`y
1'1>
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 258/846
Nr% Fëi~lr Pqa}anl Qrç`y
1'11
Zrqao` na fkzù~ayky z`r` qe ylcl zrlgankl zloc`ikle`c ± ile }`rk`ex` ω?
ilelikn` qy`enl c` nky~rkoqikùe elrg`c
Qe` fkzù~ayky ay~`nçy~ki` ay qe` `ya}ar`ikùe l ileha~qr` ile rayzai~l ` qe` l gày zloc ikleay% Ae ay~`nçy~ki`# qe` fkzù~ayky ay qe aeqeik`nl na pqa `cbl ay }arn`narl%
Ae c` }arn`n l m`cyan`n na qe` fkzù~ayky ay~`nçy~ki` ykagzra f`{ qe` keiar~knqgora#
zlrpqa el ya zqana gqay~ra`r ~ln` c` zloc`ikùe &ay~l yarç` kgzlykoca+% Ae cqb r na
ay~l# ya ~lg` qe` gqay~r` `ca`~lrk` na c` zloc`ikùe na ke~arëy { ya qy`e cly n`~ly z`r`
zrlzlrikle`r a}knaeik &ae ~ërgkely na c` zrlo`okckn`n z+ z`r` `zl{`r l ramq~ r c`
fkzù~ayky% Zlr ahagzcl# c` `iaz~`ikùe na qe` fkzù~ayky eqc` kgzcki` pqa el f`{
yqmkikae~a a}knaeik` z`r` zlnar raif`x`r c` fkzù~ayky% El loy~`e~a# yk ya raif`x` qe`
fkzù~ayky f`{ qe` a}knaeik` gày mqar~a a kgzcki` qe nkyaðl avzarkgae~`c mqar~a#
zraikyl { ileikyl% Ile~r`rk`gae~a# ac el raif`xl na qe` fkzù~ayky kgzcki` qe nkyaðl
avzarkgae~`c nëokc# ile qe` gqay~r` na keyqmkikae~a ~`g`ðl l ~ëieki`y na c`olr`~lrkl
namai~qly`y pqa ilecca}`e gqif` }`rk`ikùe%
C` ay~`nçy~ki` pqa ya qy` z`r` f`iar zrqao`y na fkzù~ayky z`r` qe ylcl zrlgankl
zloc ikle`c ±# ile }`rk`ex` ilelikn` ΰ qy`enl c` nky~rkoqikùe elrg`c# ` y`okaen`y na
pqa c` zloc`ikùe gqay~ra`n` ay elrg`c l pqa e 5 9: i`yly# ay8
x 7 & V ’ ±l+ , ω, e &1'90+
Nlena8
x 7 }`rk`oca `ca`~lrk` elrg`c ay~àen`r
V 7 zrlgankl ay~`nçy~kil
±l 7 }`clr ayzar`nl nac zrlganklω 7 nay}k`ikùe ay~àen`r ilelikn`
e 7 ~`g`ðl na c` gqay~r`
C` ~`oc` na `o`hl gqay~r` cly iàciqcly na c`y rabkleay irç~ki`y qy`enl nkmarae~ay
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 259/846
Nr% Fëi~lr Pqa}anl Qrç`y
1'1=
ek}acay na ykbekmki`eik`%
^@OC@ 1%1% ^`oc` gly~r`enl c`y rabkleay irç~ki`y na `iqarnl `c }`clr nac ek}ac na
ilemk`ex` qy`nl# ay naikr# yk c` zrqao` na fkzù~ayky ay okc`~ar`c# qekc`~ar`c naraif` lqekc`~ar`c kxpqkarn`% &Ac`olr`ikùe zrlzk`+
Ek}ac na ilemk`ex` &+ ;: ;1 ;; ;;%1
]`clray irç~kily na x
z`r` zrqao`y qekc`~ar`cay 0%?6 0%=>1 ?%99 ?%16
&naraif` l kxpqkarn`+ l '0%?6 l '0%=>1 l '?%99 l '?%16
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ ]`clray irç~kily na x z`r` °0%=>1 °0%;= °?%16 °?%60
zrqao`y okc ~ar`cay
Zlr ahagzcl yk qy`gly qe ek}ac na ilemk`ex` na ;1# ay naikr# qe ek}ac na
ykbekmki`eik` na ΰ 7 :%:1# z`r` qe` zrqao` na fkzù~ayky qekc`~ar`c kxpqkarn`# ae~leiay#
o`hl ay~`y ilenkikleay# ac }`clr irç~kil na x ay '0%=>1% Ykgkc rgae~a# yk ya qy ac ek}ac
na ykbekmki`eik` na ΰ 7 :%0:# z`r` qe` zrqao` na fkzù~ayky qekc`~ar`c naraif`#
ae~leiay# ac }`clr irç~kil na x ay na $0%?6%
I`oa el~`r pqë# z`r` c`y zrqao`y na fkzù~ayky# cly }`clray na cly ek}acay na
ykbekmki`eik` gày ilgqeay yle cly na ΰ 7 :%:1 { na ΰ 7 :%:0% Zlr ahagzcl# z`r` qe`
zrqao` okc`~ar`c ile ΰ 7 :%:1# cly }`clray irç~kily na x yle na °0%;=% El loy~`e~a# z`r`
qe` zrqao` qekc`~ar`c kxpqkarn` ile ΰ 7 :%:1# ac }`clr irç~kil na x yarç` na na '0%=>1 {
`yç yqiayk}`gae~a%
Ahagzcly na zrqao`y na fkzù~ayky qy`enl c` nky~rkoqikùe elrg`c
Ahagzcl )9=% Ya y`i` qe` gqay~r` na 9= `eàckyky na ek~r`~ly &EL9'+ z`r` ac nkyaðl na
qe` zc`e~` na ~r`~`gkae~l na `bq`y kenqy~rk`cay% Z`r` ay~l# ya i`ciqc` qe zrlgankl
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 260/846
Nr% Fëi~lr Pqa}anl Qrç`y
1'12
ay~`nçy~kil na V 7 ;? gb,C% Ay~qnkly zra}kly kenki`e qe` nay}k`ikùe ay~àen`r
ilelikn` na ω 7 ; gb,C% Zrlo`r c` fkzù~ayky na pqa ac }`clr ayzar`nl na ±l ay 0::
gb,C% @yqgkr ΰ 7 :%:1 { i`ciqc`r ac }`clr na c` zrlo`okckn`n z%Ylcqikùe8
0% C` fkzù~ayky eqc` ay Fl8± 7 0::%
?% C`y fkzù~ayky c~are`~k}`y yle F08±l ≪ 0::# F?8±l 5 0::# F98±l 3 0::%
9% C`y yqzlykikleay yle pqa c` zloc`ikle`c gqay~ra`n` ay elrg`c# ω ay ilelikn` {# c`
gqay~r` ay `ca`~lrk`%
>% Ile ac ek}ac na ykbekmki`eik` na ΰ 7 :%:1 &ek}ac na ilemk`ex` ;1+# c`y rabkleayirç~ki`y { cly ilamkikae~ay irç~kily yle na °0%;=%
1% C` ay~`nçy~ki` qy`n` ay c` na c` nky~rkoqikùe x # x 7 & V ’ ±l+ , ω, e
=% Yqy~k~q{aenl cly }`clray na V 7 ;?# ±l 7 0::# ω 7 ; { e 7 9= ae c` mùrgqc` na
`rrko` n`8
x 7 &;? ’ 0::+ , ;, 9=
7 ' 1%9
2% @flr` ilgz`r`enl c` xi`ci% 7 ’ 1%9 ile c` x ~`oqc`n` x~`o% Kbq`c ` '0%;=# ya raif`x` c`
fkzù~ayky { ely keicke`gly zlr F98±l 3 0::%
6% Ac }`clr na c` zrlo`okckn`n z ya i`ciqc` oqyi`enl ac }`clr na ’1%9 ae c` ~`oc` na c`
nky~rkoqikùe elrg`c# zarl ilgl el ay~à ~lg`gly ac }`clr na %:::9% @nagày# ilgl c`
zrqao` ay okc ~ar c# ae~leiay# gqc~kzcki`gly %:::9 zlr ?# ay naikr# &?+&%:::9+ 7 %:::=%
Ay~a }`clr ay gqifl gq{ ykbekmki`e~a { n` gqif` a}knaeik` z`r` `zl{`r ac raif`xl na
c` fkzù~ayky%
Ahagzcl )92% Qe` gqay~r` `ca`~lrk` na 9= ileiae~r`ikleay `~glymërki`y na ùvknly na
ek~rùbael &ELv+# ae gb,C# gly~rù qe zrlgankl ay~`nçy~kil l na c` gqay~r` na V 7
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 261/846
Nr% Fëi~lr Pqa}anl Qrç`y
1'16
2>%: gb,C% Yqzlekaenl pqa ω? 7 60%: gb,C# ¼kenki`rç` ay~l pqa qe cçgk~a na
ileiae~r`ikleay na ELv ay~ë `rrko` na 2: gb,C< Qy`r ΰ 7 :%:1%
Ylcqikùe80% C` zrqao` na fkzù~ayky eqc` naoa yar Fl8± ≮ 2:# zlrpqa c` fkzù~ayky `c~are`~k}`#
n`n` zlr ac zrlocag`# ay F08± 5 2:%
?% Zlr cl ~`e~l# c` zrqao` na fkzù~ayky ay qe` zrqao` qekc`~ar`c naraif`%
9% C` rabkùe irç~ki` ay xΰ 7 x:%:1 7 0%=>1
>% C` ay~`nçy~ki` qy`n` ay x 7 & V ’ ±l+ , ω, e
Yqy~k~q{aenl cly }`clray nac zrlgankl V 7 2># na c` nay}k`ikùe ay~àen`r ω 7 ;# e 79= { ±l 7 2: ae c` mqeikùe ay~`nçy~ki` x n`8
x 7 &2> ’ 2:+ , ;, 9= 7 ?%==
1% @c ilgz`r`r ac }`clr na xi`ci% 7 ?%==# ile x~`o% 7 0%=>1# ya raif`x` c` fkzù~ayky eqc` {
ya nkia pqa# F08± 5 2:# ile qe }`clr na z 7 0 ' %;;=0 7 %::9;# na f`oar faifl c`
naikykùe apqk}li`n`% @pqç# eù~aya pqa ac }`clr na z el ya gqc~kzcki` zlr ?# zlrpqa c`
zrqao` ay qekc`~ar`c naraif`% Ilgl rayqc~`nl# c` a}knaeik ` m`}lr na F08 ay gày
mqar~a pqa c` yqbarkn` zlr qe ek}ac na ykbekmki`eik` na :%:1 &zlrpqa %::9; 333 :%:1+%
Ahagzcl )96% Ae qe ay~qnkl na c` kebaekarç` `gokae~`c `~glymërki`# z`r` a}k~`r c`
ile~`gke`ikùe `gokae~`c zrlnqikn` zlr ac ileyqgl aviayk}l a keeaiay`rkl na
b`ylcke`# ae ac nkyaðl na qe gl~lr na ilgoqy~kùe ke~are`# ac naz`r~`gae~l na
kebaekarç` na qe iley~rqi~lr na `q~ly# na ikar~l glnacl# `mkrg` pqa ac raenkgkae~l nac
gkcc`ha na ay~a glnacl na `q~l ay na iq`enl gaely 91 gkcc`y zlr b`cùe% Ac
naz`r~`gae~l na ile~rlc na i`ckn`n yqbkara pqa ac }`clr na c` nay}k`ikùe ay~àen`r ay
na ω 7 > gkcc`y% C` Ae}krlegae~`c Zrl~ai~kle @baei{ na Ay~`nly Qeknly na @gërki`
pqkara zrlo`r ay~` `mkrg`ikùe z`r` }ar yk c` mkbqr` `mkrg`n` naoarç` yar gày `c~` l gày
o`h` pqa 91 gkcc`y zlr b`cùe% Z`r` ay~l# ya y`i` qe` gqay~r` `ca`~lrk` na 1: glnacly
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 262/846
Nr% Fëi~lr Pqa}anl Qrç`y
1'1;
na ay~a ~kzl { ya zrqao`e o`hl ikriqey~`eik`y kbq`cay% Cly rayqc~`nly n`e qe zrlgankl
gqay~r`c na 99%= gkcc`y zlr b`cùe% Zrlo`r ac raic`gl nac naz`r~`gae~l na kebaekarç`
qy`enl qe ek}ac na ykbekmki`eik` na :%:1% I`ciqc`r ac }`clr na z%Ylcqikùe8
0% C` zrqao` ay okc`~ar`c# zlrpqa qe zrlgankl gqay~r`c ykbekmki`e~agae~a# gày `c~l
pqa 91 &ilc` naraif`+ l gaely pqa 91 &ilc` kxpqkarn`+ ay qe` mqar~a a}knaeik` ile~r`
c` fkzù~ayky eqc` na ± 7 91%
?% C`y rabkleay irç~ki`y yle nky~rkoq{aenl ΰ 7 :%:1 kbq`cgae~a ae~ra c`y nly ilc`y z`r`
lo~aear :%:?1 ae i`n` qe`# ile ay~l# cly }`clray irç~kily yle na °0%;=%
9% Qy`gly c` mqeikùe ay~`nçy~ki` { yqy~k~qkgly cly }`clray { n`8
x 7 & V ’ ±l+ , ω, e
x 7 &99%= ’ 91+ , > 1: 7 '?%>2
>% Ae ileicqykùe# naoknl ` pqa ac }`clr na '?%>2 ya ke~rlnqia ae ac av~ragl kxpqkarnl
na c` nky~rkoqikùe elrg`c# ya raif`x` c` fkzù~ayky% Ac }`clr na z ay na %::=6# zarl
ilgl yle nly ilc`y# ae~leiay# ay~a }`clr ya gqc~kzcki` zlr ? { n` %:09=% ]ar mkbqr` na
`o`hl%
V 7 99%=x 7 '?%>2
Mkbqr` 1%0>% Bràmki` gly~r`enl cly }`clray irç~kily z`r` ac zrlocag` na `rrko`%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 263/846
Nr% Fëi~lr Pqa}anl Qrç`y
1'=:
&Ac`olr`ikùe zrlzk`+
Ahagzcl )9;% Ac ay~àen`r pqçgkil z`r` ac `bq` zl~`oca# ae iq`e~l ` yùcknly nkyqac~ly
~l~`cay &YN^+ ay na 1:: gb,C% Z`r` zrlo`r ay~` `ya}ar`ikùe ya y`i` qe` gqay~r``ca`~lrk` na 0>> i`yly { n` qe zrlgankl `rk~gë~kil ay~`nçy~kil na 1:9 gb,C# ile qe`
nay}k`ikùe ay~àen`r na 01 gb,C% Zrlo`r c` fkzù~ayky na pqa ac ay~àen`r pqçgkil# z`r`
ac `bq` zl~`oca ay na el gày na 1:: gb,C% Z`r` ay~l# qy`r ΰ 7 :%:1 { i`ciqc`r ac }`clr
na z%
Ylcqikùe8
0% C` zrqao` na c` fkzù~ayky eqc` ay Fl8±
≮1::% Ay~l pqkara naikr pqa# c` zrqao` ay
qekc`~ar`c naraif`%
?% C` fkzù~ayky c~are`~k}` ay F08± 5 1::%
9% C` rabkùe irç~ki` ay qekc`~ar`c naraif`# ay~l ay# xΰ 7 x:%:1 7 0%=>1%
>% @flr`# yqy~k~q{aenl cly }`clray ilrrayzlenkae~ay na V 7 1:9# ω 7 y 7 01# e 7 0>> {
±l 7 1::# ae c` mqeikùe na x ely n`8
xi`ci% 7 &1:9 ’ 1::+ , 01, 0>> 7 ?%>
1% Ae ileicqykùe# naoknl ` pqa ac }`clr na c` ay~`nçy~ki` ya ke~rlnqia ae ac av~ragl
naraifl na c` nky~rkoqikùe# ya raif`x` c` fkzù~ayky eqc` { ya keicke` zlr c` fkzù~ayky
`c~are`~k}`%
=% Ac }`clr na c` zrlo`okckn`n z ya i`ciqc` oqyi`enl ac }`clr na ?%> ae c` ~`oc` na c`
nky~rkoqikùe elrg`c { n` %;;06# zarl ilgl pqaragly øeki`gae~a ac }`clr na z# ca
ray~`gly 0 { ely n` z 7 0 ' %;;06 7 %::6?% Ay~a }`clr ay gq{ ykbekmki`e~a%
Ahagzcl )>:% Ya i`ciqc` ac zrlgankl gqay~r`c na 1 zzg na i`ngkl &In+# z`r` gankr c` i`ckn`n nac `kra# na ikar~` rabkùe kenqy~rk`c% Ay~l ya f`ia y`i`enl qe` gqay~r` na 9=
loyar}`ikleay na In `~glymërkil% F`iar cl ykbqkae~a8
&`+ Qe` zrqao` na fkzù~ayky ile ±l 7 >%61 zzg# ile ΰ 7 :%:1 { ΰ 7 :%:0
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 264/846
Nr% Fëi~lr Pqa}anl Qrç`y
1'=0
&o+ @nagày# i`ciqc`r ac }`clr na c` zrlo`okckn`n z a ke~arzra~`rcl `ilrnagae~a% L~rlyç#
aeile~r`r cly ke~ar}`cly na ilemk`ex` ile cly gkygly }`clray na ΰ na `rrko`# z`r` ac
zrlgankl zloc`ikle`c ±%@yqgkr pqa c` nay}k`ikùe ay~àen`r zloc`ikle`c ay na :%9%
Ylcqikùe8
&`+ C` zrqao` na fkzù~ayky eqc` ay Fl8± 7 >%61% C`y zrqao`y `c~are`~k}`y yle na F08± ≪
>%61# F?8± 3 >%61 { F98± 5 >%61%
C`y rabkleay irç~ki`y ile ΰ 7 :%:1 { ΰ 7 :%:0# z`r` qe` zrqao` na fkzù~ayky okc`~ar`c
yle# rayzai~k}`gae~a# °0%;= { °?%16%
Qy`enl c` ay~`nçy~ki` x 7 & V ’ ±l+ , ω, e ile V 7 1 zzg# ±l 7 >%61# ω 7 :%9 { e 7 9=
{ yqy~k~q{aenl ~lnly ay~ly }`clray ae c` ay~`nçy~ki` na `rrko` ely n`8
x 7 &1%: ’ >%61+ , :%9, 9= 7 9%:
Ae ileicqykùe# naoknl ` pqa ac }`clr na xi`ci% 7 9%:: ay g`{lr pqa ac }`clr irç~kil na
0%;= ya raif`x` Fl8 { ely keicke`gly zlr c` fkzù~ayky `c~are`~k}` na F98± 5 9%61% Ily`
ykgkc`r liqrra ile ac ek}ac na ykbekmki`eik` na ΰ 7 :%:0# zlrpqa ac }`clr na 9%:: ay
g`{lr pqa ac }`clr irç~kil na ?%16%
Zlr l~r` z`r~a# ile rayzai~l ` c` ay~kg`ikùe nac ke~ar}`cl na ilemk`ex` nac ;1#
pqa ilrrayzlena ` qe ek}ac na ykbekmki`eik` na ΰ 7 :%:1# cly }`clray irç~kily yle na
°0%;=% C` ay~kg`ikùe zqe~q`c na ± ay V 7 1%:% Z`r` i`ciqc`r ac ke~ar}`cl na ilemk`ex`
na ;1# ya yqy~k~q{ae cly }`clray ae aiq`ikùe &1'?>+ na o`hl z`r` n`r8
V ’ xΰ,? ω, e 3 ± 3 V $ xω,? ω, e
1%: ’ &0%;=+&:%9+, 9= + 3 ± 3 1%: $ &0%;=+&:%9, 9= +
Ac iq`c ya ykgzckmki` 8
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 265/846
Nr% Fëi~lr Pqa}anl Qrç`y
1'=?
>%;:? 3 ± 3 1%:;6
Zlr l~r` z`r~a# ac }`clr ilrrayzlenkae~a ` qe ke~ar}`cl na ilemk`ex` nac ;;# ay
naikr# ile qe ek}ac na ykbekmki`eik` na ΰ 7 :%:0# ae ay~a i`yl cly }`clray irç~kily ylena °?%121% C` ay~kg`ikùe zqe~q`c na ± ay V 7 1%:% Na `pqç pqa ac ke~ar}`cl na
ilemk`ex` nac ;;# ay8
1%: ’ &?%16+&:%9, 9= + 3 ± 3 1%: $ &?%16+&:%9, 9= +
Ac iq`c ya ykgzckmki` 8
>%620 3 ± 3 1%0?;
Ahagzcl )>0% Ae qe ay~qnkl na fkbkaea kenqy~rk`c { yabqrkn`n# c`y ~agzar`~qr`y naciqarzl na qe brqzl na 0:: ~r`o`h`nlray kenqy~rk`cay# pqa c`olr`e qe mrkblrçmkil# ya
`e`ckx`rle% C` ~agzar`~qr` zrlgankl mqa na ;6%? lM ile qe` nay}k`ikùe ay~àen`r na
:%=? lM% Aeile~r`r ac gahlr zqe~l ay~kg`nlr nac z`ràga~rl zloc`ikle`c ± na ~ln`y c`y
~agzar`~qr`y nac iqarzl% Z`r` qe ek}ac na ilemk`ex` na ;1# aeile~r`r# `goly# ac
g`rbae na arrlr A { ac ke~ar}`cl na ilemk`ex` z`r` ±%
Ylcqikùe8
Qy`enl c` mqeikùe8
V ’ xΰ,?&ω e + 3 ± 3 A $ xΰ,?&ω, e + &1'9?+
Nlena8 A 7 g`rbae na arrlr 7 xΰ,?&ω,∘ e+% @flr`# yqy~k~q{aenl cly }`clray `zrlzk`nly#
ile xΰ,? 7 0%;=# ω 7 y 7 :%=? &zlrpqa e 5 9:+# V 7 ;6%? { e 7 0::# lo~aeagly8
;6%? ’ 0%;=&:%=?+,& 0:: + 3 ± 3 ;6%? $ 0%;=&:%=?+, 0:: +
;6%? ’ :%0? 3 ± 3 ;6%? $ :%0?
;6%:6 3 ± 3 ;6%9?
Ac }`clr nac g`rbae na arrlr ay A 7 0%;=&:%=?+,& 0:: + 7 :%0?01%
Ay~a ke~ar}`cl ;6%:6 3 ± 3 ;6%9? nkia pqa yk mqër`gly ` yacaiikle`r gqif`y gqay~r`y
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 266/846
Nr% Fëi~lr Pqa}anl Qrç`y
1'=9
na qe ~`g`ðl na 0:: { iley~rq{ër`gly qe ke~ar}`cl na ilemk`ex`# ac ;1 na ay~`y
gqay~r`y ile~aenrç`e ac zrlgankl zloc`ikle`c ±% @pqç# eù~aya pqa ac ke~ar}`cl na
ilemk`ex` el ile~kaea ac }`clr na ;6%= lM# c` iq`c ay c` ~agzar`~qr` elrg`c nac iqarzl%Ahagzcl )>?% Ya y`i` qe` gqay~r` `ca`~lrk` na qe` zloc`ikùe elrg`c% Cly }`clray na
c`y loyar}`ikleay yle8 ??# ?># ??# ?1# 9:# ?6# ?;# ?6# ?># ?9# ?1# ?2# ?=# ?9# ?># ?0# ??#
?0# ?1# ?0# ?9# ?># ?0# ?:# ?0# ?:# ??# ?6# ?2% F`iar cly ykbqkae~ay iàciqcly qy`enl ac
zrlbr`g` Gkek~`o%
&`+ I`ciqc`r c` ay~`nçy~ki` nayirkz~k}` { na~argke`r ac ;1 nac ke~ar}`cl na ilemk`ex`
z`r` ac zrlgankl zloc ikle`c ±%
&o+ Na~argke`r ac ;1 nac ke~ar}`cl na ilemk`ex` z`r` c` nay}k`ikùe ay~àen`r ω# { c`
gank`e`%
&i+ F`iar qe` zrqao` na elrg`ckn`n qy`enl c` ay~`nçy~ki` na Dlcglblrl}'Ygkrel}%
Ylcqikùe8
Z`r` ay~kg`r cly keikyly &`+ { &o+ qy`r ac zrlbr`g` Gkek~`o na c` ykbqkae~a g`ear`8
Y~`~ 5 O`yki y~`~ky~kiy 5 Br`zfki`c Yqgg`r{
Ay~l baear` c` Mkbqr` 1%01 na `o`hl# c` iq`c keicq{a fky~lbr`g` ile iqr}` elrg`c
ylorazqay~`# cly ke~ar}`cly na ilemk`ex` z`r` ac zrlgankl zloc`ikle`c# c` gank`e`# c`
nay}k`ikùe ay~àen`r# c` ay~`nçy~ki` nayirkz~k}`# c` zrqao` na @enaryle'N`rckeb# cly
iq`r~kcay# a~i%%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 267/846
Nr% Fëi~lr Pqa}anl Qrç`y
1'=>
9:?6?=?>???:
Gank`e
Ga`e
?1%:?>%1?>%:?9%1?9%:??%1??%:
@ e na ryl e' N `r cke b E lrg ` ck~ { ^ ay ~
] ` rk` ei a 6% ?0 >
Y d a e a yy : % >6 6 60 9
D q r~ ly ky ': % 6? ? 90 =
E ?;
G ke kg q g ? :%: ::
@ 'Y p q` ra n
0 y~ P q ` r~ kca ? 0 %1 : :
G a n k` e ?> %:::
9 rn P q ` r~ kca ? = %1 : :
G vkg q g 9 :%: ::
;1! I lem knaeia Ke~ar} ` c mlr G a`e
?? % ;0:
:%=>
?1 % :;:
;1! I lem knaeia Ke~ar} ` c mlr G ank`e
?? %::: ?1 %:::
;1! I lemkna eia K e~ar} `c mlr Y ~N a}
? %?2> 9 %62=
Z '] cqa : %:61
G a ` e ?> %:::
Y ~N a } ? %6==
; 1 ! I l e m kn a e i a K e ~ a r } ` c y
Yqgg`r{ mlr ]`clray na c`y loyar}`ikleay
Mkbqr` 1%01% Mkbqr` gly~r`enl ac fky~lbr`g` na cly n`~ly ile iqr}` elrg`c
ylorazqay~`# cly ke~ar}`cly na ilemk`ex` z`r` ac zrlgankl { c` gank`e` { c` ay~`nçy~ki`
nayirkz~k}`%
Z`r` ac keikyl &i+# ay naikr# z`r` c` zrqao` na elrg`ckn`n na cly n`~ly ay~l ya f`ia
qy`enl c` ay~`nçy~ki` na Dlcglblrl}'Ygkrel}# nac zrlbr`g` Gkek~`o% Ykaenl `yç# ya
zrliana na c ykbqkae~a g`ear`8O`yki Y~`~ky~kiy ↟ Elrg`ck~{ ^ay~
Ae c` }ae~`ekcc` nac raiq`nrl na Elrg`ck~{ ^ay~ ke~rlnqikr c`y }`rk`ocay { zqe~a`r
Dlcglblrl}'Ygkrel}% Ay~l baear` c` mkbqr` na `o`hl%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 268/846
Nr% Fëi~lr Pqa}anl Qrç`y
1'=1
I0
Z a r i a e ~
9?9:?6?=?>???:060=
;;
;1
;:
6:
2:
=:
1:
>:
9:
?:
0:
1
0
Ga`e
5:%01:
?>
Y ~Na} ?% 6==
E ?;
DY :%:61
Z']`cqa
Mkbqr` gly~r`enl c` br`mki` na cly }`clray%Elrg`c
Mkbqr` 1%0=% Bràmki` gly~r`enl c` zrqao` na elrg`ckn`n qy`enl c` mqeikùe na
Dlcglblrl}'Ygkrel}%
Ilgl ya }a ae c` Mkbqr` 1%0=# c`y zrlo`okckn`nay &l zlriae~`hay ae ay~a i`yl+ ya
br`mki`e ae mqeikùe na cly }`clray ay~kzqc`nly zlr ac zrlocag`% Cqabl ac zrlbr`g`
~r`x` qe` cçea` na cly iq`nr`nly gçekgly# ile ac loha~l na }arkmki`r yk cly zqe~ly ay~àenae~rl na c`y o`en`y na ilemk`ex`% Yke ago`rbl# ay na el~`rya pqa# ae ilgz`r`ikùe
ile c` mqeikùe na @enaryle'N`rckeb l na Ckcckamlry# c` zrqao` na Dlcglblrl} ay
gaely zraiky` pqa c` mqeikùe na @enaryle'N`rckeb% Yke ago`rbl# c` mqeikùe na
Dlcglblrl}'Ygkrel} ya ykbqa qy`enl# ~r`nkikle`cgae~a# zlr gqifly ke}ay~kb`nlray
ay~`nçy~kily%
Ahagzcl )>9% Qe m`orki`e~a na yky~ag`y na `yzarykùe ile~r` keiaenkly# pqa ya key~`c`enae~rl na i`y`y { ankmkikly# `rbqgae~` pqa ac zrlgankl zloc`ikle`c na ~agzar`~qr` na
yqy yky~ag`y na `yzarykùe ile~r` keiaenkly ay na 1>%> lI% Z`r` ay~l ya y`i` qe`
gqay~r` `ca`~lrk` na 0= qekn`nay# c`y iq`cay# `c zrlo`rya n`e qe zrlgankl ay~`nçy~kil
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 269/846
Nr% Fëi~lr Pqa}anl Qrç`y
1'==
na 11%: lI# ile qe` nay}k`ikùe ay~àen`r na 0%: lI% Yk ya y`oa pqa c` nky~rkoqikùe na
cly ~kagzly na `i~k}`ikùe na cly yky~ag`y na `yzarykùe# ile~r` cly keiaenkly# na ay~a
m`orki`e~a# ay elrg`c# ¼ya ramq~`rç` ac `rbqgae~l nac m`orki`e~a na pqa ac }arn`narl zrlgankl ay ac pqa ya gaeikle` `rrko`< @yqgkr qe ek}ac na ykbekmki`eik` na :%:1%
0% @pqç c` zrqao` na fkzù~ayky ay Fl8± 7 1>%> ile~r` c` zrqao` na fkzù~ayky `c~are`~k}`
na F08± ≪ 1>%>%
?% Naoknl ` pqa c` zrqao` na fkzù~ayky ccae` c` ilenkikùe na kbq`cn`n# c` zrqao` ay
okc ~ar c# ay naikr# x ≡ x%:?1 { x ≮ x%:?1# ay~l ay# x ≡ 0%;= l x ≮ '0%;=%
9% Qy`gly c` nky~rkoqikùe na x# `qepqa ac ~`g`ðl na c` gqay~r` el ya` na e 5 9:
i`yly% Ay~l ay `yç# zlrpqa y`oagly na `e~ag`el pqa c` zloc`ikùe gqay~ra`n` ay
elrg`c% ^`gokëe ya zqnkar` qy`r c` nky~rkoqikùe na ~ na ay~qnk`e~a# zarl ae ay~a i`yl
ay gahlr qy`r c` nky~rkoqikùe x zlrpqa ay g`y zraiky`%
>% Ykaenl yç# ac }`clr na c` zrqao` ay~`nçy~ki` ay8
x 7 &11%: ’ 1>%>+ , 0%:, 0=
7 ?%>
1% Na `iqarnl `c keikyl &>+ ac zrlgankl gqay~r`c loyar}`nl ya aeiqae~r` ` ?%>
nay}k`ikleay ay~àen`r `rrko` na cl pqa ya fqokar` ayzar`nl# yk Fl8 mqar` }arn`nar`%
=% Ae ileicqykùe# naoknl ` pqa ac }`clr i`ciqc`nl na x i`a ae c` rabkùe irç~ki` naraif`#
ya raif`x` c` zrqao` na fkzù~ayky ~r`nkikle`c%
2% @flr`# z`r` f`iar c` zrqao` na fkzù~ayky el ~r`nkikle`c# ay naikr# i`ciqc`enl ac }`clr
na z# oqyi`gly ae c` ~`oc` na c` nky~rkoqikùe elrg`c ac }`clr na ?%> { }agly pqa ac
}`clr na c` zrlo`okckn`n z ay z 7 ?&0 ’ :%;;06+ 7 :%:0=>%6% Ac }`clr na z 7 :%:0=> ile~r`nkia c` `mkrg`ikùe nac m`orki`e~a na pqa ac }arn`narl
zrlgankl na yqy zrlnqi~ly ile~r` keiaenkly ay na 1>%> lI%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 270/846
Nr% Fëi~lr Pqa}anl Qrç`y
1'=2
Zrqao`y na fkzù~ayky z`r` c`y nkmaraeik`y na nly zrlgankly zloc`ikle`cay &±0 ’
±?+# z`r` gqay~r`y br`enay &e 9:+ qy`enl c` nky~rkoqikùe elrg`c# ile
}`rk`ex`y ilelikn`y a kbq`cay &ω?0 7 ω??+% @pqç ya `yqga pqa c`y nly gqay~r`y ylekenazaenkae~ay
F`y~` `flr`# fagly nkyiq~knl zrqao`y na fkzù~ayky na qe` ylc` gqay~r` `ca`~lrk`# ay
naikr# z`r` qe ylcl zrlgankl% @flr`# }`gly ` nkyiq~kr zrqao`y na fkzù~ayky nlena ya
ke}lcqir`e ? gqay~r`y zrl}aekae~ay na nly zloc`ikleay% Na ay~` g`ear`# ae gqifly
zrlocag y zrài~kily ay~`gly ke~aray`nly ae ilgz`r`r nly zloc`ikleay ile rac`ikùe `
`cbqe` i`r`i~arçy~ki` iq`e~k~`~k}`% Zlr ahagzcl# c` ilgz`r`ikùe na nly gë~lnly z`r`
gankr ac gkygl zrliayl iq`ck~`~k}l l iq`e~k~`~k}l%
Ae kebaekarç` `gokae~`c# zlr ahagzcl# ya zqanae ilgz`r`r nly gë~lnly z`r`
gankr c`y ileiae~r`ikleay na `ryëekil ae gqay~r`y na `bq`% L~r` `zcki`ikùe yarç`
gankr nly gë~lnly z`r` ac i`ngkl ae gqay~r`y na `bq`# { `yç yqiayk}`gae~a%
Ae ~ërgkely ay~`nçy~kily# yk ya ~kaeae nly zloc`ikleay ile gank`y ±0 { ±? { ile
}`rk`ex`y ω0 { ω? rayzai~k}`gae~a# ac ay~kg`nlr zqe~q`c na c` nkmaraeik` na cly
zrlgankly &±0 ' ±?+ cl n` ac ay~`nçy~kil & V 0 ’ V ?+% Zlr cl ~`e~l# z`r` lo~aear qe`
ay~kg`ikùe zqe~q`c na &±0 ’ ±?+ ya yacaiikle`e nly gqay~r`y `ca`~lrk`y kenazaenkae~ay#
qe` na i`n` zloc`ikùe na ~`g`ðly e0 { e? { ya i`ciqc` c` nkmaraeik`# V 0 ’ V ?% Na ay~`
g`ear`# nahagly pqa V 0 { V ? ya`e cly zrlgankly na nly gqay~r`y br`enay na
~`g`ðly e0 { e? y`i`nly na nly zloc`ikleay pqa ~kaeae zrlgankly na ±0 { ±? {
nay}k`ikleay ay~àen`r na ω0 { ω?# rayzai~k}`gae~a% Ae~leiay# yk zleagly ±0 7 ±?
ay~`gly nkikaenl pqa el f`{ nkmaraeik`y ae~ra `goly zrlgankly zloc`ikle`cay# pqaay cl gkygl pqa naikr# pqa nly gqay~r`y ya y`i`rle na zloc`ikleay pqa ~kaeae ac
gkygl zrlgankl# ±% C` ay~`nçy~ki` pqa ya qy` z`r` ay~kg`r c`y nkmaraeik`y ae~ra nly
zrlgankly ay8
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 271/846
Nr% Fëi~lr Pqa}anl Qrç`y
1'=6
x 7 T& V 0 ' V ?+ ' &±0 ' ±?+W ,ee ?
?
?
0
?
0 ω ω $ &1'99+
Nlena8
V 0# V ? 7 zrlgankly na gqay~r`y qel { nly# rayzai~k}`gae~a
ω?0# ω?
? 7 }`rk`ex`y na gqay~r`y qel { nly rayzai~k}`gae~a
& V 0 ’ V ?+ 7 ay~kg`nlr zqe~q`c na &±0 ’ ±?+
e0 { e? 7 ~`g`ðly na gqay~r`y qel { nly# rayzai~k}`gae~a
x 7 }`rk`oca elrg`c ay~àen`r
Yk ya `yqga pqa ω0 7 ω? 7 ω# c` ay~`nçy~ki` na `rrko` ya ranqia `8
x 7 & V 0 ’ V ?+ ’ &±0 ’ ±?+ , ωee ?0
00$ &1'9>+
C`y mqeikleay z`r` c`y zrqao`y na fkzù~ayky eqc`y { c`y `c~are`~k}`y# yle8
Fl8±0 ' ±? 7 : ay naikr# pqa ±0 7 ±?
F08± ' ±? ≪ : { F?8±0 ' ±? 5 ξ { F98±0 ' ±? 3 ξ
@pqç# `qepqa ξ zqana yar iq`cpqkar }`clr iley~`e~a# gqif`y }aiay ac }`clr na ξ ay na
: { ya zrqao` c` fkzù~ayky eqc` na el /nkmaraeik`/# ay naikr Fl8±0 7 ±?%
Ahagzcl )>>% Z`r` gankr c` i`ckn`n nac `kra na ikar~` xle` kenqy~rk`c# ile rac`ikùe `
cly ùvknly na `xqmra# ya y`i`rle nly gqay~r`y na ~`g`ðly 1: { 21# rayzai~k}`gae~a%
Cly zrlgankly mqarle na 2= gb,C { na 6? gb,C# rayzai~k}`gae~a% @yqgkr pqa c`y
}`rk`ex`y na ay~`y zloc`ikleay yle ilelikn`y a kbq`cay ` 0=% @yqgkr qe ek}ac na
ykbekmki`eik` na ΰ 7 %:1% Qy`enl ac }`clr na z# zrlo`r pqa el f`{ namaraeik`y ae~ra c`y
nly zloc`ikleay gqay~ra`n`y# pqa ay cl gkygl pqa ±0 7 ±?# ay~l ay# ±0 ’ ±? 7 :Ylcqikùe8
0% Qy`gly c` mqeikùe na x# zlrpqa c`y gqay~r`y yle br`enay%
?% C`y fkzù~ayky eqc`y { `c~are`~k}`y# yle# rayzai~k}`gae~a8
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 272/846
Nr% Fëi~lr Pqa}anl Qrç`y
1'=;
Fl8±0 ’ ±? 7 : { F08±0 ’ ±? ≪ :
9% Cly }`clray irç~kily ilrrayzlenkae~ay ` c`y rabkleay irç~ki`y# ile ΰ 7 %:1 yle na
°0%;=%>% Cly }`clray pqa ya yqoy~k~q{ae ae c` mùrgqc` &1'9>+ yle8
V 0 7 2=# V ? 7 6?# e0 7 1:# e? 7 21# ω0 7 ω? 7 0=% Yqoy~k~q{aenl ay~ly }`clray ae c`
mùrgqc` &1'9>+ ely n`e8
x 7 T& V 0 ’ V ?+ ’ &±0 ’ ±?+W , ω ee ?0
00$
7 T&2= ’ 6?+ ’ :W , 0=W 2101:0$ 7 ?%:1
1% Ae ileicqykùe# naoknl ` pqa ac }`clr i`ciqc`nl na x 7 ?%:1 ay g`{lr pqa c` rabkùe
irç~ki` naraif` na 0%;=# ya raif`x` c` fkzù~ayky { ya ileicq{a pqa cly zrlgankly el
yle kbq`cay%
=% Ac }`clr na c` zrlo`okckn`n z ya i`ciqc` oqyi`enl ac }`clr na x 7 ?%:1 ae c` ~`oc` na
c` nky~rkoqikùe elrg`c { n` :%;2;6% Zlr cl ~`e~l# ac }`clr na c` zrlo`okckn`n ay z 7 ?&0
' %;2;6+ 7 :%:>%Ahagzcl )>1% Qe` ilgz`ðç` m`rg`iëq~ki` pqkara zrlo`r qe` nrlb` z`r` c` mkorlyky
zqcgle`r# c` iq`c ay gq{ ilgøe ae~ra cly ~r`o h`nlray kenqy~rk cay% Z`r` ay~l ya
zrqao`e nly brqzly# ay naikr# ac na /ile~rlc/ &pqa el qy`e c` nrlb`+ { ac brqzl na
/~r`~`gkae~l/ &pqa yk qy`e c` nrlb`+% Ya ~lg` qe` gqay~r` na 1: ~r`o`h`nlray ` cly
iq`cay ya cay n` c` nrlb` { l~rl brqzl gày na 0:: zaryle`y# `c iq`c el ya cay n` c`
nrlb`% C` zraykùe `r~ark`c ya ~lg` z`r` i`n` yqha~l% C` ilgz`ðç` na nrlb`y `mkrg` pqac` nrlb` el i`qy` ekebøe amai~l yaiqen`rkl# z`r` ac brqzl na ~r`~`gkae~l% Nkifl ae
l~r`y z`c`or`y# ay~l nkia pqa ac zrlgankl ±0 nac brqzl ile~rlc { ac zrlgankl ±? nac
brqzl na ~r`~`gkae~l yle kbq`cay% Zrlo`r ac raic`gl na c` ilgz`ðç` na pqa el f`{
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 273/846
Nr% Fëi~lr Pqa}anl Qrç`y
1'2:
amai~ly yaiqen`rkly ae~ra ac brqzl pqa ~lg` c` nrlb` { ac pqa el c` ~lg`% &El~`8 Ae
ay~a zrlocag`# na `iqarnl `c `q~lr na ay~a ckorl na ay~`nçy~ki`# ac qyl na ganki`gae~ly
`r~kmkik`cay ykagzra i`qy`rà amai~ly yaiqen`rkly% Ay~l ya naoa ` pqë# ac iqarzl ay qe`ayaeik` e`~qr`c# pqa el zqana `iaz~`r `r~kmkik`ckygly# zlr yar `e~`bùekily `c nkyaðl
e`~qr`c nac lrb`ekygl fqg`el% @nagày# ac ganki`gae~l `r~kmkik`c `~`i` qe amai~l
ra`i~k}l &ac yçe~lg` na c` aemargan`n+# gày el yq lrkbae i`qy`c &}kn` `e~ke`~qr`c+% Ae
}arn`n# ac amai~l yaiqen`rkl ay qe` ra`iikùe lrbàeki` e`~qr`c# ae rayzqay~` ` c` `iikùe
keilgz`~koca nac `r~kmkik`ckygl gënkil% Na iq`cpqkar g`ear`# z`r` ay~a zrlocag` qy`r
ac ek}ac na ykbekmki`eik` naΰ
7 %:1% Cly iàciqcly na c`y }`rk`ocay { yqy }`clray ya n`e
ae c` ~`oc` na `o`hl%
^@OC@ 1%=% ^`oc` gly~r`enl cly n`~ly nac Ahagzcl )>1% \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ Brqzl na ~r`~`gkae~l Brqzl na ile~rlc
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ e0 7 1: e? 7 0:: V 0 7 ?:9%> V ? 7 06;%>ω0 7 9;%> ω? 7 9;%:
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
Eù~aya pqa ~`gokëe ya zqnkar` qy`r y ae cqb r na ω# naoknl ` pqa# ac }`clr na c`
gqay~r` e ay e 555 9:%
Ylcqikùe8
0% Ac raic`gl na c` ilgz`ðç` ya avzray` ilgl ±0 7 ±?% Ay~l pqkara naikr pqa# ae
ekebqel na cly nly brqzly f`{ qe amai~l yaiqen`rkl na `c~` zraykùe `r~ark`c%
?% Yk ac raic`gl lrkbke`c ay m`cyl# ae~leiay ±0 ≪ ±?
9% C` zrqao` na fkzù~ayky eqc` ile~kaea c` ilenkikùe na kbq`cn`n na g`ear` pqa# c`y
zrqao`y na fkzù~ayky eqc`y { c`y `c~are`~k}`y yle8
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 274/846
Nr% Fëi~lr Pqa}anl Qrç`y
1'20
Fl8&±0 ' ±?+ 7 :# F08&±0 ' ±+ ≪ :?# F?8&±0 ' ±?+ 5 :# F98&±0 ’ ±?+ 3 :
>% Ac ek}ac na ykbekmki`eik` ay na ΰ 7 %:1
1% Ac zrlocag` y`~kym`ia c`y yqzlykikleay na elrg`ckn`n%=% Qy`gly c` ay~`nçy~ki` na x &1'99+ pqa ya n` `o`hl { ya yqy~k~q{ae cly }`clray
ilrrayzlenkae~ay8
x 7 T& V 0 ' V ?+ ' &±0 ' ±?+W ,ee ?
?
?
0
?
0 ω ω $
7 T&?:9%> ' 06;%>+ ' &:+W ,0::
:?%9;
1:
>?%9;$ 7 ?%:=
2% C`y rabkleay irç~ki`y yle na xΰ,? 7 :%:1,? 7 °0%;=
6% Naoknl ` pqa c` ay~`nçy~ki` x i`a ae c` rabkùe irç~ki` naraif` ya raif`x` c` Fl8 ±0 7
±? { ya nkia pqa cly zrlgankly yle naykbq`cay%
;% Ya ileicq{a pqa yk f`{ amai~ly yaiqen`rkly { c` nrlb` yk i`qy` `c~` zraykùe `r~ark`c%
Zlr cl ~`e~l# ya raif`x` ac raic`gl na pqa# `goly brqzly ~aeb`e ac gkygl zrlgankl%
Na ay~` g`ear`# ya ileicq{a pqa F?8 ±0 ' ±? 5 :%
0:% @flr` z`r` i`ciqc`r ac }`clr na c` zrlo`okckn`n z# ya oqyi` ac }`clr na x 7 ?%:= ae
c` ~`oc` na c` nky~rkoqikùe elrg`c { ac }`clr na c` zrlo`okckn`n ilrrayzlenkae~a ay na
:%;6:9% Zlr cl ~`e~l# ac }`clr na z ay na8 z 7 :%1::: ' :%>6:9 7 :%:0;2% Yke ago`rbl#
naoknl ` pqa c` zrqao` na fkzù~ayky ay okc`~ar`c# ac }`clr na :%:0;2 ya naoa na
gqc~kzcki`r zlr ?%
C` Mkbqr` 1%02 na `o`hl gqay~r` ~ln` c` kemlrg`ikùe rapqarkn` zlr ay~a
zrlocag %
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 275/846
Nr% Fëi~lr Pqa}anl Qrç`y
1'2?
Mkbqr` 1%02% Mkbqr` gly~r`enl c` nky~rkoqikùe na c`y nkmaraeik`y na cly zrlgankly na
cly nly brqzly na ile~rlc { na ~r`~`gkae~l% &Ac`olr`ikùe zrlzk`+%
Ahagzcl )>=% Ya pqkarae zrlo`r nly `e`ckx`nlray na IL na nkmarae~ay g`ri`y# z`r`
}ar yk cly nly n`e cly gkygly rayqc~`nly ae c`y gankikleay na IL% Cc`gagly `c
zrkgar `e`ckx`nlr @ { `c yabqenl O% Zrlo`r pqa cly rayqc~ nly na c`y nly gankikleay
na IL zrl}aekae~ay na cly nly `e`ckx`nlray yle kbq`cay% @yqgkr ΰ 7 :%:1% I`ciqc`r
nac }`clr na c` zrlo`okckn`n z% Cly n`~ly ya n`e `o`hl%
^@OC@ 1%2% ^`oc` gly~r`enl cly n`~ly na ay~a zrlocag`%
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
Gqay~ra`nlr na IL &@+ Gqay~ra`nlr na IL &O+
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
e0 7 1: e? 7 0::
V 0 7 >%19 dby% V ? 7 >%:0 dby%
ω0 7 :%6: ω? 7 :%6:
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
Ylcqikùe8
Cly nly zrlgankly yle kenazaenkae~ay { ω0 { ω? yle ileliknly# zlr cl ~`e~l# qy`gly
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 276/846
Nr% Fëi~lr Pqa}anl Qrç`y
1'29
c` nky~rkoqikùe elrg`c% Qy`gly ac ek}ac na ykbekmki`eik` na ΰ 7 :%:1% C` zrqao`
ke}lcqir` nly ilc`y%
0% C`y zrqao`y na fkzù~ayky yle8Fl8±0 7 ±? &l ±0 ' ±? 7 :+
F08±0 ≪ ±? &l ±0 ' ±? ≪ :+
?% C`y rabkleay irç~ki`y yle na °0%;=
9% Qe` }ax pqa ya yqy~k~q{ae ~lnly cly }`clray ae c` aiq`ikùe na x &1'99+# ac rayqc~`nl
ay na x 7 >%:=%
>% Naoknl ` pqa >%:= i`a nae~rl na c` rabkùe irç~ki` naraif`# ya raif`x` c` fkzù~ayky
eqc` { ya ileicq{a pqa cly zrlgankly zloc`ikle`cay ilrrayzlenkae~ay ` `goly
gqay~ra`nlray na IL el yle kbq`cay% ^`c z`raia pqa ac gqay~ra`nlr @ n` rayqc~`nly
na gankikleay na IL# ile qe` zrlo`okckn`n gqifl gày ykbekmki`e~a pqa ac
gqay~ra`nlr O%
1% Z`r` i`ciqc`r ac }`clr na z oqyi`gly x 7 >%:= ae c` ~`oc` na c` nky~rkoqikùe elrg`c {
}agly pqa ac }`clr gày iari`el ay %;;;2 l ya` 0 ' %;;;2 7 %:::9# cl iq`c nkia pqa z
33 %:::9
Ke~ar}`cly na ilemk`ex` z`r` c`y nkmaraeik`y na nly zrlgankly zloc`ikle`cay# &±0
' ±?+# iq`enl c` }`rk`ex y ω?0 { ω?
? ya ileliae%
Yk V 0 { V ? yle cly zrlgankly na nly gqay~r`y ay~`nçy~ki`y kenazaenkae~ay na
~`g`ðly e0 { e? na zloc`ikleay pqa ~kaeae }`rk`ex`y ilelikn`y na ω?0 { ω?
?# ae~leiay#
ac ke~ar}`cl na ilemk`ex` ya n` ilgl8
& V 0 ' V ?+ ' xΰ,?ee ?
??
0
?0 ω ω $ 3 &±0 ' ±?+ 3 & V 0 ' V ?+ $ xΰ,?
ee ?
??
0
?0 ω ω $ &1'91+
Nlena8
xΰ,? ay ac }`clr na x ile ΰ 7 :%:1 { :%:0 ek}acay na ykbekmki`eik`
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 277/846
Nr% Fëi~lr Pqa}anl Qrç`y
1'2>
Ahagzcl )>2% Qe avzarkgae~l z`r` ranqikr ac ileyqgl na b`ylcke` &z`r` pqa ya
ile~`gkea gaely ac gankl `gokae~a# ayzaik`cgae~a# ile IL? pqa ay~à i`cae~`enl c`
~karr` { i`gok`enl ac ickg`+ ya fkxl qe ay~qnkl z`r` ilgz`r`r nly ~kzly na gàpqke`y@ { O% Ac raenkgkae~l# ae dkcùga~rly zlr ck~rl ya gknkù% Z`r` ay~l ya yacaiikleù qe`
gqay~r` na 1: qekn`nay nac ~kzl gàpqke` @ { l~r` gqay~r` na 1: qekn`nay ~kzl
gàpqke` O% Ac zrlgankl na b`ylcke` ileyqgkn` z`r` c` gàpqke` @ mqa na V 0 7 9=
Dg% zlr ck~rl { ac zrlgankl z`r` c` gàpqke` O mqa na V ? 7 >?% C`y nay}k`ikleay
ay~àen`r mqarle na = { 6 Dg% z`r` c`y gàpqke`y @ { O# rayzai~k}`gae~a% F`iar cl
ykbqkae~a8&`+ Aeile~r`r ac ;1 na ke~ar}`cl na ilemk`ex` z`r` &±@ ' ±O+%
&o+ F`iar qe` zrqao` na fkzù~ayky z`r` naiknkr yk f`{ nkmaraeik` ae~ra cly nly
zrlgankly zloc`ikle cay%
&i+ I`ciqc`r ac }`clr na c` zrlo`okckn`n z%
Ylcqikùe8
Cly zrlgankly rk~gë~kily { c`y nay}k`ikleay ay~àen`r yle na8
V 0 7 9= Dg% { V ? 7 >? Dg% ile ω0 7 = { ω? 7 6# rayzai~k}`gae~a% Cly ~`g`ðly na
c`y gqay~r`y yle e0 7 e? 7 1:
&`+ Ac zqe~l ay~kg`nlr na ±0 ' ±? ay V 0 { V ?% Qy`enl xΰ,? 7 x%:1,? ely n` ac ke~ar}`cl na
xΰ,? 7 °0%;=% @flr`# yqoy~k~q{aenl ~lnly ay~ly }`clray ae c` mqeikùe ay~`nçy~ki` nac
ke~ar}`cl na ilemk`ex`# ely n`8
9%>9 3 &±0 ' ±?+ 3 6%12
&o+ @pqç# qy`gly qe` zrqao` na fkzù~ayky z`r` zloc`ikleay elrg`cay ile }`rk`ex`yilelikn`y% Ya qy` c` ay~`nçy~ki` &1'99+ rai`zk~qc`n` `e~arklrgae~a8
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 278/846
Nr% Fëi~lr Pqa}anl Qrç`y
1'21
x 7 T& V 0 ’ V ?+ ’ &±0 ’ ±?+W ,ee ?
?
?
0
?
0 ω ω $
Yqy~k~q{aenl cly }`clray `zrlzk`nly ae ay~` mlrgqc`8
x 7 &9= ’ >?+ ’ : ,1:
6
1:
=$
7 = , :%1?;
7 00%9>
Zrqao`y na fkzù~ayky z`r` zrlzlrikleay
C`y zrqao`y na fkzù~ayky rac`ikle`n`y ile zrlzlrikleay &zlriae~`hay+ ya rapqkarae ae
gqif`y àra`y na c` kebaekarç`% Zlr ahagzcl# c`y ilgz`ðç`y iley~rqi~lr`y ay~àe
ke~aray`n`y ae y`oar# pqë zrlzlrikùe na yqy zrlnqi~ly y`cae namai~qlyly% @nagày# cly
zlcç~kily ay~àe ke~aray`nly ae y`oar pqë mr`iikùe na cly }l~`e~ay cly m`}lraiaràe%
Zlr l~rl c`nl# ae c` kebaekarç` `gokae~`c ay~`gly ke~aray`nly ae y`oar pqë
mr`iikùe na c`y kenqy~rk`y ay~àe iqgzckaenl ile c`y cabkyc`ikleay `gokae~`cay%
^`gokëe# ay na ke~arëy ylik`c y`oar pqë mr`iikùe na cly hù}aeay qek}aryk~`rkly
qy`e na~argke`n`y nrlb`y% Kbq`cgae~a# ay na ke~arëy y`oar pqë mr`iikùe l zrlzlrikùe
na zaryle`y pqa zqan`e ay~`r ileyikae~ay na c` g`bek~qn nac zrlocag` na c`
ile~`gke`ikùe `gokae~`c# a~i%
C`y zrqao`y na fkzù~ayky ile c` ay~`nçy~ki` z &pqa ay~kg` ` π+ na zrlzlrikùe
ay~àe o`y`n`y ae qe` gqay~r` `ca`~lrk` na ~`g`ðl e na c` zloc`ikùe gqay~ra`n`% Yk ac
~`g`ðl na c` gqay~r` e ay zapqaðl# ile rac`ikùe `c ~`g`ðl zloc`ikle`c# ac zrlgankl
V ~kaea `zrlvkg`n`gae~a qe` nky~rkoqikùe okelgk`c% @nagày# yk e ay br`ena# ac zrlgankl V { ac ay~kg`nlr z 7 V,e zlyaa qe` nky~rkoqikùe okelgk`c% Zarl yk e ay
br`ena# ya qy` c` nky~rkoqikùe elrg`c ilgl qe` `zrlvkg`ikùe ` c` okelgk`c%
C`y ilenkikleay z`r` qy`r c` nky~rkoqikùe okelgk`c ay ~aear qe eøgarl mkhl na
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 279/846
Nr% Fëi~lr Pqa}anl Qrç`y
1'2=
aey`{ly kenazaenkae~ay# pqa ~aeb`e zrlo`okckn`nay iley~`e~ay { na pqa# i`n` aey`{l#
~aeb` nly rayqc~`nly ic`ykmki`nly ilgl –ëvk~l‗ l –mr`i`yl‗%
Yke ago`rbl# yk c`y ilenkikleay na ez ≡ 1 { ep ≡ 1 ya y`~kym`iae# c` nky~rkoqikùe okelgk`c ya zqana `zrlvkg r zlr c` elrg`c ile ± 7 ez { ω 7 ezp # nlena# e ay ac
~`g`ðl na c` gqay~r`# z ay c` zrlo`okckn`n &+ { p ay 0 ’ z% Ae ay~a ~ùzkil#
ileyknar`ragly øeki`gae~a# c` nky~rkoqikùe elrg`c ilgl `zrlvkg`ikùe ` c` okelgk`c#
ay naikr# z`r` gqay~r`y br`enay%
Zrqao`y na fkzù~ayky z`r` zrlzlrikleay ile gqay~r`y br`enay
C` ay~`nçy~ki` qy`n` z`r` zrqao`y na fkzù~ayky z`r` zrlzlrikleay# z`r` gqay~r`ybr`enay ay8
x 7 &z ’ zl+ , e zp , 7 &z ’ zl+ , e z z ,+0& :: ∟ &1'9=+
Nlena8
z 7 zrlzlrikùe gqay~r`c 7 zrlgankl,e 7 mr`iikùe,e
πl 7 zrlzlrikùe zloc`ikle`c l }`clr ayzar`nl
p 7 0 ’ z
e 7 ~`g`ðl na c` gqay~r`
C` zrlzlrikùe gqay~r`c `cbqe`y }aiay ya n` nkrai~`gae~a% Zlr ahagzcl# yk ya n`
>:# ay~l ya ~r`nqia ae z 7 :%>: qy`enl c` mr`iikùe z 7 V,e% Zlr ahagzcl# na c`
`mkrg`ikùe ?: na 1: zlnagly i`ciqc`r ac }`clr na c` zrlzlrikùe gqay~r`c ilgl z 7
V ,e 7 ?:,1: 7 :%>:%
Ahagzcl )>6% Qe brqzl `gokae~`c `mkrg` pqa cly keiknae~ay na c`y `}ay pqa ifli`e
ile cly `}kleay yle gq{ r`rly# ay naikr# ilgl z`r` hqy~kmki`r c` g`~`ex` na `}ay ae cly
`arlzqar~ly% Yke ago`rbl# qe brqzl na zkcl~ly `}k`nlray `mkrg`e pqa# ae ac nayzabqa
na cly `}kleay# ae ac 0: na cly i`yly# c`y `}ay ifli`e ile~r` ac `}kùe% Qy`r ΰ 7 :%:1
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 280/846
Nr% Fëi~lr Pqa}anl Qrç`y
1'22
z`r` zrlo`r ay~ `mkrg`ikùe% C` gqay~r` ay na 01: nayzabqay `olr~`nly na }kleay# na
cly iq`cay 1 ya naokarle ` iflpqa ile~r` c`y `}ay%
Ylcqikùe80% Ya qy` c` nky~rkoqikùe elrg`c ilgl `zrlvkg`ikùe ` c` okelgk`c# zlrpqa ez ≡ 1 {
ep ≡ 1# ay naikr# &01:+&:%0:+ 7 01 { ep 7 &01:+&:%;:+ 7 091%
?% Naoknl ` pqa ac raic`gl ay nac 0:# ae~leiay# c` mr`iikùe 7 z 7 :%0:% Cl lzqay~l
nac raic`gl lrkbke`c ay π 7 :%0:%
9% Naoknl ` pqa π 7 :%0: ile~kaea c` ilenkikùe na kbq`cn`n# c` fkzù~ayky eqc` { c`
`c~are`~k}` yle8
Fl8π 7 :%0: { F08π ≪ :%0:
>% Ac ek}ac na ykbekmki`eik` ay na ΰ 7 :%:1%
1% C` ay~`nçy~ki` `zrlzk`n` z`r` zrlo`r yk z 7 1,01: 7 :%:99 ay qy`enl c` ay~`nçy~ki` x
pqa ya `zrlvkg` ` c` nky~rkoqikùe okelgk`c%
x 7 &z ’ zl+ ,e
zp 7 &:%:99 ’ :%0+ ,01:
&:%0+&:%;+ 7 '?%2;
=% Cly }`clray irç~kily ile ΰ 7 :%:1 yle x 7 °0%;=2% Naoknl ` pqa ac }`clr na ’?%2; i`a ae c` rabkùe irç~ki` kxpqkarn`# ya raif`x` c`
fkzù~ayky%
6% Ac }`clr na z ay na z 7 0 ' %;;2> 7 :%::?=
Ahagzcl )>;% Qe `i~k}ky~` nac gankl `gokae~a `mkrg` pqa# gaely na c` gk~`n na c`y
kenqy~rk`y# iqgzcae ile cly cçgk~ay `gokae~`cay% Zrlo`r ay~` `ya}ar`ikùe# yk qe ylenal
nkia pqa >6 na 0;;6 kenqy~rk`y yk iqgzcae# y`~kym`i~lrk`gae~a# ile cly rabc`gae~ly`gokae~`cay% Qy`r qe ek}ac na ΰ 7 :%:1 { i`ciqc`r ac }`clr na z%
Ylcqikùe8
Ae ay~a zrlocag` c` fkzù~ayky `c~are`~k}` ya ke~arzra~` ilgl F08π 3 :%1% Ykaenl `yç#
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 281/846
Nr% Fëi~lr Pqa}anl Qrç`y
1'26
zlr cl ~ e~l# c` fkzù~ayky eqc` naoa yar Fl8π ≡ :%1%
?% C` rabkùe irç~ki` ay c` kxpqkarn` { ile ΰ 7 :%:1 { ac }`clr irç~kil ay ’0%=>1
9% C` ay~`nçy~ki` qy`n` ay c` `zrlvkg`ikùe na x c` okelgk`c% @pqç# z 7 :%>6# πl 7 :%1{ e 7 0;;6% Yqy~k~q{aenl ~lnly ay~ly }`clray ae c` ay~`nçy~ki` &1'9=+ na o`hl n`8
x 7 &z ' πl+ ,e
zp
7 &:%>6 ’ :%1+ ,0;;6
&:%1+&:%1+ 7 '0%2;
>% Naoknl ` pqa ac }`clr na ’0%2; 3 0%=>1# ya ke~rlnqia ae ac av~ragl kxpqkarnl na c`
nky~rkoqikùe# ya raif`x` Fl8
1% Ac }`clr na c` zrlo`okckn`n z ay z 7 0 ' %;=9= 7 :%:9=
Ke~ar}`cl na ilemk`ex` z`r` zrlzlrikleay
Yk z ay c` zrlzlrikùe na ëvk~ly ae qe` gqay~r` `ca`~lrk` na ~`g`ðl e { p 7 0 ’ z# qe
ke~ar}`cl na ilemk`ex` `zrlvkg`nl na &0 ’ ΰ+0:: z`r` ac z`ràga~rl okelgk`c π ya n`
zlr c` mqeikùe na `o`hl8
z ’ xΰ,?e
zp 3 π 3 z $ xΰ,?e
zp &1'92+
Nlena8 xΰ,? ay ac }`clr na x nah`enl qe àra` na ΰ,? ` c` naraif`
Ahagzcl )1:% Yk z 7 ?%1# e 7 9=# p 7 2%1 { c` rabkùe irç~ki` ay na 0%=>1# ay~kg`r ac
ke~ar}`cl na ilemk`ex` z`r` ac z`ràga~rl π%
Ylcqikùe8
Qy`enl c` mqeikùe &1'92+ { yqy~k~q{aenl ~lnly cly }`clray# ely n`8
?%1 ’ 0%=>1& 1?:6%: + 3 π 3 ?%1 $ 0%=>1& 1?:6%: +
C` iq`c ya ykgzckmki` 8
:%=?; 3 π 3 >%920
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 282/846
Nr% Fëi~lr Pqa}anl Qrç`y
1'2;
Ahagzcly na zrlocag`y qy`enl ac zrlbr`g` Gkek~`o z`r` c` nky~rkoqikùe
elrg`c# c` nky~rkoqikùe avzleaeik`c# c` nky~rkoqikùe b`gg` { c` nky~rkoqikùe
Rakoqcc# z`r` baear`r }`clray na c` nky~rkoqikùe elrg`c ay~àen`r `iqgqc`n`8I`ci 5 Zrlo`okck~{ nky~rkoq~kley 5 Elrg`c…
Ae c` }ae~`e` na –Iqggqc`~k}a zrlo`okck~{‗ zqe~a`r iqggqc`~k}a zrlo`okck~{% Ae c`
}ae~`e` na –Kezq~ ilcqge‗ zlear cly }`clray na '> f`y~` $># ay naikr# ae I0% Ae c`
}ae~`e` na –Lz~kle`c y~lr`ba‗ zlear I?# z`r` rabky~r`r cly }`clray na c` zrlo`okckn`n
`iqgqc`n`%
Zlr l~rl c`nl# yk ya pqkarae f`iar c`y bràmki`y na mraiqaeik` rac`~k}` { na
mraiqaeik` `iqgqc`n`# na c` nky~rkoqikùe elrg`c zrlianar ilgl ay~à `o`hl { yabqkr c`y
key~rqiikleay8
Br`zf 5 Zcl~
@nagày# z`r` i`ciqc`r cly }`clray na c` nky~rkoqikùe elrg`c el ay~`en`rkx`n`8
I`ci 5 Zrlo`okck~{ nky~rkoq~kley 5 Elrg`c…
Ae c` }ae~`e` na –Ga`e‗ { na –Ay~àen`r na}k`~kle‗ zlear cly }`clray nac zrlgankl {
na c` nay}k`ikùe ay~`en`r naya`nly% Ae c` ilcqge` I0 zlear cly }`clray na c` }`rk`oca
`ca`~lrk` V pqa ya pqkar`e ay~`en`rkx`r% Z`r` c`y bràmki`y zrlianar ilgl `rrko`%
@nagày# z`r` i`ciqc`r cly }`clray na c` nky~rkoqikùe avzleaeik`c krya `8
I`ci 5 Zrlo`okck~{ nky~rkoq~kley 5 Avzleae~k`c%%
Ae c` }ae~`e` na –Avzleae~k`c nky~rkoq~kle‗ zqe~a`r –Iqggqc`~k}a zrlo`okck~{‗% Ae c`
}ae~`e` na –Ga`e‗ zlear ac zrlgankl naya`nl%
Z`r` i`ciqc`r cly }`clray na c` nky~rkoqikùe Rakoqcc8I`ci 5 Zrlo`okck~{ nky~rkoq~kley 5 Rakoqcc…
Ae c` }ae~`e` na –Rakoqcc nky~rkoq~kle‗ zqe~a`r –Iqggqc`~k}a nky~rkoq~kle‗% Ae c`
}ae~`e` na –Yf`za z`r`ga~ar‗ { –Yi`ca z`r`ga~ar‗ ~aica`r cly }`clray na ΰ { ο% Z`r`
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 283/846
Nr% Fëi~lr Pqa}anl Qrç`y
1'6:
ac ray~l zrlianar ilgl `rrko`% Z`r` c`y bràmki`y f`iar cl gkygl pqa `rrko`%
Ykgkc`rgae~a# z`r` i`ciqc`r cly }`clray na c` nky~rkoqikùe B`gg`8
I`ci 5 Zrlo`okck~{ nky~rkoq~kley 5 B`gg`…Zrlianar ae mlrg` `eàclb` ` ilgl ya fkxl ile c` mqeikùe Rakoqcc
Ahagzcl )10% I`ciqc`r c`y ykbqkae~ay zrlo`okckn`nay o`hl c` iqr}` elrg`c ay~àen`r
qy`enl ac z`pqa~a na ilgzq~`nlr` Gkek~`o8
&`+ Ae~ra x 7 '0%1 { x 7 '0
&o+ Z&x ≡ ?+
&i+ Ae~ra x 7 0 { x 7 '0
&n+ F`iar qe` bràmki`
Ylcqikùe8
@orkr ac zrlbr`g` Gkek~`o { yabqkr c`y key~rqiikleay ilrrayzlenkae~ay% Ay~l baear`rà
qe` ~`oc` na `o`hl%
^@OC@ 1%6% ]`clray na c` }`rk`oca `ca`~lrk` V { c` izn% &Ac`olr`ikùe zrlzk`+ \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ Ilcqge`y I0 I?
]`rk`oca `ca`~lrk` x Nky~rkoqikùe na Zrlo`okckn`n `iqgqc`n`0 '>%: %::::9?? '9%1 %:::?999 '9%: %::091:> '?%1 %::=?0:1 '?%: %:??21:= '0%1 %:==6:22 '0%: %016=116 :%: %1:::::; 0%: %6>09>1
0: 0%1 %;990;900 ?%: %;22?1:0? ?%1 %;;92;:09 9%: %;;6=1:0> 9%1 %;;;2=201 >%: %;;;;=6
&`+ Z&'0%1 ≮ x ≮ '0%:+ 7 :%0162 ’ :%:==6 7 :%:;0; &na c` ~`oc` na `rrko`+
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 284/846
Nr% Fëi~lr Pqa}anl Qrç`y
1'60
&o+ Z&x ≡ '?+ 7 0 ’ Z&x ≮ '?%1+ 7 0 ’ :%::=? 7 :%;;96
&i+ Z&0 ≮ x ≮ '0+ 7 :%=6?2 &yke ileyqc~`r c` ~`oc`% ¼Zlr pqë<+
Ahagzcl )1?% I`ciqc`r c` nky~rkoqikùe na c`y zrlo`okckn`nay `iqgqc`n`y z`r` cly
}`clray na c` }`rk`oca `ca`~lrk` V 7 ?%1# ?%=# ?%2# ?%6# ?%;# 9%:# 9%0# 9%?# 9%9# 9%>%
@nagày# i`ciqc`r cly ykbqkae~ay aeqeik`nly8
&`+ Z& V ≡ ?%;+
&o+ Z&?%= ≮ V ≮ 9%?+
&i+ Ac }`clr na V ay na iq`enl gaely 9%>
Ylcqikùe8
Qy`enl ac zrlbr`g` Gkek~`o# zrkgar`gae~a i`ciqc`gly ac zrlgankl { c` nay}k`ikùe
ay~àen`r# na cly }`clray na c` }`rk`oca `ca`~lrk` V # { n` cly rayqc~`nly na c` ay~`nçy~ki`
nayirkz~k}` na `o`hl%
Mkbqr` 1%02% _ayqc~`nly na c` ay~`nçy~ki` nayirkz~k}` qy`enl ac Gkek~`o% \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ Ay~`nçy~ki` Nayirkz~k}`8 ]`rk`oca `ca`~lrk` v
]`rk`oca E Zrlgankl Arrlr ay~àen`r y y? Ilam% na ]`r%
]`rk`oca `ca`~lrk` 0: ?%;1:: :%:;12 :%9:?6 :%:;02 0:%?=
]`rk`oca P0 Gank`e` P9 G`vkgqg Yaybl Dqr~lyky
]`rk`oca `ca`~lrk` ?%=21: ?%;1:: 9%??1: 9%>::: :%:: '0%?:
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
Nayzqëy# ~`oca`enl cly }`clray na V ae I0 ile V 7 ?%;1 { y 7 :%9:?6# ae yqy
}ae~`e`y rayzai~k}`y# ya baear` c` ~`oc` na `o`hl%
^@OC@ 1%;% ^`oc` gly~r`enl c` }`rk`oca `ca`~lrk` V { zrlo`okckn`nay `iqgqc`n`y%&Ac`olr`ikùe zrlzk`+
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 285/846
Nr% Fëi~lr Pqa}anl Qrç`y
1'6?
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ Ilcqge`y I0 I?
]`rk`oca `ca`~lrk` V Zrlo`okckn`n `iqgqc`n`
0 ?%1 :%:=6=??? ?%= :%0?96=19 ?%2 :%?:>1:6> ?%6 :%90:0=21 ?%; :%>9>>?9= 9%: :%1=11222 9%0 :%=6;6996 9%? :%2;1>;?; 9%9 :%62=091
0: 9%> :%;90926
@flr`# z`r` raylc}ar cly keikyly zanknly zlr ac zrlocag` ya zrliana ilgl8
&`+ Z&V ≡ ?%;+ 7 0 ’ :%90:? 7 :%=6;: &na c` ~`oc` na `rrko`+
&o+ Z&?%= ≮ V ≮ 9%?+ 7 :%2;1>;? ’ :%:=6=?? 7 :%2?=;
&i+ Z&V ≡ 9%>+ &z`r` raylc}arya zlr ac cai~lr+
Ahagzcl )19% Yqzleb`gly pqa ac ~kagzl zrlgankl pqa ya ~`rn` qe` yqy~`eik`
r`nk`i~k}` &qe kyù~lzl r`nk`i~k}l pqa ~kaea ac gkygl eøgarl `~ùgkil zarl nkmarae~a zayl glcaiqc r+ ae nayilgzlearya ay na ± 7 01 ðly4 ykaenl yç8
&`+ F`iar qe` ~`oc` ile cly }`clray na c` mqeikùe avzleaeik`c na naeykn`n z`r` cly
}`clray na c` }`rk`oca `ca`~lrk` V 7 :# 1# 0:# 01# ?:# ?1# 9:# 91# >:# >1 { 1: `ðly%
&o+ Br`mki`r c`y zrlo`okckn`nay kenk}knq`cay { c`y zrlo`okckn`nay `iqgqc`n`y ae
mqeikùe nac ~kagzl ae `ðly%
&i+ ¼Iqàc ay c` zrlo`okckn`n pqa ac kyù~lzl ~`rna ae nabr`n`rya ` cl gày ae 1 `ðly<
&n+ ¼C` zrlo`okckn`n na pqa ac kyù~lzl ~`rna ae lvkn`rya ae iqàenl gaely ?: `ðly<
&a+ ¼C` zrlo`okckn`n na pqa ac kyù~lzl ~`rna ae nabr`n`rya ae~ra ?: { 1: `ðly<
&m+ ¼Iqàe~` r`nk`i~k}kn`n pqanù nayzqëy na >: `ðly<
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 286/846
Nr% Fëi~lr Pqa}anl Qrç`y
1'69
&b+ ¼Iqàe~` aearbç` ya ckoarù nayzqëy na >: `ðly<
&f+ ¼Pqë i`e~kn`n nac kyù~lzl r`nk`i~k}l pqanù nayzqëy na 1: `ðly<
^@OC@ 1%0:% ^`oc` gly~r`enl cly }`clray baear`nly na c` zrlo`okckn`n `iqgqc`n` {c` zrlo`okckn`n kenk}knq`c% &Ac`olr`ikùe zrlzk`+
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
Ylcqikùe8
&`+ ]ar ^@OC@ 1%0:
&o+ Z`r` ay~a keikyl# c`y bràmki`y na `o`hl gqay~r`e ay~` yk~q`ikùe%
^kagzl ae `ðly
_ ` n k ` i ~ k } k n ` n r a y ~ ` e ~ a
1:>:9:?:0::
:%:2
:%:=
:%:1
:%:>
:%:9
:%:?
:%:0
:%::
Yi`~~arzcl~ lm _`nk`i~k}kn`n ray~`e~a }y ^kagzl ae `ðly
Mkbqr` 1%06% Bràmki` gly~r`enl c` r`nk`i~k}kn`n ray~`e~a# ae mqeikùe nac ~kagzl ae`ðly% &Ac`olr`ikùe zrlzk`+
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 287/846
Nr% Fëi~lr Pqa}anl Qrç`y
1'6>
kagzl ae `ðly
A e a r b k ` c k o a r ` n `
1:>:9:?:0::
0%:
:%6
:%=
:%>
:%?
:%:
Yi`~~arzcl~ lm Aearbk ckoar`n` }y kagzl ae ðly
Mkbqr` 1%0;% Bràmki` gly~r`enl c` aearbç` ckoar`n` &l c` i`e~kn`n na c` yqy~`eik`
aharikn`+ nac kyù~lzl r`nk`i~k}l# ae mqeikùe nac ~kagzl ae `ðly% &Ac`olr`ikùe zrlzk`+&i+ Z&V ≮ 1+ 7 :%?691 &na c` ^`oc` 1%0: na c` zrlo`okckn`n `iqgqc`n`+
&n+ Z&V ≡ ?:+ 7 0 ’ Z&V ≮ 01+ 7 0 ’ :%=9?0 7 :%9=2;
&a+ Z&?: ≮ V ≮ 1:+ 7 :%99??
&m+ Z&V 5 >:+ &raylc}arya zlr ac cai~lr+
&b+ &_aylc}arya zlr ac cai~lr+
&f+ &_aylc}arya zlr ac cai~lr+
Ae mlrg` `eàclb` z`r` f`iar bràmki`y na zrlo`okckn`n z`r` c`y nkmarae~ay
nky~rkoqikleay ilgl c` Clbelrg`c# b`gg`# Rakoqcc# Clbçy~ki`# a~i%# krya `8
Br`zf ↟ Zrlo`okck~{ Zcl~
F`ikaenl ay~l `z`raia c` }ae~`e` na –Zrlo`okck~{ Zcl~y‗ { cqabl zlear LD# cl pqa
cca}` ` c` }ae~`e` na –Zrlo`okck~{ Zcl~ Ykebca‗% Nayzqëy na krya ` –Nky~rkoq~kle‗ {
`z`raia c` }ae~`e` ile c` cky~` na ~ln`y c`y nky~rkoqikleay ilgl elrg`c# clbelrg`c#
Rakoqcc# clbçy~ki`# b`gg`# avzleaeik`c# a~i% Ya ca zkna `c cai~lr f`iar qe aharikikl
f`ikaenl bràmki`y na zrlo`okckn`n z`r` c`y nky~rkoqikleay ile~keq`y%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 288/846
Nr% Fëi~lr Pqa}anl Qrç`y
1'61
Aharikikly I`zç~qcl 1
1%0% Ae qe gqay~ral na z`r~çiqc`y `~glymërki`y# ac zrlgankl na c` gqay~r` mqa na 2?
gkir`y { c` nay}k`ikùe ay~àen`r mqa na 01 gkir`y% Na~argke`r c`y qekn`nay na
nay}k`ikùe ay~àen`r na c`y z`r~çiqc`y pqa ~q}karle }`clray na8
&`+ =: &':%6:+
&o+ ;9 &0%>+
&i+ 2? gkir`y &:+
1%?% _amkrkëenlya `c zrlocag` `e~arklr# aeile~r`r cly }`clray na c` }`rk`oca `ca`~lrk`
elrg`c x ilrrayzlenkae~ay 8&`+ x 7 '0
&o+ x 7 0%=
1%9% Ae qe ay~qnkl kenazaenkae~a# nly kenqy~rk`y ile~`gke`e~ay mqarle kemlrg`n`y na
pqa raikokarle a}`cq`ikleay ailcùbki`y na }`rk`ocay `ca`~lrk`y elrg`cay ay~àen`ray na
x na :%2 { ':%1# rayzai~k}`gae~a% Yk yqy rayqc~`nly &a}`cq`ikleay+ mqarle na ;: { 2>#
rayzai~k}`gae~a# { `yqgkaenl pqa y 7 09%9?# aeile~r`r ac zrlgankl `rk~gë~kil# z`r`
`goly i`yly% & V 7 6:%=2# V 7 =:%=2+ 1%>% Aeile~r`r ac àra` l c`
zrlzlrikùe na c` }`clray na c` }`rk oca ca ~lrk x na c iqr}` elrg c ae~ra x 7 : { x 7
0%?%
1%1% Aeile~r`r ac àra` ae~ra x 7 :%60 { x 7 0%;>% &:%06?6+
1%=% Aeile~r`r c` zrlo`okckn`n na pqa qe` x loyar}`n` ya aeiqae~ra ` c` naraif` na x 7
?%:1 { ` c` kxpqkarn` na x 7 '0%>>%
1%2% Na~argke`r ac }`clr l cly }`clray na x iq`enl8&`+ C` zrlo`okckn`n ae~ra : { x ay na :%922: &°0%0=+
&o+ C` zrlo`okckn`n ` c` kxpqkarn` na x ay na :%6=?0 &0%:;+
1%6% Ac zayl zrlgankl na rayknqly ~ùvkily zackbrlyly baear`nly zlr 1:: kenqy~rk`y ay
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 289/846
Nr% Fëi~lr Pqa}anl Qrç`y
1'6=
na 010 ~leac`n`y gë~rki`y# ile qe` nay}k`ikùe ay~àen`r na 01 ~leac`n`y% Yk cly zayly
na cly rayknqly ~ùvkily baear`nly zlr ay~`y kenqy~rk`y ay~àe elrg`cgae~a nky~rkoqknly#
aeile~r`r ~lnl cl ykbqkae~a8&`+ Iqàe~`y kenqy~rk`y baear`e ae~ra 0?: { 011 ~leac`n`y# keicqyk}a%
&o+ Iqàe~`y baear`e gày na 061 ~leac`n`y
&i+ Iqàe~`y baear`e iq`enl gaely 0?6 ~leac`n`y
&n+ Iqàe~`y baear`e kbq`c ` 0?6 ~leac`n`y
&a+ Iqàe~`y baear`e gày na 21# zarl gaely na 0:: ~leac`n`y
1%;% Yk cly nkàga~rly na qe`y ifqg`iar`y na qe` g`pqke`rk` ay~àe elrg`cgae~a
nky~rkoqknly# ile qe zrlgankl na :%=0>: zqcb`n`y { qe` nay}k`ikùe ay~àen`r na %::?1
zqcb`n`y# na~argke`r c` zrlo`okckn`n na pqa c y ifqg`iar`y ~aeb`e nkàga~rly na8
&`+ Ae~ra %=0: { %=06 zqcb`n`y keicqyk}`gae~a &:%6;:>+
&o+ 5 %=02 zqcb`n`y &:%0010+
&i+ 3 %=:6 zqcb`n`y &%:?:2+
&n+ Kbq`c ` %=01 zqcb`n`y
1%0:% Yk qe` gqay~r` `ca`~lrk` na `eàckyky na c`y ileiae~r`ikleay na nag`en`
oklpqçgki` na lvçbael na 1 nç`y &NOL1+ ay~à elrg`cgae~a nky~rkoqkn`y# ¼pqë
zrlo`okckn`n f`{ na pqa ëy~`y nkmkar`e nac zrlgankl zlr<
&`+ Gày na c` gk~`n na c` nay}k`ikùe ay~àen`r
&o+ Gaely pqa :%21 na c` nay}k`ikùe ay~àen`r%
1%00% N`n` qe` nky~rkoqikùe elrg`c na zraikzk~`ikleay zcq}k`cay ile zrlgankl na 1:
gg { y 7 0: gg% Aeile~r`r c` zrlo`okckn`n na pqa V `yqg` qe }`clr ae~ra >1 gg {=? gg na ccq}k`% &:%12=>+
1%0?% Yk ac V { y yle ac zrlgankl { c` nay}k`ikùe ay~àen`r na qe` gqay~r` `ca`~lrk` na
`eàckyky na `bq`y rayknq`cay na ileiae~r`ikleay na ek~r`~ly# ae gb,C# ¼Iqàc ay c`
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 290/846
Nr% Fëi~lr Pqa}anl Qrç`y
1'62
zrlo`okckn`n na pqa c y ileiae~r`ikleay ay~ëe<
&`+ Nae~rl nac r`ebl & V ° ?y+
&o+ @mqar` nac r`ebl & V ° 0%?y+
&i+ G`{lr pqa & V ' 0%1 y+
1%09% N`n` qe` nky~rkoqikùe elrg`c na }`clray# ae z`r~ay zlr gkccùe# na IL
`~glymërkil# ile V 7 9:: { y 7 1:% Aeile~r`r c` zrlo`okckn`n na pqa V `yqg` qe
}`clr g`{lr pqa 9=?% &:%0:21+
1%0>% N`n` qe` nky~rkoqikùe elrg`c ile ± 7 >: { ω 7 =# aeile~r`r ac }`clr na V pqa
~aeb`8&`+ >1 nac àra` ` c` kxpqkarn`
&o+ 0> nac àra` ` c` naraif`%
1%01% C` ~ac` na mkor` na }knrkl nac apqkzl na ile~rlc z`r` z`r~çiqc`y `~glymërki`y nqr`
qe zrlgankl na 9%: `ðly# ile qe` nay}k`ikùe ay~àen`r na :%1 `ðly% Yk c`y nqr`ikleay
na c`y ~ac`y ay~àe elrg`cgae~a nky~rkoqkn`y# aeile~r`r c` zrlo`okckn`n na pqa qe` ~ac`
na qe mkc~rl nqra gaely na ?%9 `ðly% &:%:6:6+
1%0=% Qe` ilgz`ðç` m`orki` acai~rlnly z`r` cly zraikzk~`nlray acai~rly~à~kily &apqkzl
na ile~rlc z`r` z`r~çiqc`y ile~`gke`e~ay ae `kra+# iq{` nqr`ikùe ay~à elrg`cgae~a
nky~rkoqkn`# ile qe zrlgankl kbq`c ` 6:: flr`y { qe` nay}k`ikùe ay~àen`r na >: flr`y%
Aeile~r`r c` zrlo`okckn`n na pqa qe acai~rlnl ya mqen` ae~ra 226 { 69> flr`y%
1%02% Ae qe zrliayl kenqy~rk`c ac nkàga~rl na qe o`carl ya ay~`ocaia ae yqy
ayzaikmki`ikleay ilgl 9%: ° :%:0 ig% Ae c` g`eqm`i~qr` na ay~ly }`carly# c`
kgzcki`ikùe ay pqa el ya `iaz~` ekebøe o`carl pqa ya y`cb` na ay~` gankn`% Ya y`i`qe` gqay~r` na 0:: }`carly `c `x`r { ya i`ciqc` ac zrlgankl `rk~gë~kil na 9%: ig%# ile
qe` nay}k`ikùe ay~àen`r na :%::1 ig% Ae zrlgankl# ¼iqàe~ly }`carly m`orki`nly ya
nayi`r~`ràe< &≆ 1 }`carly+
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 291/846
Nr% Fëi~lr Pqa}anl Qrç`y
1'66
1%06% Ya q~kckx`e ganknlray z`r` raif`x`r ~lnl cly ilgzleae~ay iq{`y nkgaeykleay el
ya aeiqae~rae nae~rl nac c` ayzaikmki`ikùe n`n` na 0%1: ° n% Yke ago`rbl# ya y`oa pqa
ay~` nkgaeykùe ay~à elrg`cgae~a nky~rkoqkn` ile qe zrlgankl na 0%1: [ qe`nay}k`ikùe ay~àen`r na :%?% Na~argke`r ac }`clr na n z`r` pqa c` ayzaikmki`ikùe iqor`
ac ;: na c`y gankikleay%
1%0;% Iqàc ay c` zrlo`okckn`n na pqa8
&`+ Z&':%1 3 x 3 0%?1+ &:%161;+
&o+ ¼Ac }`clr na x el ay~ë ae~ra ay~ly nly }`clray< &:%>0>>+
1%?:% Ae qe ay~qnkl na kebaekarç` na fkbkaea kenqy~rk`c { yabqrkn`n# ac yqzar}kylr na
zrlnqiikùe aeiqae~r` pqa# cly ~r`o`h`nlray# ae zrlgankl# ilgzca~ e qe` ~`ra` ae 0:
gkeq~ly iq`enl ay~àe avzqay~ly ` `c~`y ileiae~r`ikleay na b`yay% Cly ~kagzly
rapqarknly z`r` ilgzca~`r c` ~`ra` yle `zrlvkg`n`gae~a elrg`cay ile qe` nay}k`ikùe
ay~àen`r na 9 gkeq~ly% Aeile~r`r cl ykbqkae~a8
&`+ C` zrlzlrikùe na agzca`nly pqa ilgzca~`e c` ~`ra` ae gaely na > gkeq~ly%
&o+ Ac na agzca`nly pqa rapqkarae gày na 1 gkeq~ly ae ilgzca~`r c` ~`ra`%
&i+ C` zrlo`okckn`n na pqa qe agzca`nl# pqkae `i`o` na yar `ykbe`nl ` c` ~`ra`# c`
ilgzca~`rà nae~rl na 9 gkeq~ly%
1%?0% Ya cca}ù ` i`ol qe gqay~ral { qe `eàckyky na c`y ileiae~r`ikleay na ek~r`~ly
&EL'9+ na qe yky~ag` na ~r`~`gkae~l na `bq`y kenqy~rk`cay% C`y ileiae~r`ikleay na
ek~r`~ly ya razlr~`rle ae gb,C% Cly ykbqkae~ay n`~ly ya n`e ae gb,C ae c` ~`oc` na
`o`hl8
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 292/846
Nr% Fëi~lr Pqa}anl Qrç`y
1'6;
^`oc` gly~r`enl cly n`~ly nac zrlocag`% &Ac`olr`ikùe zrlzk`+ \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ =%; 2%6 6%; 1%? 2%2 ;%= 6%2 =%2 >%6 6%: 0:%0 6%1
=%1 ;%? 2%> =%: =%0 =%9 1%= 1%? 1%> 2%9 6%? 6%92%? 2%1 =%0 =%: ;%> 1%> 2%= 6%0 2%;
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
F`iar cly ykbqkae~ay iàciqcly ilrrkaenl qe` ay~`nçy~ki` nayirkz~k}` pqa keicq{`8
&`+ Ac zrlgankl gqay~r`c# c` }`rk`ex`# c` nay}k`ikùe ay~àen`r { ac r`ebl% & V 7 2%?=#
y? 7 ?%:?# y 7 0%>?# r`ebl 7 1%9+
&o+ Aeile~r`r ac arrlr ay~àen`r# ac yaybl# c` dqr~lyky# ac }`clr gàvkgl { ac }`clr gçekgl% &:%?1# :%:6# '%:66# 0:%0# >%6+
&i+ A}knaeik`r c` ykga~rç` na cly n`~ly%
&n+ Yk ac cçgk~a na c`y ileiae~r`ikleay na ek~r`~ly ae ac amcqae~a ay na 6%1 gb,C# na
`iqarnl ` c` cabkyc`ikùe `gokae~`c na `bq`y# f`iar qe` zrqao` na fkzù~ayky ile qe
ek}ac na ykbekmki`eik` na ΰ 7 %:1 { i`ciqc`r c` zrlo`okckn`n z a ke~arzra~`rc`
`ilrnagae~a% &Z 333 %:::9+
&a+ F`iar qe ke~ar}`cl na ilemk`ex` z`r` ac zrlgankl ±% &=%2=# 2%2=+
&m+ F`iar qe ke~ar}`cl na ilemk`ex` z`r` c` gank`e`% &=%9;# 2%;1+
&b+ F`iar qe ke~ar}`cl na ilemk`ex` ile ek}ac na ilemk`ex` na ;1 z`r` c` nay}k`ikùe
ay~àen`r zloc`ikle`c# ω% &0%0># 0%66+
&f+ Aeile~r`r ac zrkgar iq`r~kc &=%:1+
&k+ Aeile~r`r ac ~ariar iq`r~kc% &6%?1+
1%??% Ae qe ay~qnkl na ga~alrlclbç` na zraikzk~`ikùe zcq}k`c# ac zrlgankl na ccq}k`rabky~r`nl# ` c` iae~ëykg` na qe iae~çga~rl# z`r` ac gay na g`rxl mqa na ;%??
iae~çga~rly% @yqgkaenl pqa ay~ly }`clray ay~àe elrg`cgae~a nky~rkoqknly ile qe`
nay}k`ikùe ay~àen`r ilelikn` na ?%69 ig%# aeile~r`r c` zrlo`okckn`n na pqa ac
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 293/846
Nr% Fëi~lr Pqa}anl Qrç`y
1';:
ykbqkae~a gay na g`rxl nac `ðl ae~r`e~a# ya raiko`e8
&`+ Gaely na 0%6> ig% na ccq}k`%
&o+ Gày na 1 ig% na ccq}k`%&i+ Iq`enl gaely 09%6 ig% na ccq}k`%
&n+ @ cl gày 0:%: ig% na ccq}k`%
&a+ Kbq`c ` 1 ig% na ccq}k`# a%b%# Z&>%1 ≮ V ≮ 1%1+
&m+ F`iar qe` bràmki` z`r` i`n` keikyl%
1%?9% Yqzùeb`ya pqa c` mqeikùe na naeykn`n na ikar~l avzarkgae~l na gankikleay na
lvçbael nkyqac~l &LN+ ay m& v
+ 7 a v% Yk yqzleagly pqa c` }`rk`oca `ca`~lrk` ile~keq`
`yqg` }`clray ae~ra V 7 ?%: { V 7 1%:# aeile~r`r c`y ykbqkae~ay zrlo`okckn`nay%
&`+ Z&: 3 V 3 >+ &a> ’ 0+
&o+ Z& V 3 >+ &a?+ &i+
Z&? 3 V 3 1+ &a9+ &n+
Z&: 3 V 3 9+ &a9 ’ 0+ 1%?>%
N`r ac nlgkekl na i`n` qe` na c`y ykbqkae~ay }`rk`ocay { naikr yk c`y }`rk`ocay yle
ile~keq`y l nkyira~`y%
&`+ Ac eøgarl na ck~rly na `bq` ae qe r`nk`nlr na `q~lgù}kc%
&o+ Ac eøgarl na ckorly ae ac ay~`e~a na qe` ckorarç`%
&i+ Ac nkàga~rl N na qe` aymar`%
1%?1% Ya` x qe` }`rk`oca `ca`~lrk` elrg`c ay~àen`r# ae~leiay# i`ciqc`r c`y ykbqkae~ay
zrlo`okckn`nay# nkoqh`enl c`y bràmki y%
&`+ Z&: ≮ x ≮ ?%02+ &:%>61:+&o+ Z&: ≮ x ≮ 0+ &:%9>09+
&i+ Z&'?%1 ≮ x ≮ :+ &:%>;96+
&n+ Z&'?%1 ≮ x ≮ ?%1+
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 294/846
Nr% Fëi~lr Pqa}anl Qrç`y
1';0
&a+ Z&0%1 ≮ x+ &0 ' :%;99?+
&m+ Z&sxs ≮ ?%1:+
1%?=% Yk V ay qe` }`rk`oca `ca`~lrk` elrg`c ile zrlgankl na 6: { nay}k`ikùe ay~àen`r na 0:# ae~leiay# i`ciqc`r c`y ykbqkae~ay zrlo`okckn`nay# gank`e~a ay~`en`rkx`ikùe%
&`+ Z& V ≮ 0::+
&o+ Z&=1 ≮ V ≮ 0::+
&i+ Z&61 ≮ V ≮ ;1+
&n+ Z&2: ≮ V +
&a+ Z&;: ≮ V ≮ 0::+&m+ Z&6: ≮ V ≮ 00:+
&b+ Z&? 3 x 3 '?+
1%?2% C` }kn` zrlgankl na qe` z`r~çiqc` ae c` `~gùymar`# ykbqa ` c` ca{ na Y~lda% Ëy~`
}` ae mqeikùe nac nkàga~rl na yankgae~`ikùe# gkyg` pqa }` ae mqeikùe na c` naeykn`n
na c` z`r~çiqc`# c` naeykn`n nac gankl# c` }kyilykn`n `oylcq~` nac gankl# c` `iacar`ikùe
na c` br`}an`n &;60 ig,yai?+# a~i% Ile ay~l# ya zqana glnac`r c` i`çn` na c` z`r~çiqc`
qy`enl c` mqeikùe avzleaeik`c% Yqzlekaenl pqa c` }kn` zrlgankl ae c` `~gùymar` na
ay` z`r~çiqc` ya` na 0? `ðly# ae~leiay i`ciqc`r c`y ykbqkae~ay zrlo`okckn`nay8
&`+ ¼Iqàc ay c` zrlo`okckn`n na pqa c` }kn` na rayknaeik` `~glymërki` na c` z`r~çiqc`
ya` na ` cl gày = `ðly% &:%9;9>+
&o+ ¼Iqàc ay c` zrlo`okckn`n na pqa c` }kn` na rayknaeik` `~glymërki` na c` z`r~çiqc`
ya` ae~ra 1 { 0: `ðly< &:%??>>+
Yqbaraeik`8 Qy`r c`y ykbqkae~ay rac`ikleay8Z& v ≮ vl+ 7 0 ’ a' v,± { Z& v ≮ vl+ 7 Z& V ≮ 0:+ ’ Z& V ≮ 1+
1%?6% C` }kn` &ae flr`y+ na qe nkyzlyk~k}l acai~rùekil ay qe` }`rk`oca `ca`~lrk` ile c`
ykbqkae~a nky~rkoqikùe avzleaeik`c na zrlo`okckn`n8 m& v+ 7 0,1: a' v,1: z`r` v ≡ :
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 295/846
Nr% Fëi~lr Pqa}anl Qrç`y
1';?
&`+ ¼Iqàc ay c` }kn` zrlgankl nac nkyzlyk~k}l<
&o+ ¼Iqàc ay c` zrlo`okckn`n na pqa ac nkyzlyk~k}l mqeiklea 0: l gaely flr`y `e~ay na
pqa m`cca<&i+ ¼Iqàc ay c` zrlo`okckn`n na pqa ac nkyzlyk~k}l nqra ae~ra >: { =: flr`y<
&n+ F`iar qe` bràmki` ile }`clray na v 7 1# 0:# ?:# 9:# >:# 1:# =: ile yqy
zrlo`okckn`nay ilrrayzlenkae~ay m& v+%
1%?;% C` nag`en` oklpqçgki` na lvçbael &NOL1+ na 1 nç`y &¼zlrpqë na 1 nç`y<+ na
qe` gqay~r` na g`~ark` lrbàeki` ykbqa ` qe` nky~rkoqikùe avzleaeik`c na zrlo`okckn`n
ile qe zrlgankl na 6: gb,C na NOL18 m & v+ 7 0,6: a
' v,6:
&`+ F`iar qe` bràmki` ile ay~` mqeikùe qy`enl v 7 0# ?# 9# ># 1 nç`y%
&o+ I`ciqc`r c` i`e~kn`n na NOL &ae gb,C+ pqa pqanù ae~ra 0 { ? nç`y% Avzray`rcl ae
c` bràmki`% &:%:?>1+
&i+ I`ciqc`r c` ileiae~r`ikùe na NOL &gb,C+ pqa pqanù ae~ra 0 { 9 nç`y% Avzray`rcl
ae c` bràmki`% &:%:9=6+
&n+ I`ciqc`r c` ileiae~r`ikùe na NOL &gb,C+ pqa pqanù ae~ra 0 { 1 nç`y% &%:=:+
&a+ I`ciqc`r c` i`e~kn`n na NOL pqa ya aharikù &i`e~kn`n na g`~ark` lrbàeki`
lvkn`n`+ ae ` cl gày > nç`y% Avzray`rcl ae c` bràmki`% &:%:>66+
&m+ I`ciqc`r c` ileiae~r`ikùe na NOL pqa pqanù nayzqëy na 1 nç`y% &:%:=:=+
&b+ ¼Pqë zlriae~`ha na NOL ya aharikù# ` cl gày ae 1 nç`y< &:%=::=+
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 296/846
Nr% Fëi~lr Pqa}anl Qrç`y
1';9
^`oc` gly~r`enl cly rayqc~`nly nac NOL pqa }` pqan`enl { ac NOL lvkn`nl lahariknl% &Ac`olr`ikùe zrlzk`+ V NOL ray~`e~a NOL lvkn`nl
&Nç`y+ &Zrlo% kenk}knq`c+ &Zrlo% `iqgqc`n`+0 %:0?9 %:0?>? %:0?? %:?>29 %:0?: %:9=6> %:00; %:>661 %:002 %:=:=
1%9:% Ya n` c` ~`oc` na `o`hl ile cly zlriae~`hay na NOL lvkn`nl ae mqeikùe nac
~kagzl { na c` iley~`e~a na naylvkbae`ikùe d% Yk ac NOL1 øc~kgl l ~l~`c ay na Cl 79:: gb,C &nark}`nl na c` aiq`ikùe glelglcaiqc`r { 7 Cl&0 ’ 0:'d~+# f`iar cl
ykbqkae~a8
&`+ Qe` bràmki` z`r` }ar ac amai~l na c` }aclikn`n na c` iley~`e~a d z`r` qe ek}ac n`nl
na Cl na c` aiq`ikùe glelglcaiqc`r%
&o+ Qe` bràmki` pqa kenkpqa ac NOL pqa }` pqan`enl { c` i`e~kn`n na NOL pqa ya
}` aharikaenl# z`r` i`n` qel na cly ?: nç`y { z`r` i`n` qe` na c`y ~`y`y d%
&i+ ¼Iqàc ay c` zrlo`okckn`n na pqa c` ileiae~r`ikùe na NOL ay~ë ae~ra 9 { = nç`y
keicqyk}`gae~a# z`r` d 7 :%01< ¼@ pqë ileiae~r`ikùe na NOL `yiaenkù ay~l<
&n+ I`ciqc`r c` ileiae~r`ikùe na NOL pqa pqanù ae~ra 9 { 1# z`r` d 7 :%0:%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 297/846
Nr% Fëi~lr Pqa}anl Qrç`y
1';>
^`oc` gly~r`enl c` ~`y` na ra`iikùe na c` }`rk`oca d ae ac NOL }y% ~kagzl%^kagzl &nç`y+ Zlriae~`ha nac ~l~`c na NOL ahariknl
„„„„„„„„„„„„„„„„„„„„„„„„
d 7 :%:1 d 7 :%0: d 7 :%01 d 7 :%?: d 7 :%?10 0:%; ?:%= ?;%? 9=%; >9%6? ?:%= 92%: 1:%: =:%: =6%:9 ?;%: 1:%: =>%: 21%: 6?%:> 92%: =:%: 21%: 6>%: ;:%:1 >>%: =6%: 6?%: ;:%: ;>%:= 1:%: 21%: 62%: ;>%: ;2%:2 11%: 6:%: ;0%: ;=%: ;6%:0: =6%: ;:%: ;2%: ;;%: ;;%:
?: ;:%: ;;%: ;;%:$ ;;%:$ ;;%:$
&Mqae~a8 Y`{ar a~ `c% 0;=2+
1%90% Yqzùeb`ya pqa# ac ~kagzl ae flr`y# rapqarkn`y z`r` raz`r`r qe` olgo` na i`clr
ay qe` }`rk`oca `ca`~lrk` V pqa ~kaea qe nky~rkoqikùe b`gg` ile z`ràga~rly ΰ 7 ? { ο
7 :%1%
&`+ Aeile~r`r ac zrlgankl# c` }`rk`ex` { c` nay}k`ikùe ay~àen`r% &± 7 0%:# ω? 7 :%1+
&o+ ¼Iqàc ay c` zrlo`okckn`n na pqa ac ykbqkae~a yar}kikl rapqarkrà ` cl gày qe` flr`
z`r` raz r`r c olgo`<
&i+ C` zrlo`okckn`n na pqa ya rapqkar`e iq`enl gaely ? flr`y z`r` raz`r`r c` olgo`%
&:%:;0=+
1%9?% Ae ikar~` ikqn`n# ac ileyqgl nk`rkl na acai~rkikn`n# ae gkccleay na dkcl`~~'
flr`y# ay qe` }`rk`oca `ca`~lrk` V pqa ykbqa ` qe` nky~rkoqikùe b`gg` ile ± 7 = { ω?
7 0?% Aeile~r`r8&`+ Cly }`clray na ΰ { ο%
&o+ Aeile~r`r c` zrlo`okckn`n na pqa ae qe nç` n`nl ac ileyqgl nk`rkl na acai~rkikn`n
avianarà 0? gkccleay na dkcl`~~'flr`y%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 298/846
Nr% Fëi~lr Pqa}anl Qrç`y
1';1
1%99% Ya y`oa pqa c` nky~rkoqikùe na Rakoqcc ay `gzck`gae~a qy`n` ae zrlocag`y na
ay~`nçy~ki` rac`ikle`nly ile ac ae}ahaikgkae~l { na~arklrl na g`~ark`cay yùcknly
`kyc`e~ay yqha~ly ` }lc~`hay @I% Cly }`clray na cly z`ràga~rly nazaenae nac }lc~`ha {na c` ~agzar`~qr`% O`y`nl ae ay~l# yqzùeb`ya pqa ΰ 7 ?%1 { ο 7 ?::% Ykaenl `yç#
i`ciqc`r cl ykbqkae~a8
&`+ ¼Iqàc ay c` zrlo`okckn`n na pqa c` }kn` na qel na ay~ly `z`r`~ly ya` ` cl gày na
?::< &%=9?+
&o+ Gaely pqa ?:: &:%=?21+
&i+ Gày na 9:: &:%:=>+ &n+
Ae~ra 0:: { ?:: &:%?>9+
1%9>% Ae qe ay~qnkl na ga~alrlclbç`# nayzqëy na `e`ckx`r qe` gqay~r` na 0:=
~agzar`~qr`y `gokae~`cay# qe kebaekarl `gokae~`c `mkrg` pqa ac zrlgankl na
~agzar`~qr` ay gaelr pqa ;6%= lM% F`iar cl ykbqkae~a8
&`+ Knae~kmki`r c` fkzù~ayky eqc` Fl8
&o+ Knae~kmki`r c` fkzù~ayky c~are`~k}` F08
&i+ Knae~kmki`r yk ay~` zrqao` ay na nly ilc`y# na c` ilc` kxpqkarn` l na c` ilc` naraif`%
1%91% C` `mkrg`ikùe na pqa ac zrlgankl zloc`ikle`c nac zayl na c`y ~`oca~`y na
`yzkrke` ay zrlo`n` ile qe ek}ac na ykbekmki`ikùe na ΰ 7 %:1% C`y ilenkikleay yle na
pqa ya zqana qy`r c` nky~rkoqikùe elrg`c zlrpqa e 5 9:% Aeile~r`r c`y rabkleay
irç~ki`y l cly }`clray irç~kily na x# yk c` zrqao` ay na8
&`+ Nly ilc`y% &x~`o% 7 °0%;=+
&o+ Na c` ilc` kxpqkarn`% &x%:1 7 '0%=11+&i+ Na c` ilc` naraif`% &x~`o% 7 0%=>1+
1%9=% Ile~ay~`r c`y zrabqe~`y ae cly aharikikly nac 0 `c =%
0% Ac zrlgankl nac ilamkikae~a na cly key~rqi~lray na ay~`nçy~ki` ay na 061%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 299/846
Nr% Fëi~lr Pqa}anl Qrç`y
1';=
?% Ac zrlgankl nac zayl nac z`zac nayi`r~`nl i`n` yag`e` ae qe ay~qnkl na raikic`ha
na z`zac ay gaelr pqa 0: dkclbr`gly%
9% Ac ~kagzl zrlgankl rapqarknl z`r` cly ay~qnk`e~ay zqan`e `npqkrkr yq ~ç~qcl ayg`{lr pqa 1 `ðly%
>% Ac zrlgankl `eq`c na kebrayly na cly gënkily ay na .9::#::: nùc`ray%
1% Ac zrlgankl na c` an`n na cly `}kleay ilgarik`cay ay na iq`enl gaely 0: `ðly%
=% C` ~`y` nac zrlgankl na ileyqgl na cly q~lgù}kcay Ifa}rlca~ ay na el gày na 02
gkcc`y,b`cùe%
Z`r` i`n` qel na ay~ly ahagzcly nac 0 `c =# ile~ay~`r cl ykbqkae~a8
&`+ Knae~kmki`r c` fkzù~ayky eqc`# Fl8
&o+ Knae~kmki`r c` fkzù~ayky `c~are`~k}`# F08
&i+Knae~kmki r c` zrqao` yk ay okc`~ar`c# l qekc`~ar c kxpqkarn` l naraif`%
&b+ @yqgkr pqa c` ileicqykùe ay na raif`x`r c` fkzù~ayky eqc`% Naic`r`r c` ileicqykùe
ae ~ërgkely el ~ëiekily% @yabqr`rya na aecky~`r ac raic`gl lrkbke`c%
&f+ @yqgkr pqa c` ileicqykùe ay c` na m`cc`r ae raif`x`r c` fkzù~ayky eqc`% Naic`r`r c`
ileicqykùe ae ~ërgkely el ~ëiekily% @yabqr`rya na aecky~`r ac raic`gl lrkbke`c%
1%92% I`ciqc`r ac }`clr na c` zrlo`okckn`n z ile ek}acay na ykbekmki`ikùe na ΰ 7 :%:1 {
ΰ 7 :%:0% Ya n`e cly ykbqkae~ay }`clray8 e 7 1:# V 7 90%6 { ω 7 :%21% Zrlo`r c`
fkzù~ayky eqc` Fl8± ≡ 9? ile~r` F08 3 9?% &x 7 '0%6;# z 7 %:?;>+
1%96% Qe` gqay~r` `ca`~lrk` na 9= i`yly na `eàckyky na `bq`y ile~aekaenl iclr`~ly
&gb,C na IcL9'+# ya qy` ac gë~lnl `rbae~lgë~rkil na ~k~qc`ikleay & Ay~àen r Ga~flny
mlr ~fa Av`gke ~kle lm R`~ar `en R`y~a`~ar# 0;20+% Zrlo`r c` fkzù~ayky na pqa ac zrlgankl zloc ikle`c ay kbq c ` 0>1 gb,C% Ya i`ciqc` ac zrlgankl ay~`nçy~kil { ely
n` V 7 096%6> ile qe` nay}k`ikùe ay~àen`r na ?:% Zrlo`r c` fkzù~ayky na Fl8± 7 0>1
ile cly ek}acay na ykbekmki`eik` na :%:1 { :%:0% ^`gokëe i`ciqc`r ac }`clr na z% 1%9;%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 300/846
Nr% Fëi~lr Pqa}anl Qrç`y
1';2
Ac zrlgankl na qe` gqay~r` `ca`~lrk` na 0:: `eàckyky na yqcm`~ly &YL>'?+ ay na 012:
gb,C# ile qe` nay}k`ikùe ay~àen`r na 0?: gb,C% Yk ± ay ac zrlgankl na ~lnly cly
i`yly na yqcm`~ly# zrlo`r c` fkzù~ayky eqc` na Fl8± 7 0=:: gb,C# ile~r` c` fkzù~ayky`c~are`~k}` na F08± ≪ 0=:: gb,C qy`enl cly ek}acay na ykbekmki`ikùe na :%:1 { :%:0%
I`ciqc`r ac }`clr na c` zrlo`okckn`n# z% &x 7 '?%1# z 7 %::=?+
1%>:% Ae qe ay~qnkl na lia`elbr`mç` &ac ay~qnkl mçykil# pqçgkil { oklcùbkil na c`y
`bq`y na cly lië`ely+# yqzùeb`ya pqa qe lia`eùbr`ml# `c ra}ky`r c` zrlmqenkn`n
zrlgankl nac lië`el# ae ikar~` z`r~a aeiqae~r pqa ay na =?%9 or`x`y% Ay~l cl f`ia
z`r` }ar c` m`i~kokckn`n na f`iar ikar~ly `eàckyky oklcùbkily% Ay~a ke}ay~kb`nlr naiknkù
qy`r ek}acay na ykbekmki`eik` na :%:1 { :%:0% Z`r` ay~l# ~lgù qe` gqay~r` na ylenaly
na zrlmqenkn`n ae >: cli`ckx`ikleay g`rke`y { aeile~rù pqa ac zrlgankl na c`
gqay~r` ay~`nçy~ki` ar` na =>%6 or`x`y# ile qe` nay}k`ikùe ay~àen`r na 1%0% Naikr yk ya
raif`x` Fl8 { i`ciqc`r ac ek}ac na zrlo`okckn`n%
1%>0% Ay~a ay qe ay~qnkl rac`ikle`nl ile ac `eàckyky na `bq`y kenqy~rk`cay na i`cikl
&gb,C+ qy`enl ac gë~lnl br`}kgë~rkil% Z`r` ay~l# ya y`i` qe` gqay~r` na >6 `eàckyky {
ya i`ciqc` qe zrlgankl ay~`nçy~kil na 2=%> gb,C ile qe` nay}k`ikùe ay~àen`r na 9%=%
Qy`enl qe ek}ac na ykbekmki`eik` na :%:1 zrlo`r c` fkzù~ayky na pqa ac zrlgankl
zloc ikle`c ay g`{lr pqa 21 gb,C { i ciqc r ac }`clr na z% &x 7 ?%=;# z 7 %::9=+
1%>?% Qe` gqay~r` `ca`~lrk` na 0= loyar}`ikleay na `eàckyky na ilora# ae ac `bq`# ya
y`i`rle na qe` zloc`ikùe elrg`c% Ya i`ciqc` qe zrlgankl na V 7 >;%21 { qe`
nay}k`ikùe ay~àen`r na 0:% @yqgkr qe` zrqao` okc`~ar`c% C` gqay~r` na cly rayqc~`nly
na cly `eàckyky ya n` `o`hl8
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 301/846
Nr% Fëi~lr Pqa}anl Qrç`y
1';6
^`oc` gly~r`enl cly n`~ly% &Ac`olr`ikùe zrlzk`+ \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ =? >9 =: >; 2? 1= >1 >= 92 1= >0 >9 9= >1 1= >;
Qy`r ac ek}ac na ykbekmki`eik` na %:1 { zrlo`r c`y ykbqkae~ay fkzù~ayky eqc`y8&`+ Fl8± 7 >:
&o+ Fl8± 7 >;
&i+ Fl8± 7 1:
&n+ Fl8± 7 10
&a+ Fl8± 7 =:
Naoknl ` pqa V 7 >;%21 yk zqana na~argke`r yk qe` ileicqykùe ay ilrrai~` l arrùea`%
Z`r` i`n` qel na cly ikeil keikyly naikr yk c` ileicqykùe ay ilrrai~` l yk ac arrlr K l ac
arrlr KK ya f`e faifl ilga~knl% Ay~a aharikikl ya f`ia z`r` nagly~r`r pqa ac ~kzl na
arrlr KK ya zqana ilga~ar yk ac zrlgankl zloc`ikle`c fkzl~ë~kil ay~à iari`el `c
}arn`narl zrlgankl zloc`ikle`c%
^`oc` gly~r`enl cly n`~ly% &Ac`olr`ikùe zrlzk`+ \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
Keikyl ±l V ' ±l x Fl8 _abkùe irç~ki` ^kzl na arrlr \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
` >: ;%21 9%;: raif`x`r °0%;= ekebqel o >; :%21 :%9: `iaz~`r / KKi 1: ':%?1 ':%0: `iaz~`r / KKn 10 '0%?1 ':%1: `iaz~`r / KKa =: '0:%?1 '>%0: raif`x`r / ekebqel
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
1%>9% Qe` gqay~r` `ca`~lrk` na ?1:: loyar}`ikleay na ~agzar`~qr`y# avzray`n`y ae
br`nly M`fraefak~ &lM+# ya y`i`rle { ya i`ciqcù qe zrlgankl na kbq`c ` >;%;# ile qe`
nay}k`ikùe ay~àen`r na ;%;?% Qy`r c`y gkyg`y fkzù~ayky eqc`y pqa ae ac zrlocag`
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 302/846
Nr% Fëi~lr Pqa}anl Qrç`y
1';;
`e~arklr ` qe ek}ac na ykbekmki`ikùe na ΰ 7 :%:0% C` ke~aeikùe na ay~a aharikikl ay z`r`
nagly~r`r pqa c` zrlo`okckn`n na ilga~ar `goly arrlray ~kzl K { ~kzl KK ya zqanae
ranqikr# `c gkygl ~kagzl# `qgae~`enl ac ~`g`ðl na c` gqay~r`% Cly n`~ly zar~keae~ayya n`e ae c` ~`oc` na `o`hl%
^`oc` gly~r`enl cly n`~ly% &Ac`olr`ikùe zrlzk`+ \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ Keikyl ±l V ' ±l x Fl8 _abkùe irk~ki` ^kzl na arrlr
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ ` >: ;%; >;%1 °?%16 raif`x`r ekebqel
o >; :%; >%1 / / /
i 1: ':%0 ':%1 / / /n 10 '0%0 '1%1 / / /a =: '0:%0 '1:%1 / / /
1%>>% Yqzlekaenl pqa ya loyar}` qe }`clr na x 7 0%62 ile qe ek}ac na ykbekmki`ikùe na
ΰ 7 :%:1 { ile Fl8 ± 7 0:# ae~leiay# i`ciqc`r ac }`clr na c` zrlo`okckn`n# z%
1%>1% Yk ac }`clr na c` }`rk`oca `ca`~lrk` ay na x 7 ?%29# ΰ 7 :%:1 { Fl8 ± 7 0:#
aeile~r`r ac }`clr na z% &:%::9?+1%>=% Ae qe ay~qnkl na fkbkaea kenqy~rk`c { yabqrkn`n# &yabqrkn`n z`r` cly gl~lrky~`y+#
ile c` illzar`ikùe nac naz`r~`gae~l na zlckiç`# ya rapqkara pqa cly yqr~knlray na
cc`e~`y ~aeb`e qe }kn` zrlgankl na iq`enl gaely 9:#::: gkcc`y% Z`r` `yabqr`rya na
ay~a kgzankgae~l# ac naz`r~`gae~l na zlckiç` zrqao` qe` gqay~r` `ca`~lrk` na 9=
cc`e~`y { lo~kaea qe zrlgankl ay~`nçy~kil na ?1#6:: gkcc`y ile qe` nay}k`ikùe
ay~àen`r na 6#::: gkcc`y% Zargk~kaenl qe` zrlo`okckn`n na %:1 nac ~kzl arrlr K# qy`r clyn`~ly z`r` zrlo`r c` fkzù~ayky eqc` na pqa ac }arn`narl zrlgankl ay na iq`enl gaely
9:#::: gkcc`y%
1%>2% Ya n`e cly ykbqkae~ay n`~ly zrlianae~ay na qe ay~qnkl na zraikzk~`ikùe zcq}k`c ae
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 303/846
Nr% Fëi~lr Pqa}anl Qrç`y
1'0::
gankikleay na gkcçga~rly na ccq}k`8 Fl8± 7 :%9>: gg# F08± ≪ :%9>:# nlena :%9> 7 ±l#
ΰ 7 %:1# V 7 :%9>9 gg%# ω 7 %:0# e 7 91% &Eù~aya pqa `pqç ay qe` zrqao` na ? ilc`y#
zlrpqa Fl8± 7 :%9>: raøea c` ilenkikùe na kbq`cn`n+% C`y rabkleay irç~ki`y yle °0%;=# zlrpqa ΰ 7 :%:1% ]ar ~`oc` na `o`hl% &x 7 0%22% Ya ra~kaea Fl8 ile z 7 %:2=6+
^`oc` gly~r`enl cly n`~ly% &Ac`olr`ikùe zrlzk`+ \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ Fkzù~ayky `c~are`~k}`y _aif`x`r Fl8± 7 :%9>: yk8
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ F08 ± 3 ±l 7 :%9>9: x 3 ' xΰ F08 ± 5 ±l 7 :%9>9 x 5 $ xΰ
F08 ±≪
±l 7 %:%9>9 x 3 ' xΰ,? l x 5 xΰ,? \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
1%>6% Qe` agzray` na i`gkleay na i`rb` ylyzaif` na c` `mkrg`ikùe na pqa ac ikicl na
}kn` na ikar~ly eaqgà~kily ay na `c gaely ?6#::: gkcc`y &≡ ?6#:::+% Z`r` }arkmki`r
ay~a `rbqgae~l# c` agzray` key~`c` >: na ay`y cc`e~`y ae yqy i`gkleay { lo~kaea qe
ikicl na }kn` zrlgankl na ?2#>=9 ile ω 7 09>6 gkcc`y% ¼Pqë ya zqana ileicqkr# yk c`
zrlo`okckn`n na qe arrlr ~kzl `cm` ya mkh` ae :%:0< @yqgkr qe` zrqao` na fkzù~ayky
qekc`~ar`c kxpqkarn`%
1%>;% Z`r` qe `eàckyky na zay~kikn`y iclrke`nly fknrli`role`nly ae `bq`y rayknq`cay
&qy`enl irlg`~lbr`mç` na b`y+# ya nkl qe` gqay~r` ile~aekaenl ay~a zay~kikn` ` nly
c`olr`~lrkly% Cly ~`g`ðly na c`y gqay~r`y mqarle na >: { 1: i`yly# rayzai~k}`gae~a%
Yk c`y gqay~r`y ~kaeae zrlgankly na V 0 7 2> ile nay}k`ikùe ay~àen`r na ω0 7 6# { na
zrlgankl na V ? 7 26 ile qe` nay}k`ikùe ay~àen`r na ω? 7 2# naikr yk f`{ qe`
nkmaraeik` ykbekmki`e~a ae~ra cly rayqc~`nly na cly nly c`olr`~lrkly% @yqgkr ek}acay naykbekmki`eik` na ΰ 7 %:1 { ΰ 7 %:0% &x 7 '?%>;# z 7 %::=>+
1%1:% Qe` gqay~r` `ca`~lrk` na 0:: gqar~ay ae A% Q% gly~rù qe` }kn` zrlgankl na
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 304/846
Nr% Fëi~lr Pqa}anl Qrç`y
1'0:0
20%6 `ðly ile qe` nay}k`ikùe ay~àen`r na 6%; `ðly% ¼Zqnkar` ay~l kenki`r pqa c` }kn`
zrlgankl na fl{ ae nç ay g`{lr pqa 2: `ðly< Qy`r ΰ 7 %:1%
1%10% Qe m`orki`e~a na i`ocay na `iarl `mkrg` pqa yq zrlnqi~l ~kaea qe` rayky~aeik` narqz~qr` na 6%: Db% Zrlo`r c` fkzù~ayky eqc` na pqa Fl8± 7 6%: Db%# ile~r` c` zrqao`
`c~are`~k}` na pqa F08± ≪ 6%: Db% Z`r` ay~l# ya y`iù qe` gqay~r` `ca`~lrk` na 1:
i`ocay { ya aeiqae~r` pqa ~kaea qe` rayky~aeik` zrlgankl na V 7 2%6 Db%# ile qe`
nay}k`ikùe ay~àen`r na :%1 Db% Z`r` ay~` zrqao` qy`r ΰ 7 %:1 { ΰ 7 %:0% &z 7 %::>=+
1%1?% Ae qe ay~qnkl na c` `zcki`ikùe nac zF &zl~aeik`c fknrùbael pqa ~kaea qe` ayi`c`
na : ` 0># nlena 2 ay eaq~r`c { `o`hl na 2 ay àiknl { `rrko` na 2 ay `ci`ckel+ z`r`gankr c` `ci`ckekn`n { c` `iknax na ylcqikleay# qe ikae~çmkil# nanki`nl `c ay~qnkl na c`
ile~`gke`ikùe `gokae~`c# `yabqr` pqa nly gqay~r`y na ylcqikleay &@ { O+ zrl}kaeae
nac gkygl cqb`r na qe rçl# nlena yqzqay~`gae~a fqol qe nayi`rb` kenqy~rk`c na àiknl
iclrfçnrkil &FIc+% Yk ay~l mqar` ikar~l# ae~leiay ac zF na c`y nly gqay~r`y na
ylcqikleay yarç`e kbq`cay% @yqgkaenl pqa c`y loyar}`ikleay zrl}kaeae na
zloc ikle`cay elrg`cay# zrlo`r c` fkzù~ayky eqc` na kbq`cn`n na cly zrlgankly na zF%
@yqgkr ΰ 7 :%:1% F`iar c`y ykbqkae~ay ay~kg`ikleay8
&`+ F`iar ay~ly iàciqcly qy`enl c` nky~rkoqikùe elrg`c { c` nky~rkoqikùe na ~ na
Ay~qnk`e~a%
&o+ ^`gokëe# i`ciqc`r ac }`clr na c` zrlo`okckn`n z ae `goly i`yly { }ar pqa
nkmaraeik`y f`{%
&i+ F`iar ke~ar}`cly na ilemk`ex` qy`enl c`y mùrgqc`y z`r` c` nky~rkoqikùe x { z`r` c` ~
na Ay~qnk`e~a%&n+ ¼Nay`zrqao`e cly n`~ly c` `mkrg`ikùe nac ikae~çmkil< C` ~`oc` na `o`hl gqay~r` c`
kemlrg`ikùe rapqarkn` z`r` ay~a zrlocag`%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 305/846
Nr% Fëi~lr Pqa}anl Qrç`y
1'0:?
^`oc` gly~r`enl cly n`~ly na c`y gankikleay nac zF% &Ac`olr`ikùe zrlzk`+
Gankikleay nac zF na ylcqikùe @ Gankikleay nac zF na ylcqikùe O
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ =%?> =%?2=%90 =%?1=%?6 =%99=%9: =%?2=%?1 =%?>=%?= =%90=%?> =%?6=%?; =%?;
=%?? =%9>=%?6 =%?2 \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
1%19% Qe` ilgz`ðç` ay~à ae ac zrliayl na naiknkrya yk }` ` zrlnqikr qe eqa}l
ilgzleae~a acai~rùekil% Ae c` zc`e~` f`{ nly gàpqke`y pqa zqanae yar `n`z~`n`y
z`r` f`iar ay~a ilgzleae~a% Z`r ay~l# ya f`ia qe` zrqao` ae c gàpqke` 0 { ya gkna
ac ~kagzl na zrlnqiikùe zlr ilgzleae~a { n` qe zrlgankl na V 0 7 1%?9 gkeq~ly
z`r` qe` gqay~r` na 0:: ilgzleae~ay% Ae c` gàpqke` ? ac zrlgankl na ~kagzl mqa na
V ? 7 1%92 gkeq~ly z`r` qe` gqay~r` na => ilgzleae~ay% Ae z`y`n`y avzarkaeik`y# ya
y`oa pqa c`y nay}k`ikleay ay~àen`r mqarle na :%01 { :%0: gkeq~ly# rayzai~k}`gae~a
@yqgkr ΰ 7 :%:1% F`iar cly ykbqkae~ay iàciqcly8
&`+ Zrlo`r c` fkzù~ayky na pqa el f`{ nkmaraeik`y ae~ra c`y nly zloc`ikleay na
ilgzleae~ay gqay~ra`n`y% &x 7 '?%11 ya raif`x` Fl8+
&o+ F`iar qe ke~ar}`cl na ilemk`ex` z`r` ac }arn`narl zrlgankl ±%
&i+ I`ciqc`r ac }`clr na z% &z 7 :%:00+
1%1>% Ae qe` ke}ay~kb`ikùe rac`ikle`n` ile c`y ileiae~r`ikleay na zclgl &Zo+# ya
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 306/846
Nr% Fëi~lr Pqa}anl Qrç`y
1'0:9
y`oa pqa ac zclgl ay qe }aeael gq{ zackbrlyl# ae ac iq`c ac iqarzl ya `n`z~`
irùeki`gae~a ` c`y `iqgqc`ikleay na ay~a ga~`c zay`nl% C` zrayaeik` na Zo ae ac `bq`
zl~`oca zqana }aekr na nayi rb`y kenqy~rk cay# na gke`y { na mqenkikleay na ga~ cay%F`{ `cbqely gë~lnly z`r` na~argke`r c`y ileiae~r`ikleay na Zo ae ac `bq`% Qel na
accly ay ac gë~lnl na `oylrikùe `~ùgki` ayzai~rlgë~rkil &gë~lnl @+ { ac gë~lnl
i`clrkgë~rkil &gë~lnl O+% Ae ay~` ke}ay~kb`ikùe ya zra~aena ilgz`r`r cly rayqc~`nly
na cly gë~lnly na `oylrikùe `~ùgki` { ac na nk~kxlea% Ac gë~lnl na `oylrikùe `~ùgki`
ayzai~rlgë~rki` ileyky~a ae `yzkr`r c` gqay~r` zraz`r`n` ae qe` mc`g` {
`~lgkxàenlc`% Ac gë~lnl nk~kxlea ileyky~a ae av~r`ar ae ~a~r`iclrqrl na i`rolel
&IIc>+# ac Zo ae qe` ylcqikùe ckbar`gae~a oàyki`% Cly n`~ly nao`hl n`e c`y
ileiae~r`ikleay &ae gb,C+ na nly gqay~r`y na gë~lnl @ { gë~lnl O% @yqgkr qe
ek}ac na ykbekmki`ikùe na :%:1% ^`gokëe# `yqgkr pqa c`y zloc`ikleay gqay~ra`n`y yle
elrg`cay% F`iar cly ykbqkae~ay iàciqcly8
&`+ Zrlo`r pqa el f`{ nkmaraeik` ae~ra c`y nly zloc`ikleay `e`ckx`n`y%
&o+ I`ciqc`r ac }`clr na z%
&i+ F`iar qe ke~ar}`cl na ilemk`ex` ile ΰ 7 :%:1
C` ~`oc` na `o`hl gqay~r` cly rayqc~`nly na c`y ileiae~r`ikleay na cly nly gë~lnly%
^`oc` gly~r`enl c`y gankikleay na Zo% &Ac`olr`ikùe zrlzk`+
Gë~lnl @ s %:11# %:10# %:1?# %:19# %:11# %:19# %:11# %:>;# %:>6# %:>;# %:1# %:19# %:1?# %:1># %:1=# %:1># %:12# %:>;#
%:>6# %:1# %:12# %:1;# %:>:# %:>?# %:>9# %:>=# %:11# %:9# %:2# %:21# %:6# %:6=# %:1=# %:26# %:2=# %:22
Gë~lnl O s%:12# %:=# %:2# %:12# %:1;# %:1;# %:>;# %:=# %:2# %:21# %:=# %:=2# %:=6# %:=># %:=;# %:26# %:2#
%:2;# %:2># %:1# %:=# %:2# %:6# %:60# %:2?# %:6?# %:2;# %:62# %:># %:># %:># %:>9# %:>># %:>=# %:60# %:69
1%11% Nly `y~rùelgly rabky~r`rle loyar}`ikleay na ikar~` ay~racc` ae ac mkrg`gae~l%
Ya lo~q}karle 0? loyar}`ikleay zlr ac zrkgar `y~rùelgl { nkl qe zrlgankl na 0%?:
gankikleay% Ac yabqenl `y~rùelgl y`iù qe` gqay~r` na 6 loyar}`ikleay { lo~q}l qe
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 307/846
Nr% Fëi~lr Pqa}anl Qrç`y
1'0:>
zrlgankl na 0%01 gankikleay% C` avzarkaeik` z`y`n` kenkiù pqa ay~ly `y~rùelgly
lo~q}karle gankikleay ile }`rk`ex`y na :%>: gankikleay% @yqgkr pqa c` zloc`ikùe
gqay~ra`n` ay elrg`c% Qy`r ac ek}ac na ykbekmki`ikùe na :%:1 { zrlo`r c`y fkzù~ayky8Fl8±0 ' ±? 7 : ile~r` c`y fkzù~ayky `c~are`~k}`y na F?8± 5 : { F98± 3 :% &x 7 :%02+
1%1=% Naikr na iqàe~`y ilc`y ya f`ràe c`y ykbqkae~ay zrqao`y na fkzù~ayky { naikr c`y
zrqao`y c~are`~k}`y8
&`+ Yk c` zrqao` na fkzù~ayky eqc` ay na Fl8 ± 7 0>%::# ae~leiay c`y zrqao`y
`c~are`~k}`y yle na8
&o+ Yk c` zrqao` na fkzù~ayky eqc` ay na Fl8 ± ≡ 0>%::# ae~leiay c`y zrqao`y
`c~are`~k} y yle na8
&i+ Yk c` zrqao` na fkzù~ayky ay na Fl8 ± ≮ 0>%::# ae~leiay c`y zrqao`y `c~are`~k}`y
yle na8
1%12% Qe` gqay~r` na >; loyar}`ikleay na `eàckyky na rqknly &ae naikoacay# nO+ ya qyù
z`r` zrlo`r c` fkzù~ayky eqc` na pqa ac zrlgankl zloc`ikle c ay na ± 7 0>1 nO% Ya
i`ciqcù qe zrlgankl gqay~r`c na V 7 096%:: nO ile qe` nay}k`ikùe ay~àen`r na ?:%
F`iar cly ykbqkae~ay iàciqcly8
&`+ Ay~`ocaiar c`y zrqao`y `c~are`~k}`y% &F08± ≪ 0>1+
&o+ Yk ac ek}ac na ykbekmki`eik` ay ΰ 7 :%:1 ay~`ocaiar c` rabkùe irç~ki`%&~irç~ki` 7 °0%;=+
&i+ Yk ya raif`x` c` fkzù~ayky eqc`# i`ciqc`r ac }`clr na z% &z 7 %:0>?+
1%16% Nayzqëy na `e`ckx`r c`y ~agzar`~qr`y na 1: ~r`o`h`nlray na qe mrkblrçmkil# ac
gënkil na c` agzray` `mkrg` pqa# c` ~agzar`~qr` zrlgankl zloc`ikle`c nac iqarzl# ay
kbq`c ` ;6%= lM% Ac zrlgankl ay~`nçy~kil na ay~a brqzl mqa na V 7 ;6%? lM ile qe`nay}k`ikùe ay~àen`r na ω 7 :%=?% F`iar cl ykbqkae~a8
&`+ Knae~kmki`r c` fkzù~ayky eqc` Fl8
&o+ Knae~kmki`r c` l c`y fkzù~ayky c~are`~k}`y F08
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 308/846
Nr% Fëi~lr Pqa}anl Qrç`y
1'0:1
&i+ Ay~`ocaiar c`y rabkleay irç~ki`y qy`enl ac ek}ac na ykbekmki`ikùe na ΰ 7 :%:1 { ΰ 7
:%:0%
&n+ I`ciqc`r ac }`clr na c` ay~`nçy~ki` x%&a+ Yk ya raif`x` c` fkzù~ayky eqc`# i`ciqc`r ac }`clr na c` zrlo`okckn`n# z%
1%1;% Yk ya qy` ac }`clr ykbekmki`e~a na ΰ 7 :%:0 aeile~r`r cly }`clray irç~kily na x
&x~`o%+ yk ya qy`8
&`+ Zrqao` okc`~ar`c# ay naikr na nly ilc`y% &°?%99+
&o+ Zrqao` okc`~ar`c ile ΰ 7 :%0:% &°0%?6+
&i+ Zrqao` okc`~ar`c ileΰ
7 :%::1% &°?%60+
1%=:% Ya n`e cly ykbqkae~ay n`~ly8 Zrlgankl `rk~gë~kil# V 7 90%6# ω 7 :%?1# e 7 1:#
Fl8 ± ≡ 9?% F`iar cly ykbqkae~ay iàciqcly8
&`+ Naikr iqàc ay c` zrqao` na fkzù~ayky c~are`~k}`
&o+ Ay~`ocaiar c`y rabkleay irç~ki`y qy`enl ΰ 7 :%:1 { ΰ 7 :%:0
&i+ I`ciqc`r ac }`clr na c` ay~`nçy~ki` x &xi`ci +%
&n+ Yk ya raif`x` c` fkzù~ayky ae iq`cpqkar` na cly nly ek}acay na ykbekmki`ikùe na :%:1
{,l :%:0# i`ciqc`r ac }`clr na c` zrlo`okckn`n z%
1%=0% Ya y`i` qe }`clr na e 7 ?1 na qe` zloc`ikùe elrg`c# ile y? 7 9% Qy`r ΰ 7 %:1%
F`iar cl ykbqkae~ay8
&`+ I`ciqc`r ς ?% &ς ? 7 ;%=+
&o+ Ay~kg`r c`y rabkleay irç~ki`y ς ?ΰ,?4e'0 { ς ?0'ΰ,?4e'0 &0?%> { 9;%>+
&i+ Zrlo`r Fl8ω? 7 21# { F08ω
? ≪ 21 &ya raif`x` Fl8+
&n+ I`ciqc`r ac }`clr na z% &z ≆ :%:0+1%=?% Ae qe ay~qnkl `gokae~`c faifl ae }`rkly c`bly na Elrqab`# `iari` nac zF nac
`bq`# ae rayzqay~` ` c` zraliqz`ikùe na cly amai~ly na c` zraikzk~`ikùe zcq}k`c àikn`#
ya fkikarle nly gqay~raly faifly ae cly `ðly na 0;2= { 0;60% Ya pqkara y`oar yk fqol
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 309/846
Nr% Fëi~lr Pqa}anl Qrç`y
1'0:=
nkmaraeik`y ae c`y nly gankikleay na zF na ayly `ðly 0;2= { 0;60% Cly n`~ly ya n`e
ae c` ~`oc` na `o`hl% @yqgkr pqa c`y }`rk`ex`y na c`y nly zloc`ikleay yle kbq`cay%
Qy`r qe ek}ac na ykbekmki`ikùe na :%:1 { i`ciqc`r ac }`clr na c` zrlo`okckn`n z ae c`~lg` na naikykleay% &Y~`~ky~kiy mlr Ae}krlegae~`c Yikaeiay `en G`e`bagae~ # zlr
Or{`e G`ec{# z% 6+%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 310/846
Nr% Fëi~lr Pqa}anl Qrç`y
1'0:2
^`oc` gly~r`enl c`y gankikleay na zF z`r` 0;21 { 0;60%
El% na c bl zF &0;21+ zF &0;60+0 >%1; >%=9? >%;2 >%;69 >%9? >%>;> >%;2 1%?01 >%;6 1%::= >%16 >%;>2 >%2? >%;:6 >%19 >%1>; >%;= 1%=;0: >%;= 1%2100 1%90 1%>90? 1%>? 1%0;
09 1%=: 1%2:0> 1%92 1%9601 >%62 >%;:0= 1%62 =%:?02 =%?: =%?106 =%=2 =%=20; =%:= =%:;?: 1%96 1%10?0 1%=: 1%;6?? 1%=: 1%==?9 1%92 1%=2?> 1%:2 1%06
?1 =%?9 =%?;?= =%?> =%92?2 1%01 1%=6?6 >%6? 1%>1?; 1%>? 1%1>9: >%;; 1%?190 1%90 1%119? 1%;; =%0999 >%=9 >%;?9> >%>2 >%1:91 >%=: >%==9= >%66 >%;?
92 >%=: >%6>96 >%61 >%6=9; 1%:= 1%00>: 1%;2 =%02>0 1%>2 1%6?
&Mqae~a8 Y~`~ky~kiy mlr Ae}krlegae~`c Yikaeia `en G`e`bagae~ % G`ec{# ?::0+
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 311/846
Nr% Fëi~lr Pqa}anl Qrç`y
1'0:6
Yqbaraeik`8 Qy`r c` mqeikùe ay~`nçy~ki` z`r` zrqao`y na fkzù~ayky z`r` c`y nkmaraeik`y
na nly zrlgankly%
1%=9% Ac zrayknae~a na ikar~` ilgz`ðç` m`orki`e~a na z`r~ay na `q~lgù}kc `mkrg` pqë#ac eøgarl zrlgankl na z`r~ay }aenkn`y# nk`rk`gae~a# ay na 01::% Ac nkrai~lr baear`c
na ~ln` c` i`nae` na ay~`ocaikgkae~ly pqkara ilgzrlo`r ay~` `mkrg`ikùe% Z`r` ay~l# ya
~lg` qe` gqay~r` `ca`~lrk` ileyky~ae~a ae 9= nç`y# c` iq`c gly~rù qe zrlgankl na
0>1: z`r~ay% @yqgkr pqa ya ilelia ac }`clr na ω 7 0?: z`r~ay% Qy`r ΰ 7 :%:1% I`ciqc`r
ac }`clr na c` zrqao` el ~r`nkikle`c# ay naikr# qy`enl ac }`clr na z% ¼Pqë ya zqana
ileicqkr `iari` na ay~` yk~q`ikùe< &x 7 '?%1 { ya raif`x` Fl8+
1%=>% H`{ Na}lra `q~lr nac ckorl Zrlo`okckn`n { Ay~ nçy~ki` z`r` Kebaekarç` {
Ikaeik`y &??:0+ nkyiq~a ac zrlocag` rac`ikle`nl ile ac `eàckyky na qe` gqay~r`
`ca`~lrk` na e0 7 ?: ayzaikgaeay na `iarl c`gke`nl ae mrçl# z`r` na~argke`r yq
rayky~aeik`# n`enl# ilgl rayqc~`nl# qe` rayky~aeik` zrlgankl gqay~r`c na V 0 7 ?;%6
dyk% Qe` yabqen` gqay~r` `ca`~lrk` na e? 7 ?1 ayzaikgaeay na `iarl b`c}`ekx`nl na
nly c`nly nkl qe` rayky~aeik` zrlgankl gqay~r`c na V ? 7 9>%2 dyk% Yk ya yqzlea pqa
c`y nly nky~rkoqikleay na rayky~aeik` na cly `iarly yle elrg`cay ile ω0 7 >%: { ω? 7 1%:
dyk &yqbarkn`y zlr qe` bràmki` ae ac `r~çiqcl –Ykei'Il`~an Yfaa~ Y~aac8 @e L}ar}ka‗#
@q~lgl~k}a Aebr%# nkikagora na 0;6># zz% 9;'>9+%
&`+ ¼Ykbekmki`e ay~ly n`~ly pqa c`y }arn`nar`y rayky~aeik`y zrlgankl ±0 { ±? yle
nkmarae~ay<
&o+ I`ciqc`r ac }`clr na z%
&i+ ^`gokëe f`iar qe ke~ar}`cl na ilemk`ex` z`r` cly nly zrlgankly zloc`ikle`cay%_a`ckx`r c` zrqao` na fkzù~ayky ile ΰ 7 :%:0%
1%=1% Ae qe ay~qnkl na fkbkaea kenqy~rk`c { yabqrkn`n ae i`rra~ar`y ay~`~`cay# `c
yacaiikle`r qe ileira~l na `xqmra z`r` iley~rqkr qe` i`rra~ar`# ay kgzlr~`e~a ayilbar
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 312/846
Nr% Fëi~lr Pqa}anl Qrç`y
1'0:;
qe ileira~l ile o`hl }`clr na ilenqi~k}kn`n ~ërgki`# z`r` ranqikr `c gçekgl cly
n`ðly li`ykle`nly zlr i`gokly na ~agzar`~qr` {# `yç# a}k~`r `iiknae~ay
`q~lgl}kcçy~kily ae c`y i`rra~ar`y% Yqzùeb`ya pqa f`{ nly ~kzly na ileira~l# qel ayqe `brab`nl ayi`cle`nl { ac l~rl el ~kaea `brab`nly mkely ileyknar`nly z`r` ikar~`
i`rra~ar`% C` ~`oc` na `o`hl rayqga cly n`~ly na qe avzarkgae~l ra`ckx`nl z`r`
ilgz`r`r cly nly ~kzly na ileira~l% ¼Yqbkara ay~` kemlrg`ikùe pqa ac }arn`narl
zrlgankl na ilenqi~k}kn`n nac ileira~l# ile `brab`nl ayi cle`nl yqzar` `c nac
ileira~l yke `brab`nl mkel< & Zrlo`okckn`n { Ay~ nçy~ki` z r` Kebaekarç { Ikaeik y#
H% C%Na}lra# ?:::+% &9%9=# z 7 %:::>+
^kzl na ileira~l ^`g`ðl Zrlgankl gqay~r`c Nay}k`ikùegqay~r`c na ilenqi~k}kn`n ay~àen`r
Ayi`cle`nl >? %>6= %062
Yke `brab`nly mkely >? %91; %016
&Mqae~a8 Na}lra# ?:::+
1%==% Ac barae~a na qe` i`nae` na fl~acay ay~à ileyknar`enl iley~rqkr qe gl~ac ` cl
c`rbl na qe` `q~lzky~`% Ac nqaðl pqa ay~à }aenkaenl ac ~arrael `c barae~a# z`r` c`
iley~rqiikùe nac gl~ac# `yabqr` pqë# zlr `fç z`y`e 00:: }afçiqcly zlr nç`% Yke
ago`rbl# ac barae~a na c` i`nae` na fl~acay nkia pqa# qe` ikmr` g`{lr pqa 00::
}afçiqcly# yarç` `naiq`n` z`r` c` iley~rqiikùe nac gl~ac ae aya yk~kl% Z`r` ay~l ya
~lg` qe` gqay~r` `ca`~lrk` nqr`e~a 06 nç`y% ¼Cly rayqc~`nly ra`mkrg`e l nay`zrqao`e
c` `mkrg`ikùe nac nqaðl nac ~arrael< C` ~`oc` na `o`hl n` c` kemlrg`ikùe rapqarkn`8
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 313/846
Nr% Fëi~lr Pqa}anl Qrç`y
1'00:
^`oc` gly~r`enl cly n`~ly nac zrlocag`% &Ac`olr`ikùe zrlzk`+%
Nç` s 0 ? 9 > 1 = 2 6 ; 0: 00 0? 09 0> 01 0= 02 06
El% na }afçiqcly s001: 0??1 00;1 00;1 0?0: 00:: 001: 00;1 00:1 0?:1 00?0 00;: 00;1 00;? 00:: 0?:0 0:;: 0:;1
1%=2% Aeile~r`r c`y ykbqkae~ay zrlo`okckn`nay8
&`+ Z&'0%: 3 X 3 ?%:+ &:%6061+
&o+ C` zrlo`okckn`n na pqa c` }`rk`oca `ca`~lrk` X el ya aeiqae~ra ae~ra ay~ly nly
}`clray% &:%0601+
&i+ Yk V 7 > { y 7 0# aeile~r`r cly }`clray na c` }`rk`oca `ca`~lrk` V z`r` ac ke~ar}`cl
nac keikyl &`+% & V 7 =+
1%=6% C` }kn` zrlgankl na qe zay~kikn` ùrb`el iclr`nl nazlyk~`nl ae c` ~karr` ay qe`
}`rk`oca `ca`~lrk` ile c` ykbqkae~a nky~rkoqikùe na zrlo`okckn`n8
m & v+ 7 0,=: a' v,=: z`r` v ≡ :
&`+ N`r ac zrlgankl nac zay~kikn` ae iqay~kùe%
&o+ Ay~kg`r c` zrlo`okckn`n na pqa ac zay~kikn`# ae iqay~kùe# nqra 0:: nç`y%
1%=;% Ya y`iù qe` gqay~r` `c `x`r na >; `eàckyky na `bq`y rayknq`cay { ya i`ciqcù V 7
6:: gb,C ile y 7 =:%: gb,C% Zrlo`r c` fkzù~ayky eqc` na pqa ac }arn`narl zrlgankl
ay na 61: gb,C% @yqgkr ΰ 7 :%:1% I`ciqc`r ac }`clr na z% &x 7 '1%69# z 7 %:::9+%
1%2:% Ya y`iù qe` gqay~r` `ca`~lrk` na YL9 `~glymërkil ae qekn`nay na zzg
zrl}aekae~ay na qe ilgzcahl kenqy~rk c% Zrlo`r pqa ± 7 1?% Ya y`oa pqa c` zloc`ikùe
gqay~ra`n` ay elrg`c% I`ciqc`r ac }`clr na z ile ΰ 7 :%:0%
^`oc` gly~r`enl cly n`~ly% &Ac`olr`ikùe zrlzk`+ \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ YL9 &zzg+ s 1: 1? 1= 12 11 11 1> 11 1= 12 1= 1>
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 314/846
Nr% Fëi~lr Pqa}anl Qrç`y
1'000
1%20% Ae qe ay~qnkl rac`ikle`nl ile ac `flrrl na ilgoqy~koca# ya y`oa pqa ac >: na
cly ilifay el `garki`ely na > ikckenrly# ac ileyqgl na b`ylcke` ya ranqia
ileyknar`ocagae~a# ay naikr# ile rac`ikùe ` cly ilifay `garki`ely na = q 6 ikckenrly%Yk ya y`i` qe` gqay~r` `ca`~lrk` na 01 ilifay na > ikckenrly# i`ciqc`r c` zrlo`okckn`n
na pqa > na ay~ly ilifay ya`e amkikae~ay ae ac `flrrl na ilgoqy~koca% F`iar ay~l
qy`enl c` nky~rkoqikùe okelgk`c { c` elrg`c% Ilgz`r`r cly rayqc~`nly% &Qy`enl c`
nky~rkoqikùe elrg`c n` :%0?0>4 qy`enl c` nky~rkoqikùe okelgk`c n` :%0?=6+
1%2?% Yqzùeb`ya pqa ac ~kagzl na ra`iikùe V ` ikar~l ay~çgqcl ae qe kenk}knql
yacaiikle`nl `ca`~lrk`gae~a# ~kaea qe` nky~rkoqikùe b`gg` ay~àen`r ileΰ
7 ?y
&Na}lra# ?::0+% Yqbaraeik`8 qy`r c` rac`ikùe Z&` ≮ V ≮ o+ 7 M&o+ ’ M&`+% Qy`r c` ~`oc`
na c` mqeikùe na b`gg` keilgzca~`%
1%29% Ay~a ay qe zrlocag` pqa ke}lcqir` ac qyl na c` nky~rkoqikùe b`gg` ae nlena
`z`raiae nky~rkoqikleay pqa el yle ay~àen`r% Ay~a zrlocag` nkia `yç &Na}lra ?::0# z%
020+8 Yqzùeb`ya pqa ac ~kagzl V na yqzar}k}aeik`# ae yag`e`y# na qe r`~ùe g`ifl#
yacaiikle`nl `c `x`r { avzqay~l ` ?>: r`ny na r`nk`ikùe b`gg`# ~kaea qe` nky~rkoqikùe
b`gg` ile ΰ 7 6 { ο 7 01% Ac ~kagzl ayzar`nl na yqzar}k}aeik` ay na A& V + 7 &6+&01+
7 0?: yag`e`y# ae ~`e~l pqa# ]& V + 7 &6+&01+? 7 06:: { ωv 7 ∘06:: 7 >?%>9 yag`e`y%
Ykaenl `yç# aeile~r`r c` zrlo`okckn`n na pqa qe r`~ùe ylora}k}`8
&`+ Ae~ra =: { 0?: yag`e`y% &%>;=+
&o+ Zlr cl gaely 9: yag`e`y% &%;;;+
1%2>% Ya` V c` rayky~aeik` mke`c ` c` ~aeykùe &dyk+ ` '?:: lM na qe ~kzl na ga~`c pqa
zrayae~` zrlocag`y na rayky~aeik` ` ~agzar`~qr`y o`h y% Yqzùeb`ya pqa V ~kaea qe`nky~rkoqikùe Rakoqcc ile z`ràga~rly ΰ 7 ?: { ο 7 0::% I`ciqc`r cl ykbqkae~a8
&`+ C` zrlo`okckn`n na pqa c` rayky~aeik` mke`c ` c` ~aeykùe &dyk+ ` '?:: lM ya na ` cl
gày 0:1% &%;9:+
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 315/846
Nr% Fëi~lr Pqa}anl Qrç`y
1'00?
&o+ Ae~ra ;6 { 0:?
1%21% Ac aei`rb`nl na c` i`ya~` na ilorl na qe` i`rra~ar` f` loyar}`nl pqa cly
}afçiqcly ccab`e `ca`~lrk` a kenazaenkae~agae~a# ile qe zrlgankl na 9:: }afçiqcly zlr flr`% Yqbaraeik`8 qy`r Z& V ≡ `+ 7 a'δ` z`r` &`+ { z& v+ 7 a'±v, v! z`r` &o+% Ykaenl `yç#
raylc}ar cly ykbqkae~ay aeqeik`nly8
&`+ Qy`r c` mqeikùe avzleaeik`c z`r` i`ciqc`r c` zrlo`okckn`n na iq`enl gaely 0
gkeq~l z`y`rà `e~ay na pqa ac ykbqkae~a gl~lrky~` ccabqa% &Z& V ≡ 0%:+ 7 a'1&0++
&o+ Qy`r c` nky~rkoqikùe na Zlkyyle z`r` ilgz`r`r ac }`clr na c` zrlo`okckn`n lo~aekn`
ae &`+%
1%2=% C` nqr`ikùe na ikar~` ram`iikùe z`r` `q~lgù}kcay ykbqa ` qe` nky~rkoqikùe
Rakoqcc ile qe` ~`y` na m`cc` @&~+ 7 0,∘ ~% Ykaenl `yç# aeile~r`r c`y ykbqkae~ay
zrlo`okckn`nay8
&`+ C` zrlo`okckn`n na pqa c` ram`iikùe ae iqay~kùe ya na~arklra `e~ay na > `ðly%
&o+ C` zrlo`okckn`n na pqa c` ram`iikùe el ya nayb`y~a nayzqëy na > `ðly%
1%22% Qe` o`~arç` ylc`r ~kaea qe` }kn` zrlgankl pqa ay~à avzleaeik`cgae~a
nky~rkoqkn` ile qe zrlgankl na }kn` na 0: flr`y% Qy`enl iàciqcl ke~abr`c# na~argke`r
c`y ykbqkae~ay zrlo`okckn`nay8
&`+ C` gank`e` na c`y }kn`y na c`y o`~arç`y% &=%;9 flr`y+
&o+ C` zrlo`okckn`n na pqa c` }kn` na qe` o`~arç` ay~ë ae~ra 6 { 0? flr`y% &:%0>6
flr`y+
&i+ C` zrlo`okckn`n na pqa c` }kn` na qe` o`~arç` avianarà 01 flr`y% &:%222 flr`y+
&n+ C` zrlo`okckn`n na pqa c` }kn` na qe` o`~arç` ylc`r ay~ë ae~ra =: { 0?: gkeq~ly%1%26% Naikr iqàc na c`y ykbqkae~ay `mkrg`ikleay ay ilrrai~`8
&`+ @ gankn` pqa ac ~`g`ðl na c` gqay~r`# e nkygkeq{a { c` nay}k`ikùe ay~àen`r# y
`qgae~`# ac }`clr na c` zrlo`okckn`n# z nkygkeq{a%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 316/846
Nr% Fëi~lr Pqa}anl Qrç`y
1'009
&o+ @ gankn` pqa e nkygkeq{a { y nkygkeq{a# ac }`clr nac arrlr ay~àen`r `qgae~` {#
zlr cl ~`e~l# ac }`clr na z nkygkeq{a%
&i+ @ gankn` pqa e `qgae~` { c`y ~ëieki`y nac c`olr`~lrkl ya ramke`e i`qy`enl qe`}`rk`ex` zapqað`# ac arrlr ay~àen`r nac zrlgankl o`h` {# zlr ileykbqkae~a# ac }`clr
na z `qgae~` { c` fkzù~ayky eqc` ya raif`x`%
&n+ @ gankn` pqa ac arrlr ay~àen`r nac zrlgankl nkygkeq{a zlr ~`g`ðly na
gqay~r` br`enay# ile zapqað`y }`rk`ikleay# ay~l ilecca}` ` qe }`clr zapqaðl na z
gqifl gq{ ykbekmki`e~a# cl iq`c ely cca}` ` ra~aear c` fkzù~ayky eqc`%
&a+ @ gankn` pqa c` }`rk`ex` nkygkeq{a# ile e iley~`e~a# ac }`clr na z nkygkeq{a {
c` fkzù~ayky eqc` ya raif`x`%
&m+ @ gankn` pqa e `qgae~` { c`y ~ëieki`y nac c`olr`~lrkl ya ramke`e i`qy`enl qe`
}`rk`ex` zapqað`# ac arrlr ay~àen`r o`h` {# zlr ileykbqkae~a# ac }`clr na z
nkygkeq{a { ya ra~kaea Fl8
&b+ @ gankn` pqa e `qgae~` { c`y ~ëieki`y nac c`olr`~lrkl ya ramke`e i`qy`enl qe`
}`rk`ex` zapqað`# ac arrlr ay~àen`r o`h` {# zlr ileykbqkae~a# ac }`clr na z
nkygkeq{a { ya `iaz~` F@8
&f+ cly keikyly &n+# &a+ { &m+ yle ilrrai~ly
&k+ Cly keikyly &a+ { &b+ yle ilrrai~ly
1%2;% @i~q`cgae~a# f`{ gqifl nao`~a# zlr y`oar yk c`y agkykleay na i`gzly
acai~rlg`beë~kily zrlnqiknly zlr ~acëmlely gù}kcay &iacqc`ray+ { yqy ay~`ikleay na
`e~ae`y o`ya zqan`e ay~`r `mai~`enl c` y`cqn% Ile gày na 1:: gkccleay na
~acëmlely gù}kcay ae ~lnl ac gqenl# na `iqarnl ` `c `r~çiqcl Av`gkekeb ~fa ammai~y
lm acai~rlg`bea~ki mkacny agk~~an o{ BYG glokca zfleay le fqg`e a}ae~'rac`~an
zl~ae~k`cy `en zarmlrg`eia nqrkeb `e `qnk~lr{ ~`yd zqocki`nl ae Ickeki`c
Eaqrlzf{yklclb{ 001 &?:>+ 020' 026 &f~~z8,,%l'ilg%ilg,kenqy~r{,y~`~y+#
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 317/846
Nr% Fëi~lr Pqa}anl Qrç`y
1'00>
ac naygayqr`nl keiragae~l nac qyl na c` ~acamleç` iacqc`r { yqy ileykbqkae~ay
amai~ly ae c`y mqeikleay ilbek~k}`y { mkyklcùbki`y naoknl ` c`y r`nk`ikleay
acai~rlg`beë~ki`y &_A+# ay qe` yk~q`ikùe pqa ay~à i`qy`enl zraliqz`ikùe ae~ra c`y zaryle`y ilelianlr`y na ay~a zrlocag`% Ya f`e faifl gqif`y ke}ay~kb`ikleay ile
rac`ikùe ` cly amai~ly ae c` y`cqn zrlnqiknly zlr c` r`nk`ikùe na gkirllen`y
naokn`y ` c` zrlvkgkn`n na cly ~acëmlely iacqc`ray ` c` i`oax` nac qyq`rkl { na c`
zrlvkgkn`n ` c`y ay~`ikleay na `e~ae`y o`ya na ~acamleç` iacqc`r# ` ay~`ikleay
acëi~rki`y# ` cçea`y na na `c~` ~aeykùe# flrely na gkirllen`y# a~i% C` g`{lrç` na
ay~`y ke}ay~kb`ikleay ilkeiknae ae pqa cly amai~ly na ay~ly ~kzly na _A ay~àe
`mai~`enl ac iaraorl { `c yky~ag` ear}klyl ae g`{lr l gaelr br`nl% F`{ ay~qnkly
pqa f`e rac`ikle`nl c`y agkykleay acai~rlg`beë~ki`y ile i`yly na iàeiar ae ac
iaraorl# amai~ly ae c` `i~k}kn`n aexkgà~ki` { ayzargà~ki`# amai~ly }kyq`cay {
`qnk~lrkly# zra}`caiç` na nlclray na i`oax` ae~ra cly qyq`rkly na ~acëmlely gù}kcay#
zrlocag`y ile ac yqaðl# amai~ly ae c`y iëcqc`y ckemà~ki`y fqg`e`y# gq~`ikleay# a~i%#
na c`y zaryle`y avzqay~`y% Ae iq`e~l ` c` zrlvkgkn`n na c`y `e~ae`y o`ya na
~acamleç` gù}kc# { yqy amai~ly ae c` y cqn# `cbqely z`çyay f`e ay~kzqc`nl# ilgl qe
irk~arkl yabqrl# ac ay~`ocaikgkae~l na c`y `e~ae`y na gkirllen`y ` nky~`eik`y
gçekg`y na =:: ga~rly na ilgzcahly f`ok~`ikle`cay% Ykaenl `yç# ya nkyað` qe
ahagzcl fkzl~ë~kil rac`ikle`nl ile c`y gankikleay na r`nk`ikùe acai~rlg`beë~ki` {
c` zrlvkgkn`n ` c` mqae~a agkylr`# ay naikr# na gankikleay ` nkmarae~ay nky~`eik`y
na c`y `e~ae`y o`ya na ~acamleç` iacqc`r% Z`r` ay~a ahagzcl# ae z`r~kiqc`r# ya i`ciqc`e
cly zrlgankly na qe` gqay~r` na 9: gankikleay na r`nk`ikùe acai~rlg`beë~ki`# z`r` i`n` qe` na c`y ykbqkae~ay nky~`eik`y8 ?1# 1:# 0::# ?::# 9::# >::# 1:: { =::
ga~rly na c` `e~ae` o`ya na ~acëmlely gù}kcay% Cly zrlgankly na c` r`nk`ikùe
acai~rlg`beë~ki` z`r` i`n` nky~`eik` yle8 ;1: GFx# 6:: GFx# 11: GFx# >::
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 318/846
Nr% Fëi~lr Pqa}anl Qrç`y
1'001
GFx# 0;1 GFx# 6:#::: Fx# 9:#::: Fx { 1:: Fx# rayzai~k}`gae~a% Yqy rayzai~k}`y
nay}k`ikleay ay~àen`ray mqarle 1: GFx# >: GFx# 91 GFx# 6: GFx# 0:: GFx#
?:#::: Fx# 0:#::: Fx { 0:: Fx% Ac ay~qnkl ya cca}ù ` i`ol nqr`e~a ~lnl qe `ðl# aeqe aymqarxl zlr a}`cq`r }`rk`ocay# ilgl c` nky~`eik`# c` `c~qr`# ëzli` nac `ðl#
m`i~lray ga~alrlcùbkily &ilgl ~agzar`~qr`# zraykùe `~glymërki`# ke~aeykn`n {
nkraiikùe nac }kae~l# fqgan`n rac`~k}`+# ile~`gke`ikùe nac `kra zlr z`r~çiqc`y {
b`yay# a~i%# pqa zqnkar`e `mai~`r ac zlnar na c` naeykn`n na c` r`nk`ikùe
acai~rlg`beë~ki` agk~kn`% Z`r` raylc}ar ay~a zrlocag` ay~kg`r ac glnacl
g`~agà~kil pqa gahlr `hqy~a cly n`~ly% Qe` }ax pqa ya a}`cøa ac glnacl
`ilrnagae~a# zranaikr c` r`nk`ikùe na gkirllen`y ` qe` nky~`eik` na 0: { 0:::
ga~rly na c` `e~ae` o`ya% Yk fqokaya }`clray `~çzkily av~ragly# aecky~`r ~ray
zlykocay m`i~lray pqa zqan`e avzcki`r ay~`y yk~q`ikleay%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 319/846
Nr% Fëi~lr Pqa}anl Qrç`y
='0
I@ZK^QCL =
Nky~rkoqikleay na ~ na Ay~qnk`e~a# HK iq`nr`n` { M
Zrlzkan`nay na c` nky~rkoqikùe na ~ na Ay~qnk`e~a%' Ke~ar}`cly na ilemk`ex` z`r`
ac zrlgankl zloc`ikle`c ±%' Zrqao` na fkzù~ayky z`r` ±%' Zrqao` na ~ z`ra`n`
z`r` na~ai~`r nkmaraeik`y ae~ra nly ~r`~`gkae~ly%' Zrqao` na ~ z`r` zrlo`r c`
fkzù~ayky na nly zrlgankly# iq`enl c`y }`rk`ex`y yle kbq`cay%' Zrqao` na ~ z`r`
zrlo`r c` fkzù~ayky na nly zrlgankly iq`enl c`y }`rk`ex`y yle naykbq`cay%'
Gai`ekygly z`r` i`ciqc`r ac }`clr na z iq`enl ya f`iae zrqao`y na fkzù~ayky el
~r`nkikle`cay%' Ke~ar}`cly na ilemk`ex` { zrqao`y na fkzù~ayky ile c` HK iq`nr`n`#
&ς ?+%' @zcki`ikùe na c` HK iq`nr`n` ae iq`e~l ` c` zrqao` na olen`n na `hqy~a
ilgz`r`enl c`y mraiqaeik`y loyar}`n`y { c`y mraiqaeik`y ~aùrki`y%' Nky~rkoqikùe
M { yq `zcki`ikùe ae c` ilgz`r`ikùe na }`rk`ex`y gqay~r`cay%'
@pqç# nkyiq~kragly c` nky~rkoqikùe na ~ na Ay~qnk`e~a# pqa ay~à rac`ikle`n` ile c`
~alrç` na gqay~ral zapqaðl% ^`gokëe# nkyiq~kragly c` nky~rkoqikùe na HK iq`nr`n` {
c` nky~rkoqikùe na M%
Ae cly i`zç~qcly `e~arklray fkikgly fkei`zkë na pqa# z`r` gqay~r`y pqa mqar`e
≡ 9: i`yly# ya qy` c` nky~rkoqikùe elrg`c% Yke ago`rbl# z`r` gqay~r`y gaelray pqa
9: loyar}`ikleay ya qy` cl pqa ya cc`g` ~alrç` na gqay~ral zapqaðl# pqa ay~à
rac`ikle`n` ile c` nky~rkoqikùe na ~ na Ay~qnk`e~a# ile c` HK iq`nr`n` l ile c`
nky~rkoqikùe M% C` nky~rkoqikùe na ~ ya elgorù nayzqëy na R%Y% Blyya~# pqkae qyù ac
yaqnùekgl na ay~qnk`e~a%
Zlr ahagzcl# iq`enl qy`gly c` nky~rkoqikùe elrg`c ykagzra ya ilelia ac }`clr
na ω# ac ~`g`ðl na c` gqay~r` ay 5 9: { ya y`oa pqa c` nky~rkoqikùe gqay~ra`n` ay
elrg`c% Zarl iq`enl qy`gly c` nky~rkoqikùe na ~ na Ay~qnk`e~a# el ya ilelia ω { ac
~`g`ðl na c` gqay~r` ay gaelr pqa 9: i`yly# yke y`oar yk c` nky~rkoqikùe gqay~ra`n`
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 320/846
Nr% Fëi~lr Pqa}anl Qrç`y
='?
ay elrg`c l el% Ay~`y yk~q`ikleay ya avzcki`e qy`enl c`y mùrgqc`y na c` nky~rkoqikùe
elrg`c { c` na ~ na Ay~qnk`e~a%
Nac I`zç~qcl 1# {` y`oagly pqa# z`r` `zcki`r c` nky~rkoqikùe elrg`c ya qy` c`}`rk`oca `ca`~lrk` elrg`c ay~`en`rkx`n` x# n`n` ilgl x 7 & V ' ±+ , ω,∘e% Yke
ago`rbl# ay~` mqeikùe ~kaea qe qyl ckgk~`nl# zlrpqa c` }`rk`ex` ω? na c` zloc`ikùe
r`r` }ax ya ilelia { zlrpqa c` zloc`ikùe gqay~ra`n` naoa yar elrg`c l
`zrlvkg`n`gae~a elrg`c% C` nky~rkoqikùe na ~ na Ay~qnk`e~a el ~kaea ay~` ckgk~`ikùe#
zlrpqa `øe# z`r` gqay~r`y na e 3 9: i`yly# ya `yqga pqa ω 7 y% @yç yqy~k~q{aenl ac
}`clr na ω zlr y c` mqeikùe na ~ na Ay~qnk`e~a ely n`8~ 7 & V ' ±+ , y,∘e &='0+
Nlena8
V 7 zrlgankl gqay~r`c
± 7 zrlgankl zloc`ikle`c pqa ya pqkara zrlo`r
y 7 nay}k`ikùe ay~àen`r gqay~r`c
e 7 ~`g`ðl na c` gqay~r`
y,∘ e 7 arrlr ay~àen`r nac zrlgankl
Zrlzkan`nay na c` nky~rkoqikùe na ~ na Ay~qnk`e~a
C` nky~rkoqikùe na ~ na Ay~qnk`e~a ay qe` m`gkck` na nky~rkoqikleay# i`n` qe`
i`r`i~arkx`n` zlr ac eøgarl na br`nly na ckoar~`n η% Ay ykgkc`r ` c` nky~rkoqikùe na x
elrg`c# ile zrlgankl kbq`c ` iarl { ay ykgë~rki` ae mlrg` na i`gz`e`% Yq mlrg`
nazaena ae ac ~`g`ðl na c` gqay~r`% Ile ~`g`ðly na gqay~r`y zapqað`y# c` mlrg` na
ay~` iqr}` ay gaely zkiqn` pqa c` elrg`c# zarl ` gankn` pqa e ccab` ` 9: i`yly l ya}` ` kemkek~l# y? ya `zrlvkg` ` ω? { c` ~ na Ay~qnk`e~a ya `zrlvkg` ` c` nky~rkoqikùe
elrg`c% C` bràmki` na `o`hl gqay~r` c` nky~rkoqikùe na ~ na Ay~qnk`e~a# ile nkmarae~ay
br`nly na ckoar~`n%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 321/846
Nr% Fëi~lr Pqa}anl Qrç`y
='9
Mkbqr` =%:% Bràmki` gly~r`enl m`gkck`y na iqr}`y na c` nky~rkoqikùe na ~ na Ay~qnk`e~a
ile nkmarae~ay br`nly na ckoar~`n η# nagly~r`enl iùgl# ` gankn` pqa η ↟ ∕# c`
nky~rkoqikùe ~ ya `zrlvkg` ` c` nky~rkoqikùe elrg`c%
Nkmaraeik`y ae~ra c` nky~rkoqikùe na ~ na Ay~qnk`e~a { c` nky~rkoqikùe elrg`c
C` nky~rkoqikùe na ~ ya qy` ae cqb`r na c` nky~rkoqikùe elrg`c# iq`enl ac ~`g`ðl na c`
gqay~r` ay gaelr pqa 9: i`yly% Iq`enl f`oc`gly na c` nky~rkoqikùe elrg`c# ëy~`
rapqkara pqa c` gqay~r` ya` na e ≡ 9: loyar}`ikleay l pqa# c` zloc`ikle`c gqay~ra`n`ya` elrg`c% Ay~a ~`g`ðl na gqay~r` ya ileyknar` ilgl qe` gqay~r` br`ena% Zarl
iq`enl c` gqay~r` na i`yly ay e 3 9: loyar}`ikleay# el ya zqana qy`r c` iqr}` elrg`c
{ ~aeagly pqa qy`r cl pqa ya cc`g` /~alrç` na gqay~ral zapqaðl%/ Z`r` ~`cay amai~ly
ya qy` c` nky~rkoqikùe na ~ na Ay~qnk`e~a# c` HK iq`nr`n` l c` nky~rkoqikùe# M% C`
ay~`nçy~ki` ~ ya qy` z`r` ilgz`r`r cly zrlgankly na nly nky~rkoqikleay# gkae~r`y pqa c`
zrqao` na M ya qy` z`r` ilgz`r`r c`y }`rk`ex`y na nly nky~rkoqikleay%
Na faifl# c`y nkmaraeik`y ae~ra c` nky~rkoqikùe na ~ { c` nky~rkoqikùe elrg`c yle
pqa c` nky~rkoqikùe ~ el eaiayk~` ac z`ràga~rl na zloc`ikùe# ω# gkae~r`y pqa c` elrg`c
yk cl rapqkara% @nagày# c` mqeikùe ~ el rapqkara na gqay~r`y br`enay% L~rlyç# c`
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 322/846
Nr% Fëi~lr Pqa}anl Qrç`y
='>
}`rk`ex`# ω? 5 0 {# ylc`gae~a# iq`enl e ↟ ∕ ae~leiay# `go`y nky~rkoqikleay yle
kbq`cay &zrài~ki`gae~a# iq`enl e ≡ 9: i`yly+%
Mqeikleay qy`n`y ile c` nky~rkoqikùe na ~ na Ay~qnk`e~a
0% Ya qy` z`r` f`iar ke~ar}`cly na ilemk`ex` z`r` ±%
?% Ya qy` z`r` zrlo`r c` fkzù~ayky na pqa ± ~kaea qe }`clr na~argke`nl# ilgl zlr
ahagzcl# Fl8± 7 ±l%
9% Ya qy` z`r` zrlo`r nkmaraeik`y ae~ra nly ~r`~`gkae~ly nackoar`n`gae~a agz`rah`nly#
ay~l ay# Fl8±0 ' ±? 7 :% @pqç# cly ~`g`ðly na nky~rkoqikleay naoae yar kbq`cay%
>% Ya `zcki` z`r` zrlo`r nkmaraeik`y ae~ra nly zrlgankly qy`enl ac gë~lnl na
yacaiikùe ilgzca~`gae~a `c `x`r &`ca`~lrkl+# { ile }`rk`ex`y kbq`cay% @pqç cly
~`g`ðly na c`y nky~rkoqikleay zqanae yar kbq`cay l naykbq`cay%
1% Ya `zcki` z`r` yacaiikleay ilgzca~`gae~a `ca`~lrk`y &`c `x`r+ ile }`rk`ex`y
naykbq`cay% Ac ~`g`ðl nac c`y nky~rkoqikleay zqana yar kbq`c l naykbq`c%
@zcki`ikleay na c` nky~rkoqikùe na ~ na Ay~qnk`e~a
C`y `zcki`ikleay na c` ~ na Ay~qnk`e~a yle }`rk`y% Zlr ahagzcl# zqana qy`rya z`r` ac
ile~rlc na c` i`ckn`n kenqy~rk`c% ^`gokëe ay gq{ ø~kc z`r` ac ile~rlc na c` i`ckn`n na
qe yky~ag` na ~r`~`gkae~l na `bq`y rayknq`cay ae ac i`gzl na c` kebaekarç` `gokae~`c%
Zlr l~r` z`r~a# l~r` `zcki`ikùe gq{ kgzlr~`e~a na c` ~ na Ay~qnk`e~a# ay c`
nky~rkoqikùe z`ra`n`% Ay~l ay# z`r` ilgz`r`r ac zrlgankl na nly nky~rkoqikleay l
~r`~`gkae~ly# ilgl# zlr ahagzcl# z`r` zrlo`r c` fkzù~ayky eqc` na Fl8±0 7 ±?# ay naikr#
pqa el f`{ nkmaraeik`y ae~ra cly nly zrlgankly% @pqç# zqnkër`gly ay~kg`r nly ~kzly
na `eàckyky qy`enl nly gë~lnly { ~r`~`gkae~ly# nkb`gly na lvçbael nkyqac~l &LN+ lc` ilgz`r`ikùe na nly gë~lnly ae c` kebaekarç` nac `bq` ilgl ac gë~lnl Rkedcar {
ac na acai~rlnly { }ar yk f`{ nkmaraeik`y ae~ra cly nly gë~lnly qy`enl c` zrqao` na ~
`z`ra`n`y% ^`gokëe ya zqana qy`r z`r` ilgz`r`r nly nky~rkoqikleay yacaiikle`n`y#
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 323/846
Nr% Fëi~lr Pqa}anl Qrç`y
='1
`ca`~lrk`gae~a# {# ile }`rk`ex`y kbq`cay l naykbq`cay% @pqç# i`oa el~`r pqa# yk ya ~r`~`
na ilgz`r`r cly zrlgankly na gày na ? nky~rkoqikleay# ae~leiay ya qy` ac `eàckyky na
}`rk`ex` ykgzca l gøc~kzca%Nayirkzikùe na c`y mqeikleay qy`n`y ile c` nky~rkoqikùe na ~ na Ay~qnk`e~a
Ay~`nçy~ki` nayirkz~k}`8
e
Zrlgankl8 V 7 Ω Vk , e &='?+
v7:
]`rk`ex` gqay~r`c8 y? 7 T∐ V ? ’ &∐ V +? ,e W , e ' 0 &='9+
Nay}k`ikùe ay~àen`r8 y 7 ∘ y? &='9`+
Ke~ar}`cly na ilemk`ex` z`r` ac zrlgankl zloc`ikle`c ±8
Zrlot V ' ~T0 'ΰ,?4 ηW y,∘e 3 ± 3 V $ ~T0 ' ΰ,?4 ηw y,∘ew 7 0 ' ΰ &='>+
Nlena8
V 7 zrlgankl gqay~r`c
~T0’ ΰ,?4 ηW 7 }`clr zlriae~q`c na ~ ile qe ek}ac na ykbekmki`eik` ΰ# ile η br`nly na
ckoar~`ny 7 nay}k`ikùe ay~àen`r
e 7 ~`g`ðl na c` gqay~r`
y,∘ e 7 arrlr ay~àen`r nac zrlgankl
Zrqao` na fkzù~ayky z`r` ac zrlgankl zloc`ikle`c ±
~ 7 & V ’ ±l+ , y,∘ e &='1+
Nlena8 V 7 zrlgankl gqay~r`c
±l 7 zrlgankl zloc`ikle`c pqa ya naya` zrlo`r
y 7 nay}k`ikùe ay~àen`r na c` gqay~r`
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 324/846
Nr% Fëi~lr Pqa}anl Qrç`y
='=
Zrqao` na ~ z`r` loyar}`ikleay z`ray# z`r` na~ai~`r nkmaraeik`y ae~ra nly
~r`~`gkae~ly
~ 7 & N ' ±n + , &y n ,∘e+ &='=+Nlena8
N 7 Zrlgankl na c` gqay~r` na c`y nkmaraeik`y na c`y loyar}`ikleay nac z`r na
nky~rkoqikleay
yn 7 Nay}k`ikùe ay~àen`r na c`y nkmaraeik`y na c`y loyar}`ikleay nac z`r na
nky~rkoqikleay
e 7 eøgarl na loyar}`ikleay±n 7 :
Zrqao` na fkzù~ayky z`r` c` nkmaraeik` ae~ra nly zrlgankly zloc`ikle`cay% Ay~`
mqeikùe ~`gokëe `zcki` iq`enl c`y }`rk`ex`y na c`y nly nky~rkoqikleay yle kbq`cay
{ elrg`cay%
& V 0 ' V ?+ ' &±0 ' ±?+
~ 7 ┊┊┊┊┊┊┊┊┊┊┊┊┊┊┊┊┊┊┊┊┊┊┊┊ &='2+y?
z &0,e0 $ 0,e?+
Nlena8
V 0# V ? 7 zrlgankly rk~gë~kily na c`y nly nky~rkoqikleay
e0# e? 7 ~`g`ðly na c`y nly gqay~r`y
±0# ±? 7 z`ràga~rly na zloc`ikùe qel { nly ` ay~kg`rya
y? z 7 & η0 y0
?
$ η? y??
+ , & η0 $ η?+Nlena8
y? z 7 c` }`rk`ex` ilgoke`n` na c`y nly gqay~r`y
η0# η? 7 br`nly na ckoar~`n na gqay~r`y qel { nly
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 325/846
Nr% Fëi~lr Pqa}anl Qrç`y
='2
y0?# y?
? 7 }`rk`ex`y na gqay~r`y qel { nly# rayzai~k}`gae~a
Mqeikùe na ~ z`r` c` gkyg` yk~q`ikùe pqa c` mqeikùe `e~arklr# zarl `zckiàenlc`
iq`enl c`y }`rk`ex`y na c y nly nky~rkoqikleay yle naykbq`cay { `yqgkaenl pqa
c`y zloc`ikleay yle elrg`cay
& V 0 ' V ?+ ' &±0 ' ±?+
~ 7 ┊┊┊┊┊┊┊┊┊┊┊┊┊┊┊┊┊┊┊┊┊┊ &='6+ ∘&y0? , e0+ $ &y?
? , e?+
Z`r` i`ciqc`r cly br`nly na ckoar~`n# ya qy` c` mùrgqc`8
η 7 &y?0,e0 $ y?
?,e?+? , T&y?
0,e0+?,e'0 $ &y?
?,e?+?,e'0W &=';+
Nlena8
y?0 { y?
? 7 }`rk`ex`y na c`y gqay~r`y qel { nly
e0 { e? 7 ~`g`ðly na c`y gqay~r`y qel { nly
El~` kgzlr~ e~a8 c`y nkmaraeik`y ae~ra c`y mqeikleay na ~ &='=+# &='2+# { &='6+ ya o`y`e
ae ac gë~lnl na c` yacaiikùe `c `x`r pqa ya ykbqa% Zlr ahagzcl# ae c` mqeikùe &='=+# ac
gë~lnl na yacaiikùe ae ac agz`rah`gkae~l na cly z`ray na c`y loyar}`ikleay na c`ynky~rkoqikleay ay nackoar`nl% Yke ago`rbl# ae ac i`yl na c`y mqeikleay &='2+ { &='6+#
ile rac`ikùe ` c` mqeikùe &='=+# c` yacaiikùe ay ilgzca~`gae~a `ca`~lrk`# yke f`iar
agz`rah`gkae~ly% @nagày# c`y nkmaraeik`y ae~ra ac qyl na c`y mqeikleay &='=+# &='2+# {
&='6+ ay na pqa ae ac i yl na c` &='=+# ac ~`g`ðl na c`y gqay~r`y z`ray naoa na yar
kbq`c% Ae ile~r`y~a# c`y mqeikleay &='2+ { &='6+ zqanae qy`rya ile ~`g`ðly na
gqay~r`y naykbq`cay% ^`gokëe# ile rayzai~l ` c` qyl na c`y mqeikleay &='2+ { &='6+#
ëy~`y ay~àe rac`ikle`n`y ile c` ilenkikùe na kbq`cn`n l naykbq`cn`n na c`y }`rk`ex`y%
C` mqeikùe &='2+ rapqkara pqa c`y }`rk`ex`y ya`e kbq`cay { c` mqeikùe &='6+ el% @flr`
okae# z`r` f`iar qe naikykùe ylora iq`c na c`y nly mqeikleay# &='2+ l &='6+ ya }`{` `
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 326/846
Nr% Fëi~lr Pqa}anl Qrç`y
='6
qy`r# c` g`ear` na y`oar yk c`y }`rk`ex`y yle kbq`cay l naykbq`cay# ya zqana nanqikr
f`ikaenl qe` zrqao` na kbq`cn`n na }`rk`ex`y ile c` nky~rkoqikùe M# ay~l ay# qy`enl c`
mqeikùe na M 7 y?0 ,y??%^kzly na irk~arkly pqa ya ykbqae z`r` ay~`ocaiar c`y zrqao`y na fkzù~ayky
&`eàclbly ` cly na c` nky~rkoqikùe elrg`c+
0% C` fkzù~ayky eqc` ya zqana f`iar ilgl8 Fl8± 7 ±l% O`hl ay~`y ilenkikleay na
kbq`cn`n# c`y fkzù~ayky `c~are`~k}`y yle8
F08± ≪ ±l# F?8± 3 ±l { F98± 5 ±l% Nlena ±l ay ac zrlgankl zloc`ikle`c pqa ya pqkara
zrlo`r% @pqç# i`oa el~`r pqa ae ay~a i`yl# c` zrqao` na fkzù~ayky ay okc ~ar c l na nly
ilc`y%
?% ^`gokëe c` fkzù~ayky eqc` ya zqana f`iar ilgl8 Fl8± ≡ ±l% Ae ay~a i`yl# c`
fkzù~ayky `c~are`~k}` ay Fl8± 3 ±l% @pqç# c` zrqao` na fkzù~ayky ay qekc`~ar`c kxpqkarn`%
9% Kbq`cgae~a# c` fkzù~ayky eqc` ya zqana f`iar ilgl8 Fl8± ≮ ±l% Ae ay~a i`yl c`
fkzù~ayky `c~are`~k}` ay F08± 5 ±l% @pqç# c` zrqao` na fkzù~ayky ay qekc`~ar`c naraif`%
>% Yacaiikle`r qe ek}ac na ykbekmki`ikùe na ~`g`ðl ΰ# ay~l ay# ΰ 7 %:1 l ΰ 7 %:0 ile yqy
rayzai~k}ly ek}acay na ilemk`ex` na ;1 { ;;% ^`gokëe# ya zqanae qy`r l~rly
ek}acay na ykbekmki`ikùe# ilgl ac %0:# %?:# a~i%# zarl cly gày ilgqeay yle cly na :%:1
{ %:0%
1% Yacaiikle`r c` ay~`nçy~ki` `zrlzk`n` &zlr ahagzcl# yk e 5 9: i`yly ya qy` c`
nky~rkoqikùe x% Yk c` gqay~r` ay e 3 9: i`yly { c` zloc`ikùe gqay~ra`n` el ay elrg`c
ya qy` c` nky~rkoqikùe na ~ na Ay~qnk`e~a# c` nky~rkoqikùe na Hk iq`nr`n`# c` nky~rkoqikùe
M# a~i%=% Ya ay~`ocaiae c`y rabkleay irç~ki`y qy`enl ek}acay na ilemk`ex` nac ;1# ;;# ;:#
6: a~i% &;1 { ;; cly gày ilgqeay+
2% Ya ay~kg` ac }`clr na c` zrqao` na ay~`nçy~ki` na c` gqay~r` { ya ilgz`r` ile ac
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 327/846
Nr% Fëi~lr Pqa}anl Qrç`y
=';
}`clr na c` ay~`nçy~ki` i`ciqc`n`# ay naikr# xi`ci% l ~i`ci% &Na c`y rabkleay irç~ki`y+ { ya
ilgz`r`e ile x~`o% l ~~`o% Yk c` ay~`nçy~ki` i`ciqc`n` ay g`{lr pqa c` ay~`nçy~ki` ~`oqc`n`
&na c`y rabkleay irç~ki`y+ ya raif`x` c` fkzù~ayky eqc`+% Na l~r` g`ear`# ya `iaz~` c`fkzù~ayky l el ya f`ia ekebqe` naikykùe% Na ay~` g`ear`# yk ac }`clr na c` ay~`nçy~ki`
i`ciqc`n` ya ga~a ae c`y rabkleay irç~ki`y ya raif`x` c` fkzù~ayky eqc` &l ~`gokëe yk ac
}`clr na z ay gaelr l kbq`c `c ek}ac na ykbekmki`ikùe# ΰ naya`nl+%
El~ 8 @pqç ay kgzlr~`e~a railrn`r pqa# c` zrqao` na fkzù~ayky eqc` ay~`nçy~ki` ya
nkyaðù ac ykbcl `e~az`y`nl% Ae ~kagzly glnarely na c` ar` ikoareë~ki`# avky~a c`
zrqao` el ~r`nkikle`c rac`ikle`n` ile ac }`clr na c` zrlo`okckn`n z% ^`gokëe ay
kgzlr~`e~a el~`r pqa gqifly zrlbr`g`y na ilgzq~`nlr` n`e øeki`gae~a ac }`clr na
z { ac ke}ay~kb`nlr ~kaea pqa ke~arzra~`rcl `ilrnagae~a%
Gai`ekygly pqa ya ykbqae z`r` i`ciqc`r ac }`clr na c` zrlo`okckn`n z qy`enl c`y
~`oc`y na c`y nky~rkoqikleay na ~ na Ay~qnk`e~a# c` HK iq`nr`n` l c` nky~rkoqikùe M
@pqç# z`r` i`ciqc`r ac }`clr na c` zrlo`okckn`n z ya zqana f`iar qy`enl c` mqeikùe ~ ay
naikr# f`ikaenl ke~arzlc`ikleay `zcki`enl qe` mùrgqc` agzçrki` nkyað`n` zlr ac `q~lr
na ay~a ckorl# ac Nr% Fëi~lr Pqa}anl Qrç`y { qvkck`nl zlr c` Nr`% Ylilrrl @r~a`b`%
&δ? ’ δ0+ , &~? ’ ~0+ 7 &δ? ' V+ , &~? ’ ~i`ci%+ &='0:+
Nlena8
δ? 7 ac ek}ac na ilemk`ex` gày `c~l na c` ~`oc` na c` ~ na Ay~qnk`e~a
δ0 7 ac ek}ac na ilemk`ex` gày o`hl na c` ~`oc` na c` nky~rkoqikùe na ~
~0 7 c` zrlo`okckn`n ilrrayzlenkae~a ` δ0
~? 7 c` zrlo`okckn`n ilrrayzlenkae~a ` δ?
V 7 }`clr nayileliknl na δ
~i`ci%7 }`clr na c` ay~`nçy~ki` na c` nky~rkoqikùe na ~# ile ac ek}ac ykbekmki`e~a naya`nl#
a%b%# ΰ 7 %:1 l ΰ 7 %:0
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 328/846
Nr% Fëi~lr Pqa}anl Qrç`y
='0:
Nlena8 &e ' 0+ 7 η 7 br`nly na ckoar~`n
El~ 8 yk ya qy` c` nky~rkoqikùe na HK iq`nr`n` l c` na Mkyfar# øeki`gae~a ya yqoy~k~q{a ~
zlr ς ? l zlr M# rayzai~k}`gae~a%Ahagzcl )0% Yqzleb`gly pqa pqaragly f`iar c` zrqao` na fkzù~ayky el ~r`nkikle`c
ile c` mqeikùe ~# ay naikr# qy`enl ac }`clr na c` zrlo`okckn`n z% Ae~leiay# yk ac }`clr na
c` ~i`ci% 7 ?%69 ile > br`nly na ckoar~`n# ile ΰ 7 :%:1 z`r` Fl8± 7 ±l oqyi`gly ac }`clr
na ?%69 ae c` ~`oc`# zarl el cl aeile~r`gly% Yke ago`rbl# }agly pqa ay~à ae~ra ?%22=
{ 9%2>2# ile yqy rayzai~k}ly }`clray na δ na %;; { %;21% Ae~leiay z`r` aeile~r`r V#
zrlianagly qy`enl c` mùrgqc` &='0:+ na `rrko`# nlena cly }`clray ilrrayzlenkae~ay
yle8
δ? 7 %;;# δ0 7 %;21# ~? 7 9%2>2# ~0 7 ?%22=# ~i`ci% 7 ?%69%
@flr`# yqy~k~q{aenl ay~ly }`clray ae c` mùrgqc` na ke~arzlc`ikùe { yqy~k~q{aenl8
&&δ? ' δ0+ , &~? ’ ~0+ 7 &δ? ’ V+ , &~? ’ ~i`ci%+
&%;; ' %;21+ , &9%2>2 ’ ?%22=+ 7 &%;; ’ V+ , &9%2>2 ’ ?%69+
_aylc}kaenl zlr V n` V 7 :%;2=% Zlr cl ~`e~l# z 7 0 ’ :%;2= 7 :%:?># zarl ilgl yle
nly ilc`y# ae~leiay# gqc~kzcki`gly aya }`clr zlr ? { n` z 7 %:>6%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 329/846
Nr% Fëi~lr Pqa}anl Qrç`y
='00
Ahagzcl )?% Ya y`i` qe` gqay~r` `ca`~lrk` na 6 loyar}`ikleay na zF iq{ly }`clray
yle8 1# ># =# 1# ># =# 1# 1% Zrlo`r c` fkzù~ayky eqc` na pqa ac }`clr ayzar`nl nac zF ay na
iq`enl gaely =%1 qy`enl ΰ 7 :%:1% I`ciqc`r ac }`clr na c` zrlo`okckn`n z%Ylcqikùe8
0% C` fkzù~ayky eqc` ay Fl8± ≡ =%14 { c` fkzù~ayky `c~are`~k}` ay F08± 3 =%1% Ay~l nkia
pqa c` zrqao` ay qekc`~ar`c kxpqkarn`%
?% Qy`gly c` ay~`nçy~ki`8 ~ 7 & V ’ ±l+ , y,∘ e
7 &1%: ’ =%1+ , :%21=,∘ 6
7 ' 1%=9% C` rabkùe irç~ki` kxpqkarn` ay ~Tΰ4 ηW 7 ~T:%:142W 7 ' 0%6;1
>% Naoknl ` pqa ~i`ci%7 ' 1%= 3 ~~`o% 7 ' 0%6;1# ya raif`x` c` fkzù~ayky { ely keicke`gly
zlr c` fkzù~ayky c~are`~k}`%
1% Ac }`clr na c` zrlo`okckn`n z ya i`ciqc` oqyi`enl s'1%=s ile η 7 2 ae c` ~`oc` na c`
nky~rkoqikùe ~ { ya yqy~k~q{ae cly }`clray na δ? 7 %;;;21# ~? 7 2%661# δ0 7 %;;;1 ~0 7
1%>:6 { ~i`ci% 7 '1%=0 ae c` mùrgqc` na ke~arzlc`ikùe { raylc}kaenl zlr c` }`rk`oca V n`8
&%;;;21 ' %;;;1+,&2%661 ’ 1%>:6+ 7 &%;;;21 ’ V+,&2%6661 ’ 1%=+
Ac }`clr na c` zrlo`okckn`n ay z 7 :%:::>6# ac iq`c ay gqifl# gq{ ykbekmki`e~a
Ahagzcl )9% Qe m`orki`e~a na ikb`rrkccly `mkrg` pqë# ac zrlgankl na ekil~ke` na yqy
zrlnqi~ly ay na iq`enl gqifl 1 gkckbr`gly zlr ikb rrkccl mqg`nl% Z`r` ilgzrlo`r
ay~` `ya}ar`ikùe# ya y`iù qe` gqay~r` `ca`~lrk` na ?1 ikb`rrkccly { ya aeile~rù qe
zrlgankl ay~`nçy~kil na V 7 1%1 gkckbr`gly na ekil~ke` zlr ikb`rrl mqg`nl# ile qe`
nay}k`ikùe ay~àen`r na y 7 :%1% Zrlo`r c` `ya}ar`ikùe nac m`orki`e~a pqa ac }arn`narl zrlgankl ay na ` cl gày 1 gkckbr`gly zlr ikb rrkccl mqg`nl% @yqgkr qe
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 330/846
Nr% Fëi~lr Pqa}anl Qrç`y
='0?
}`clr ykbekmki`e~a na ΰ 7 :%:1%
Ylcqikùe8
0% C` zrqao` na fkzù~ayky eqc` ay8 Fl8± ≮ 1%:% C` zrqao` na fkzù~ayky `c~are`~k}` ay8 {F08± 5 1%:%
?% C` rabkùe irç~ki` ay ~:%:14?> 7 0%200%
9% Qy`enl c` mqeikùe na ~ { yqy~k~q{aenl cly }`clray n`8
~ 7 &1%1 ’ 1%:+ , :%1,1 7 1%:
>% Naoknl ` pqa 1%: 5 0%200 ya raif`x` c` fkzù~ayky eqc`%
1% El loy~`e~a# ay~` zrqao` na fkzù~ayky ~r`nkikle`c el n` qe` kna` na c` mqarx` na
ile}kiikùe na pqa c` naikykùe ~lg`n` ay# ae }arn`n# ilrrai~`% Yke ago`rbl# qy`enl c`
zrqao` na fkzù~ayky el ~r`nkikle`c nac }`clr na z# ay~a }`clr yk na~argke`# pqë ~`e
}arlyçgkc ay gqay~ra`r qe }`clr nac z`ràga~rl pqa ya` kbq`c l gaelr pqa V 7 1%1#
iq`enl ± 7 1%:%
=% Ac }`clr i`ciqc`nl na z ay na `zrlvkg`n`gae~a %::::?%
Ahagzcl )>% Ya n`e cly ykbqkae~ay n`~ly na qe` gqay~r` `ca`~lrk` na 01 gankikleay na
z`r~çiqc y `~glymërki`y ae zzg8 99%96# 9?%01# 99%;;# 9>%0:# 99%;2# 9>%9># 99%;1#
99%61# 9>%?9# 9?%29# 99%>=# 9>%09# 9>%>1# 9>%0;# 9>%:1% F`iar cly ykbqkae~ay iàciqcly
na ay~`nçy~ki` nayirkz~k}`%
&`+ Ay~kg`r ac ~`g`ðl na c` gqay~r` e
&o+ Ay~kg`r ac zrlgankl V # c` gank`e` V u { c` gln` V ˍ
&i+ Ay~kg`r c` }`rk`ex` { c` nay}k`ikùe ay~àen`r gqay~r`cay
&n+ Ac }`clr gàvkgl# gçekgl# ac r`ebl { ac arrlr ay~àen`r &a+ Ac yaybl
&m+ Ac eøgarl na br`nly na ckoar~`n# η
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 331/846
Nr% Fëi~lr Pqa}anl Qrç`y
='09
&b+ Ac ke~ar}`cl na ilemk`ex` nac ;1# ay naikr# ac ek}ac na ykbekmki`eik` na ΰ 7 %:1+
z`r` ac zrlgankl zloc ikle`c ±%
^`gokëe# f`iar cly ykbqkae~ay iàciqcly na ay~`nçy~ki` na kemaraeik`8&`+ Zrlo`r c` fkzù~ayky eqc` na Fl8± 7 9>%1 ile~r` c` fkzù~ayky `c~are`~k}` na F08± ≪
9>%1% I`ciqc`r ac }`clr na c` zrlo`okckn`n z%
&o+ Zrlo`r c` fkzù~ayky na Fl8± ≡ 9>%1 ile~r` c` fkzù~ayky `c~are`~k}` na F08 ± 3 9>%1%
I`ciqc`r ac }`clr na z%
&i+ Zrlo`r c` fkzù~ayky eqc` na Fl8± ≮ 99%? ile~r` F08± 5 99%?%
&n+ I`ciqc`r ac }`clr na c` zrlo`okckn`n z
Ylcqikùe8
Cly iàciqcly na c` ay~`nçy~ki` nayirkz~k}` yle8
&`+ Ac ~`g`ðl na c` gqay~r` ay e 7 01
&o+ Ac zrlgankl `rk~gë~kil# c` gank`e` { c` gln` yle8
V 7 Ω V , e 7 &99%96 $ 9?%01 $%%%$ 9>%:1+,01 7 99%6
C` gank`e` ay8 99%;;% C` gln` el avky~a%
&i+ C` }`rk`ex` 7 y? 7 TΩ V ? ’ &Ω V + ?,eW,e'0
7 T02#0?1%2= ’ &1:=%2=+?,01W , 01'0
7 :%96
C` nay}k`ikùe ay~àen`r 7 y 7 ∘ y? 7 ∘:%96 7 :%=?
&n+ Ac }`clr gàvkgl# gçekgl { ac r`ebl yle8
]`clr gàvkgl 7 9>%>1% ]`clr gçekgl 7 9?%01
_`ebl 7 }`clr gàvkgl ’ }`clr gçekgl 7 ?%9Ac arrlr ay~àen`r nac zrlgankl ay8
Arrlr ay~àen`r 7 ω,∘ e 7 :%=?,∘ 01 7 :%0=
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 332/846
Nr% Fëi~lr Pqa}anl Qrç`y
='0>
&a+ Ac yaybl nael~` c` ykga~rç` na c` nky~rkoqikùe { ae ay~a i`yl ay na ?%11# ac iq`c
ilgz`r`nl ile ac yaybl na c` nky~rkoqikùe elrg`c ay~`en`rkx`n`# pqa ay na :# kenki`
pqa c` nky~rkoqikùe na cly n`~ly ay lockiq` ` c` naraif` l ile yaybl zlyk~k}l%&m+ Ac eøgarl na br`nly na ckoar~`n yle8 η 7 e ’ 0 7 01 ’ 0 7 0>
&b+ Ac ke~ar}`cl na ilemk`ex` nac ;1 l ΰ 7 %:1# ilrrayzlena ` cly }`clray irç~kily na
°?%0>1# ile η 7 0> br`nly na ckoar~`n%
V ’ ~T0'ΰ,?4 ηW &y,∘ e+ 3 ± 3 V $ ~T0'ΰ,?4 ηW &y,∘ e+
99%6 ’ ~T%;2140>W &:%0=+ 3 ± 3 99%6 $ ~T%;2140>W &:%0=+
99%6 ' ?%0>1 &:%0=+ 3 ± 3 99%6 $ ?%0>1 &:%0=+99%>1 3 ± 3 9>%01
Cly iàciqcly na c` ay~`nçy~ki` na kemaraeik` yle8
&`+ Ay~` ay qe` zrqao` na fkzù~ayky okc`~ar`c ile rabkleay irç~ki`y na ~~`o% 7 ~T%;2140>W 7
°?%0>1 ile 0> br`nly na ckoar~`n# ile qe ek}ac na ykbekmki`eik` na ΰ 7 %:1 &na c` ~`oc`
na c` nky~rkoqikùe na ~+% C` ay~`nçy~ki` qy`n` ay c` mqeikùe ~ na `o`hl8
~i`ci% 7 & V ' ±l+ , y,∘ e
7 &99%6 ’ 9>%1+ , :%=9,∘ 01
7 ' >%9
@flr` ya ilgz`r` c` ~ i`ciqc`n` ile c` ~ ~`oqc`n`# ay naikr# ile cly }`clray irç~kily% Ac
irk~arkl pqa ya ykbqa ay na pqa yk c` ~ i`ciqc`n` ya ke~rlnqia ae c`y rabkleay irç~ki`y#
ae~leiay# ya raif`x` c` fkzù~ayky yqy~ae~`n` na pqa Fl8± 7 9>%1 { ya keicke` zlr c`
fkzù~ayky `c~are`~k}`% Ae ileicqykùe }agly pqa ’ >%9 3 ' ?%0>1# ay naikr# ya ke~rlnqia
ae ac av~ragl kxpqkarnl na c` iqr}`% Ac }`clr na c` zrlo`okckn`n z ya i`ciqc` qy`enl c`mùrgqc` na ke~arzlc`ikùe &='0:+ 8
&δ? ’ δ0+,&~? ’ ~0+ 7 &δ? ’ V+ , &~? ’ ~i`ci%+
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 333/846
Nr% Fëi~lr Pqa}anl Qrç`y
='01
Nlena8
δ? 7 %;;;21# ~? 7 >%>;;# δ0 7 %;;;1# ~0 7 >%0># ~i`ci% 7 '>%9 &`pqç ae ay~a i`yl# ya ~lg`
ac }`clr `oylcq~l+# V kbq`c ` }`clr oqyi`nl ac iq`c ilrrayzlenae ` c` ke~arzlc`ikùena ~ 7 '>%9 ile η 7 0> b%c%
Yqy~k~q{aenl cly }`clray ae c` mùrgqc` na rrko` n`8
&%;;;21 ’ %;;;1+,&>%>;; ’ >%0>+ 7 &%;;;;; ' V+,&>%>;; ’ >%9+
V 7 :%;;;62 { ac }`clr na z ay z 7 ?&0 ' %;;;;;+ 7 :%::::?% Ay~a }`clr ay gqifl gq{
ykbekmki`~k}l { `zl{`# gq{ ile~qenae~agae~a# c` ile~aeikùe na pqa ac zrlgankl el ay
g`{lr pqa 9>%1%
&o+ Zrlo`enl c` fkzù~ayky eqc` na Fl8± ≡ 9>%1 ile~r` F08± 3 9>%1
C` ~ i`ciqc`n` ay c` gkyg` pqa ae c` z`r~a &`+# ay naikr# ' >%9% Ay~` ay qe` zrqao`
qekc`~ar`c kxpqkarn` ile ΰ 7 :%1 ile ac }`clr zlriae~q`c na ~%;140> 7 ' 0%2=0 l ya` pqa c`
rabkùe irç~ki` kxpqkarn` ay ’ 0%2=0 &na c` ~`oc` na c` nky~rkoqikùe na ~+% Z`r` f`iar qe`
naikykùe na raif`x`r l na `iaz~`r Fl8 ya ilgz`r` ac }`clr na ~%;140> 7 ’ 0%2=0 ile ~i`ci% 7
’ >%9 { }agly# eqa}`gae~a# pqa ya ke~rlnqia ae ac av~ragl kxpqkarnl na c`
nky~rkoqikùe# zlr cl ~`e~l# ya raif`x` c` fkzù~ayky% Ac }`clr na c` zrlo`okckn`n z ya
i`ciqc` oqyi`enl ac }`clr `oylcq~l na s'>%9s ae c` ~`oc` ile ΰ 7 :%:1 { }agly pqa ay~à
ae~ra >%>;; { >%0> ile yqy rayzai~k}ly }`clray na δ kbq`c ` %;;;21 { %;;;1% Ay naikr
pqa ac }`clr na z ay~à ae~ra %:::?1 3 z 3 %:::1# ile qe }`clr na z ≆ %:::?%
&i+ Z`r` zrlo`r c` fkzù~ayky na Fl8± ≮ 99%? ile~r` c` fkzù~ayky `c~are`~k}` na F08± 5
99%?# ya qy` c` ay~`nçy~ki` na ~ na Ay~qnk`e~a# ay naikr8
~ 7 &99%6 ’ 99%?+,:%=9,9%62 7 9%=6C` rabkùe irç~ki` naraif` ay ~%;140> 7 0%2= { }agly pqa 9%=6 ay g`{lr pqa ay~a }`clr {
ya raif`x` c` fkzù~ayky eqc`% O`hl ay~`y ilenkikleay# ac }`clr na c` zrlo`okckn`n z ay
:%::0%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 334/846
Nr% Fëi~lr Pqa}anl Qrç`y
='0=
Ahagzcl )1% Qe m`orki`e~a na cc`e~`y `mkrg` pqë# c` }kn` zrlgankl na ikar~l ~kzl na
eaqgà~kil# ay g`{lr pqa ?1#::: dkcùga~rly# o`hl ilenkikleay elrg`cay na g`eahl {#
z`r` }afçiqcly na ikar~l zayl% Z`r` ay~l# ya y`i` qe` gqay~r` `ca ~lrk na 01 cc`e~`y {ya i`ciqc` qe zrlgankl `rk~gë~kil { qe` nay}k`ikùe ay~àen`r na ?2#::: { 9#:::#
dkcùga~rly# rayzai~k}`gae~a% @yqgkr pqa ΰ 7 :%:1 { pqa c` zloc`ikùe na cc`e~`y ay~à
elrg`cgae~a nky~rkoqkn` ¼Ya zqana ileicqkr na ay~` kemlrg`ikùe pqa c` ile~aeikùe
nac m`orki`e~a na cc`e~`y ay cabç~kg`< Z`r` raylc}ar ay~a zrlocag` f`iar cl ykbqkae~a8
&`+ Ay~`ocaiar c`y zrqao`y na fkzù~ayky eqc` { `c~are`~k}`
&o+ Ay~`ocaiar c`&y+ rabkùe&ay+ irç~ki`&y+
&i+ I`ciqc`r ac }`clr na c` ay~`nçy~ki`
&n+ I`ciqc`r { br`mki`r ac }`clr na z%
Ylcqikùe8
&`+ Ac zrlocag` ay~à zrabqe~`enl yk ya zqana ileicqkr pqa ± ay g`{lr pqa ?1#:::
dkcùga~rly% Zlr cl ~`e~l# qe` `mkrg`ikùe na ay~a amai~l naoarà kr ae c` zrqao` na
fkzù~ayky `c~are`~k}`% C`y fkzù~ayky `zrlzk`n`y yle8
Fl8± ≮ ?1#::: { F08± 5 ?1#:::
&o+ C` rabkùe irç~ki` ile ΰ 7 :%:1 ay8 ~:%;140> 7 0%2=09
&i+ Ac }`clr i`ciqc`nl na c` ay~`nçy~ki` ~ ile V 7 ?2#:::# arrlr ay~àen`r 7 22>%=0# e 7
01 { ±l 7 ?1#::: ay8
~ 7 &?2#::: ’ ?1#:::+ , 9:::,∘01
7 ?%16
&n+ Z`r` aeile~r`r ac }`clr na c` zrlo`okckn`n z ya zrliana na c` ykbqkae~a g`ear`8 Ya oqyi` ~ 7 ?%16 ae c` ~`oc na c` nky~rkoqikùe na ~ ile η 7 0> br`nly na ckoar~`n# {
}agly pqa ay~a }`clr ay~à ae~ra ?%=?> { ?%0>>6# ile yqy rayzai~k}ly zariae~kcay na
:%0: { :%:?1% Na ay~` g`ear`# yk c` fkzù~ayky eqc` Fl8 ay ikar~`# ae~leiay# c`
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 335/846
Nr% Fëi~lr Pqa}anl Qrç`y
='02
zrlo`okckn`n na lo~aear qe }`clr na ~ ~`e br`ena l gày br`ena pqa ?%0>>6 ay :%:?1%
Ykgkc`rgae~a# c` zrlo`okckn`n na lo~aear qe }`clr ~`e br`ena l gày br`ena pqa ?%=?>
ay na :%0:% Zlr cl ~`e~l# yk Fl8 ay }arn`nar`# c` zrlo`okckn`n na lo~aear qe }`clr na ~~`e br`ena l gày br`ena pqa ~ 7 ?%16 ay~à ae~ra :%:0: { :%:?1# ay naikr# :%0: 3 z 3
:%:?1% C`y mkbqr`y na `o`hl gqay~r`e ay~` yk~q`ikùe%
Mkbqr` =%0% Mkbqr`y &`+ { &o+ gly~r`enl ac ke~ar}`cl na c` zrlo`okckn`n z { ac }`clr na
c` zrlo`okckn`n z# rayzai~k}`gae~a%
Ahagzcl )=% Z`r` zrlo`r c` amkikaeik` na qe` zc`e~` na ~r`~`gkae~l clnly `i~k}`nly ya
gknkù c` ileiae~r`ikùe nac NOL1 ae c` ae~r`n` { ae ac amcqae~a &y`ckn`+% Ya rapqkara
y`oar pqë ~`e amkikae~a ay ay~a yky~ag` na ~r`~`gkae~l nac nrae`ha%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 336/846
Nr% Fëi~lr Pqa}anl Qrç`y
='06
^@OC@ =%:% ^`oc` gly~r`enl c`y ileiae~r`ikleay na NOL% &Ac`olr`ikùe zrlzk`+
Ileiae~r`ikleay na Ileiae~r`ikleay na Nkmaraeik`y na c`y
NOL ae c` ae~r`n` NOL ae ac amcqae~a ileiae~r`ikleay
&gb,C+ &gb,C+ &gb,C+
02:%1 0>:%> 9:%0
?:2%> 02>%2 9?%2
?01%; 02:%? >1%2
?:;%: 02>%= 9>%>
020%= 01>%= 02%:?:0%? 061%: 0=%?
?:;%; 006%; ;0%:
?09%9 0=;%6 >9%1
06>%0 02>%2 ;%>
??:%> 02=%2 >9%2
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
Ylcqikùe8
Qy`enl cly }`clray na c` ^@OC@ =%: y`i`gly c`y nkmaraeik`y ae~ra c`y
ileiae~r`ikleay ae c` ae~r`n` { ae ac amcqae~a% Ay~l ya gqay~r` ae c` ~ariar` ilcqge`
na c` ~`oc`% Qe` }ax faifl ay~l# ya i`ciqc` ac zrlgankl `rk~gë~kil na c`y nkmaraeik`y
&pqa ay kbq`c ` N + { c` nay}k`ikùe ay~àen`r &pqa ay kbq`c yn+# ac arrlr ay~àen`r# a~i%
0% Qy`enl qe z`pqa~a na ilgzq~`nlr` ya i`ciqc` ac }`clr nac zrlgankl N 7 V 7
9=%92# c` nay}k`ikùe ay~àen`r pqa ay kbq`c ` yn 7 ??%;1# e 7 0:# arrlr ay~àen`r 7 2%?=
?% C` zrqao` na fkzù~ayky eqc` ay na Fl8± 7 : l ya` pqa el f`{ nkmaraeik`y ae~ra ac
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 337/846
Nr% Fëi~lr Pqa}anl Qrç`y
='0;
NOL na c` ae~r`n` { nac amcqae~a% C` fkzù~ayky `c~are`~k}` ay F08± ≪ :%
9% Ac ek}ac na ykbekmki`eik` ay ΰ 7 :%:1% C` rabkùe irç~ki` ay nac av~ragl naraifl { ay
kbq`c ` ~%;14; 7 ?%?=?# ay naikr# ile ; br`nly na ckoar~`n%>% Ya qy` c` mqeikùe &='=+ z`r` agz`rah`gkae~l nackoar`nl# { yqy~k~q{aenl cly }`clray
n`8
~ 7 & N ’ ±l+ , yn,∘ e
~ 7 &9=%92 ’ :+,2%?=
7 1%:0
1% @c ilgz`r`r ac }`clr na ~i`ci% 7 1%:0 ile ac }`clr na c` ~ irç~ki` na ~%;14; 7 ?%?=?# yaraif`x` c` fkzù~ayky eqc` { naikgly pqa yç f`{ nkmaraeik`y ae~ra c`y ileiae~r`ikleay na
c` ae~r`n` { na c` y`ckn` nac nrae`ha%
=% Z`r` i`ciqc`r ac }`clr na c` zrlo`okckn`n z qy`gly c` mùrgqc` na ke~arzlc`ikùe% Z`r`
ay~l oqyi`gly 1%:0 ae c` ~`oc` na c` ~ na Ay~qnk`e~a { }agly pqa ay~à ae~ra cly }`clray
zlriae~q`cay na δ? 7 %;;;21 ile ~? 7 1%?;0 { δ0 7 %;;;1 ile ~0 7 >%260% @flr` qy`enl
c` mùrgqc` na ke~arzlc`ikùe { yqy~k~q{aenl ~lnly cly }`clray n`8
&%;;;21 ' %;;;1+,&1%?;0 ’ >%260+ 7 &%;;;21 ’ V+,&1%?;0 ’ 1%:0+
_aylc}kaenl zlr V n` V 7 %;;;=# zlr cl ~`e~l# z 7 0 ' %;;;= 7 %:::9;%
Ay~a }`clr na z ay gqifl gq{ ykbekmki`e~a { `zl{`# ae mlrg` gq{ ile~qenae~a# c`
naikykùe na f`oar raif`x`nl c` fkzù~ayky# na pqa el f`{ nkmaraeik`y ae~ra c`y
ileiae~r`ikleay na c` ae~r`n` ` c` zc`e~` { na c` y`ckn`%
=% Ae ileicqykùe raif`x`gly c` fkzù~ayky Fl8± 7 :# ay~l ay# na pqa el f`{ nkmaraeik`y
ae~ra c`y ileiae~r`ikleay na c` ae~r`n` { nac amcqae~a &ae }arn`n yk f`{ gqif`nkmaraeik`# c hqxb`r zlr ac }`clr na c` zrlo`okckn`n z+%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 338/846
Nr% Fëi~lr Pqa}anl Qrç`y
='?:
Ahagzcl )2% Ay~a ay qe ahagzcl na kebaekarç` `gokae~`c &kebaekarç` y`ek~`rk`+
rac`ikle`nl ile ac qyl na c` nky~rkoqikùe ~# iq`enl c`y }`rk`ex`y na c`y nky~rkoqikleay
yle naykbq`cay# `yqgkaenl pqa c`y zloc`ikleay yle elrg`cay% @zcki`enl ay~aileiaz~l# ae qe ay~qnkl ya y`oa pqa# ac na~arklrl na gqif`y ranay na ~qoarç`
gqekikz`c na `bq` { nrae`ha ae ~lnl ac z`çy ay qe `yqe~l pqa zraliqz` i`n` }ax gày `
c`y `q~lrkn`nay% Qe`y na c`y ~aielclbç`y zrlzqay~`y z`r` c` raf`okck~`ikùe na c`y
~qoarç`y ileyky~ae ae qy`r qe mlrrl mcavkoca `crananlr nac ~qol avky~ae~a% Ac `r~çiqcl
–Ammai~ lm Racnkeb le ` Fkbf Naeyk~{ Zlc{a~f{caea Ckear‗ &H% lm G`~ark`cy ke Ik}kc
Aebkeaarkeb# 0;;=# zz% ;>'0::+# kemlrg` cly n`~ly ykbqkae~ay na rayky~aeik` ` c`
~aeykùe# ae coy,ke? &zyk+# l ya` ckor`y zlr zqcb`n` iq`nr`n`# na ayzaikgaeay na mlrrl#
~`e~l ae ac i`yl ae pqa ikar~l zrliayl na mqykùe ya qy`# ilgl iq`enl el ya qy`% C`
~`oc` na `o`hl n` cly n`~ly irqnly { zrliay`nly% Qy`r ΰ 7 %:1% Z`r` ay~l f`iar cly
ykbqkae~ay iàciqcly8
&`+ Ay~`ocaiar c` zrqao` na fkzù~ayky eqc` { c` zrqao` na fkzù~ayky `c~are`~k}`% Ay~l ay#
f`ikaenl qe` zrqao` na fkzù~ayky eqc` na pqa el f`{ nkmaraeik`y ae c`y rayky~aeik`y `
c` ~aeykùe z`r` cly nly ~r`~`gkae~ly%
&o+ Ay~`ocaiar c` rabkùe irç~ki`%
&i+ Qy`r c` ay~`nçy~ki` gày `zrlzk`n` z`r` ac`olr`r ay~a zrlocag`%
&n+ F`iar qe` naikykùe ay~`nçy~ki` qy`enl ac irk~arkl ~r`nkikle`c# ay naikr# na raif`x`r
l na ra~aear c` fkzù~ayky eqc`%
&a+ F`iar qe` zrqao` na fkzù~ayky el ~r`nkikle`c# ay naikr# i`ciqc`enl ac }`clr nac ek}ac
na z%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 339/846
Nr% Fëi~lr Pqa}anl Qrç`y
='?0
^@OC@ =%0% ^`oc` gly~r`enl cly n`~ly nac zrlocag`% &Ac`olr`ikùe zrlzk`+
Yke mqykùe &ae ckor`y zlr zqcb`n` iq`nr`n`+
?2>6 ?2:: ?=11 ?6?? ?100 90>; 9?12 9?09 9??: ?219
e0 7 0: V 0 7 ?;:?%6 y0 7 ?22%? y?0 7 2=#621%;;
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
Ile mqykùe &ae ckor`y zlr zqcb`n` iq`nr`n`+
9:?2 991= 991; 9?;2 90?1 ?;0: ?66; ?;:?
e? 7 6 V ? 7 90:6%0 y? 7 ?:1%; y?? 7 >?96?%>0%
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
Ylcqikùe8
&`+ C` zrqao` na fkzù~ayky eqc` ay8 Fl8± 7 : l ya` pqa el f`{ nkmaraeik`y ae~ra c`y
~aeykleay# z`r` cly nly ~r`~`gkae~ly%
C`y zrqao`y na fkzù~ayky c~are`~k}`y yle F08± 5 : { F?8± 3 :%
&o+ C` rabkùe irç~ki` ay qekc`~ar`c kxpqkarn` ay kbq`c '0%21
&i+ Ya qy` c` mqeikùe ay~`nçy~ki` na ~ z`r` }`rk`ex`y naykbq`cay% Ay naikr# iq`enl ya
qy`e nly gqay~r`y `ca`~lrk`y kenazaenkae~ay na zloc`ikleay elrg`cay# ile }`rk`ex`y
naykbq`cay% Ay~` ay~`nçy~ki` na c` mqeikùe na ~# `cbqe`y }aiay ya cc`g` zrqao` na
Ygk~f'Y`~~ar~f``ka `o`hl gly~r`n`% &Gkccar a~ `c% 0;2=# z% ?=0+
Yqy~k~q{aenl cly }`clray ae c` aiq`ikùe &='2+ n`8
90:6%0: ’ ?;?1%99
~ 7 „„„„„„„„„„„„ ∘&?22%9+?,0: $ &?:1%;+?,6
7 ' 0%6=
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 340/846
Nr% Fëi~lr Pqa}anl Qrç`y
='??
@flr`# qy`enl c` mùrgqc` na cly br`nly na ckoar~`n# rac`ikle`n` ile c` mqeikùe na ~
pqa ~kaea }`rk`ex`y naykbq`cay &Ck# 0;=>+# ay naikr8
&y?0,e0 $ y??,e?+? η 7 „„„„„„„„„„„„„„„„„ &='00+
T&y?0,e0+
?,&e0'0+W $ Ty??,e?+
?,&e?'0+W
[ yqy~k~q{aenl ~lnly cly }`clray na8 y?0 7 2=#621%;=# e0 7 0:# y?
? 7 >?#96?%>0# e? 7 6
n`8
T&2=#621%;=+,&0:+ $ >?#96?%>0,&6+W? η 7 „„„„„„„„„„„„„„„„„
T&2=#621%;=+,0:+?,;+ $ &>?#96?%>0,6+?,2W
7 0= br`nly na ckoar~`n
&n+ Ileicqykùe8 ya raif`x` c` zrqao` na fkzù~ayky eqc` na el nkmaraeik`y ae c`y
rayky~aeik`y ` c` ~aeykùe naoknl ` pqa ac }`clr na c` ay~`nçy~ki` ~ 7 '0%6= ay gaelr pqa
c` rabkùe irç~ki` kxpqkarn` na '0%21%
&a+ Z`r` f`iar c` zrqao` na fkzù~ayky el ~r`nkikle`c ya oqyi` ac }`clr `oylcq~l na c` ~
i`ciqc`n`# ay naikr# s'0%6=s ae c` ~`oc` na c` nky~rkoqikùe na ~ na Ay~qnk`e~a ile 0=br`nly na ckoar~`n { }agly pqa cly }`clray zariae~kcay yle na :%:?1 { :%:1 ile yqy
zqe~ly zlriae~q`cay na 0%2>= { ?%0?:% Ae~leiay# ac r`xle`gkae~l pqa ya ykbqa z`r`
i`ciqc`r ac }`clr na z ay ilgl ykbqa% Yk Fl8 ay }arn`nar`# c` zrlo`okckn`n na lo~aear
qe }`clr na ~ ~`e br`ena l gày br`ena pqa 0%2>= ay :%:?1% @nagày# c` zrlo`okckn`n na
lo~aear qe }`clr ~`e br`ena l gày br`ena pqa ?%0?: ay na :%:1% Zlr cl ~`e~l# yk Fl8 ay
}arn`nar`# c` zrlo`okckn`n na lo~aear qe }`clr ~`e br`ena l gày br`ena pqa ac }`clr
na '0%6= ay~à ae~ra :%:?1 { %:1% Z`r` ay~` zrqao` ae z`r~kiqc`r# :%:1 5 z 5 :%:?1%
Ahagzcl )6% Yqzùeb`ya pqa ya y`i` qe` gqay~r` na 6 gankikleay na ek~r`~ly &EL9'+ {
ya i`ciqc` qe }`clr na ~ 7 ' 9%=?# ile qe ek}ac na ykbekmki`eik` na ΰ 7 :%:1% Zrlo`r c`
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 341/846
Nr% Fëi~lr Pqa}anl Qrç`y
='?9
fkzù~ayky eqc` na Fl8± 7 9?%:% I`ciqc`r ac }`clr na c` zrlo`okckn`n z%
Ylcqikùe8
@pqç c` zrqao` ay okc`~ar`c% C`y rabkleay irç~ki`y yle na ~T%:142W 7 ?%9=1% Ac }`clr na c` ~i`ciqc`n` ay na ~ 7 '9%=?% Ya qy` c` mqeikùe na Z z`r` nly ilc`y n`n` ilgl8
Z 7 Z&~%:?1 3 's~s+ $ Z&~%:?1 5 s~s+ &='0?+
?% Ay~`gly oqyi`enl c` zrlo`okckn`n na y`i`r qe }`clr na ~ pqa avian` 9%=? ile η 7 2
br`nly na ckoar~`n# zarl }agly pqa ay~a }`clr el ay~à ae c` ~`oc` na c` nky~rkoqikùe ~%
Ae~leiay ~aeagly pqa ke~arzlc`r ay~a }`clr { cl oqyi`gly ae c` ~`oc` { }agly pqa
ay~à ae~ra δ?
7 %;;21 ile ~?
7 >%:?; { δ0
7 %;;1 ile ~0
7 9%>;;% @nagày# y`oagly pqa
~T%:142W 7 '?%921 &zlrpqa ay na c` ilc` kxpqkarn`+% @flr` ya yqy~k~q{ae ~lnly ay~ly
}`clray ae c` mùrgqc` na ke~arzlc`ikùe &1'?2+ rai`zk~qc`n` o`hl8
Z 7 &δ? ’ δ0+ , &~? ’ ~0+ 7 &δ? ’ V+ , &~? ’ ~i`ci%+
Aeyabqkn` yqy~k~q{aenl cly }`clray na rrko` n`8
z 7 T&%;;21 ' %;;1+,&>%:?; ’ 9%>;;+ 7 &%;;21 ’ V+,&>%:?; ’ 9%=?+W
7 &%::?1+,&:%19+ 7 &%;;21 ’ V+,&:%>:;+
7 %;;261%
C` zrlo`okckn`n z ay 0 ' %;;261 7 %::?% Yke ago`rbl# naoknl ` pqa c` zrqao`
ke}lcqir` nly av~ragly# zlr cl ~`e~l# ac }`clr na c` zrlo`okckn`n z ya gqc~kzcki` zlr ?
z`r` n`r z 7 %::>9% Ay~a }`clr ay gqifl gq{ ykbekmki`~k}l% C`y mkbqr`y na `o`hl
gqay~r`e ay~` yk~q`ikùe%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 342/846
Nr% Fëi~lr Pqa}anl Qrç`y
='?>
Mkbqr` =%?% Mkbqr`y gly~r`enl c`y rabkleay irç~ki`y% &Ac`olr`ikùe zrlzk`+
C` nky~rkoqikùe na HK iq`nr`n` &ς ?+
C` nky~rkoqikùe na HK iq`nr`n` ay~à rac`ikle`n` ile c` }`rk`ex`% Ay~` nky~rkoqikùe ya
qy` z`r` f`iar ke~ar}`cly na ilemk`ex` z`r` c` }`rk`ex` zloc`ikle`c { zrqao`y nafkzù~ayky z`r` c` }`rk`ex` zloc`ikle`c% Ay~` ay~`nçy~ki` na ς ? ~`gokëe ya qy` z`r`
f`iar zrqao`y na olen`n na `hqy~a% Ay~l ya f`ia z`r` }ar yk cly n`~ly zrl}kaeae na
qe` zloc`ikùe pqa ykbqa `cbqe` nky~rkoqikùe ayzaikmki`n`# ilgl nkyira~` l
ile~keq`# ay naikr# ilgz`r`enl cly n`~ly ~aùrkily ile cly loyar}`nly% Mke`cgae~a#
c` HK iq`nr`n` ~`gokëe ya qy` z`r` f`iar zrqao`y na kenazaenaeik`# a~i%
C` nky~rkoqikùe na HK iq`nr`n` ay~à irç~ki`gae~a ilenkikle`n` ` gqay~raly na
zloc`ikleay elrg`cay# zlrpqa na l~r` g`ear` zqana ilenqikr ` arrlray gq{
br`enay% @nagày# qe ~`g`ðl na gqay~r` br`ena# el b`r`e~kx` qe` zrqao` ilemk`oca%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 343/846
Nr% Fëi~lr Pqa}anl Qrç`y
='?1
Zrlzkan`nay na c` nky~rkoqikùe na HK iq`nr`n` &ς ?+
0% C` nky~rkoqikùe na HK iq`nr`n` el ay ykgë~rki`# ilgl c` nky~rkoqikùe elrg`c l c`
nky~rkoqikùe na ~% Cly }`clray na c` HK iq`nr`n` zqanae yar na iarl l zlyk~k}ly# zarlel eab`~k}ly%
9% C` nky~rkoqikùe na HK iq`nr`n` ay qe` m`gkck` na iqr}`y { f`{ qe` nky~rkoqikùe
nkmarae~a z`r` i`n` eøgarl na br`nly na ckoar~`n# η% Zarl# ` gankn` pqa ac eøgarl
na br`nly na ckoar~`n `qgae~`# c` nky~rkoqikùe na c` HK iq`nr`n` ya `zrlvkg` ` c`
nky~rkoqikùe elrg`c%
Mkbqr` =%9% Nky~rkoqikùe na HK iq`nr`n` &ς ?+ ile }`rkly br`nly na ckoar~`n# η ae
mqeikùe m &ς ?+ 7 T&ς ?+ η,&?'0+ a'ς ?,?W , t? η,? T η ' ? η+ , ?W!w% &Nqee a~ `c% 0;2>+
Ahagzcly z`r` na~argke`r c`y rabkleay irç~ki`y na c` HK iq`nr`n` qy`enl cly
}`clray zlriae~q`cay na ς ?z { na ς
?Tδ # ηW%
Ahagzcl );% Aeile~r`r cly }`clray irç~kily na ς ?
pqa na~argkeae c`y rabkleayirç~ki`y pqa ile~aeb`e qe àra` na :%:?1 ae i`n` ilc`% @yqgkr pqa e 7 0:# zlr cl
~`e~l# cly br`nly na ckoar~`n yle na η 7 0: ’ 0 7 ;%
Ylcqikùe8
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 344/846
Nr% Fëi~lr Pqa}anl Qrç`y
='?=
C` mkbqr` na `o`hl gqay~r` qe` zrqao` okc`~ar`c% Z`r` aeile~r`r ac }`clr irç~kil
kxpqkarnl# ya ramkara ` c` ~`oc` na c` HK iq`nr`n` { ya oqyi` η 7 ; ae c` ilcqge`
kxpqkarn` na c` ~`oc` { ya }` f`y~` c` ilcqge` :%;21# zlrpqa ac àra` ~l~`c ` c`naraif` na ay~a }`clr ay :%;21 pqa cl y`i`gly ray~`enl :%:?1 na 0 { ely n` ς ? 7
?%2::% Ykgkc`rgae~a# z`r` c` rabkùe irç~ki` naraif`# ya cli`ckx` ac }`clr na η 7 ; {
ely gl}agly f`ik` ac }`clr na :%:?1 { n` ς ? 7 0;%:?9% C` Mkbqr` =%> na `o`hl
gqay~r` ay~` yk~q`ikùe%
← ← ς?:%;214; 7 ?%2: ς?:%:?14; 7 0;%:?9
Mkbqr` =%>% Bràmki` gly~r`enl cly }`clray irç~kily na c` nky~rkoqikùe# ile qe àra` na
:%:?1 ae i`n` ilc`# ile e 7 0: { η 7 e ’0 7 0: ’ 0 7 ;% Mqae~a8 ^rklc` &0;;1+
Zlr ahagzcl# rai`zk~qc`enl ac r`xle`gkae~l `e~arklr# na c` Mkbqr` =%># ya zqana }ar
pqa# z`r` lo~aear ac }`clr irç~kil l cçgk~a kxpqkarnl na ?%2:# f`{ pqa cli`ckx`r ; ae
c` ilcqge` kxpqkarn` na br`nly na ckoar~`n { cqabl cli`ckx`r :%;21 `rrko` na c`
~`oc`% Ac àra` ~l~`c ` c` naraif` na ay~a }`clr irç~kil ay :%;21# ac iq`c ya ay~kg` na 0 ’ :%:?1% Ykgkc`rgae~a# z`r` lo~aear ac }`clr irç~kil na 0;%:?9# cli`ckx`r ; ae c`
ilcqge` na br`nly na ckoar~`n { cqabl cli`ckx`r :%:?1 `rrko` na c` ~`oc`%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 345/846
Nr% Fëi~lr Pqa}anl Qrç`y
='?2
Ahagzcl )0:% Aeile~r`r cly }`clray irç~kily na ς ? zlr cly iq`cay ac àra` nac av~ragl
naraifl na c` nky~rkoqikùe ay na :%:1# yk8
&`+ η 7 01&o+ η 7 ?0
Ylcqikùe8
&`+ Ac }`clr na c` ilc` naraif` na c` nky~rkoqikùe na HK iq`nr`n` ya oqyi` ae c` ~`oc`
na ay~` nky~rkoqikùe { ay8 ς ?ΰ4e'0 7 ς ?%:140='0 7 ς ?%:1401 7 ?>%;;=
&o+ Ac }`clr na c` ilc` naraif` ay na ς ?%:14?0 7 9?%2
Ahagzcl )00% Z`r` qe` nky~rkoqikùe na HK iq`nr`n` ile 0? br`nly na ckoar~`n#
aeile~r`r ac }`clr na ς ? na ~`c g`ear` pqa8
&`+ Ac àra` ` c` naraif` na ς ? ay %:1#
&o+ Ac àra` ` c` kxpqkarn` na ς ? ay %;;
Ylcqikùe8
&`+ ς ?%:140? 7 ?0%:?=
&o+ ς ?%:040? 7 ?=%??
Ahagzcl )0?% Aeile~r`r cly }`clray irç~kily na ς ? zlr cly iq`cay ac àra` ` c` naraif`
na c` nky~rkoqikùe ay na ΰ 7 %:0# yk η 7 18
Ylcqikùe8
Yk ac àra` ylgora`n` ylora c` naraif` ay %:0:# ac àra` ` c` kxpqkarn` na ς ?? ay %;; {
ς ?? razrayae~` ac ;;`}l zariae~kc# ς ?%;;# ac iq`c ay kbq`c ` 01%0%
Ke~ar}`cly na ilemk`ex` { zrqao`y na fkzù~ayky qy`enl c` nky~rkoqikùe na HK
iq`nr`n` ς ?
Ac ke~ar}`cl na ilemk`ex` 0 ’ ΰ z`r` c` }`rk`ex` zloc`ikle`c# ω? ya n` ilgl8
&e ’ 0+ y? , ς ?T0'ΰ,?4e'0W 3 ω? 3 &e ’ 0+y? , ς ?Tΰ,?4e'0W &='0>+
Ahagzcl )09% Yk qe` gqay~r` `ca`~lrk` ay~`nçy~ki` na 02 gankikleay ~kaea qe`
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 346/846
Nr% Fëi~lr Pqa}anl Qrç`y
='?6
}`rk`ex` na y? 7 0;=%96# aeile~r`r ac ke~ar}`cl na ilemk`ex` z`r` ω? qy`enl cly
ek}acay na ilemk`ex` yle na8
&`+ ΰ 7 :%:1&o+ ΰ 7 :%:0
Ylcqikùe8
&`+ Ya rapqkara c` mqeikùe Z&ς ?0'ΰ,? 3 ω? 3 ς ?ΰ,?+ 7 0 ’ ΰ% Ya i`ciqc`e cly cçgk~ay
yqzarklray a kemarklray { cqabl ya yqy~k~q{ae cly }`clray ilrrayzlenkae~ay%
Z`r` ac cçgk~a yqzarklr8 ς ?T0'ΰ,?4e'0W 7 ς ?T0'%:1,?402'0W 7 ς ?%;2140= 7 =%;0
Z`r` ac cçgk~a kemarklr8 ς ?Tΰ,?4e'0W
7 ς ?%:1,?402'0
7 ς ?%:?140=
7 ?6%6
@flr`# yqy~k~q{aenl ay~ly }`clray ae c` mqeikùe &='0>+ ely n`8
&02 ’ 0+&0;=%96+ , =%;0 3 ω? 3 &02 ’ 0+&0;=%96 , ?6%6+
>1>%2 3 ω? 3 0:;%0
C` iq`c ya ykgzckmki` `8 >19%2 5 ω? 5 0:;%0
&o+ Z`r` i`ciqc`r cly yqzarklray a kemarklray# ile qe ek}ac na ykbekmki`eik` na :%:0 ya
zrliana ilgl ykbqa8
Z`r` ac cçgk~a kemarklr8 ς ?Tΰ,?4e'0W 7 ς ?T%:0,?02'0W 7 ς ?%::140= 7 9>%09
Z`r` ac cçgk~a yqzarklr8 ς ?T0'ΰ,?4e'0W 7 ς ?T0'%:0,?402'0W 7 ς ?%;;140= 7 1%0>
Aeyabqkn`# qy`enl c` mùrgqc` nac ke~ar}`cl { yqy~k~q{aenl n`8
&02 ’ 0+0;=%96 , 1%0> 3 ω? 3 &02 ’ 0+0;=%96 , 9>%09
Pqa ya ykgzckmki` `8
;?%:= 3 ω? 3 =00%9
Ay~a ke~ar}`cl nkia pqa ay~`gly ilemk`nly ae qe ;1 na pqa c` }`rk`ex` zloc`ikle`c ay~à ae~ra ;?%:= { =00%9%
Ac ke~ar}`cl na ilemk`ex` z`r` ω ya i`ciqc` y`i`enl c` r`çx iq`nr`n`# cl iq`c
n` &0:%>1# ?0%9?+ { &;%1;# ?>%2?+# z`r` ΰ 7 %:1 { ΰ 7 %:0# rayzai~k}`gae~a%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 347/846
Nr% Fëi~lr Pqa}anl Qrç`y
='?;
Ahagzcly na zrqao`y na fkzù~ayky z`r` c` }`rk`ex` qy`enl c` nky~rkoqikùe na
c` HK iq`nr`n`# ς ? `yqgkaenl pqa c` zloc`ikùe gqay~ra`n` ay elrg`c
C` mqeikùe ay~`nçy~ki` qy`n` z`r` f`iar zrqao`y na fkzù~ayky z`r` c` }`rk`ex` ay c`mqeikùe &='01+ nayirk~` `o`hl8
ς ? 7 &e ’ 0+y? , ω? &='01+
Nlena8
ς ? 7 ay~`nçy~ki` na c` nky~rkoqikùe na c` HK iq`nr`n`
y? 7 }`rk`ex` gqay~r`c
ω? 7 }`rk`ex` zloc`ikle`c &c` n`n` ae c` fkzù~ayky eqc`+
e 7 ~`g`ðl gqay~r`
Ahagzcl ) 0>% Qe m`orki`e~a na ganknlray na IL `mkrg` pqa c` nay}k`ikùe ay~àen`r
zloc`ikle`c na ay~ly `z`r`~ly ay gaelr pqa 9 zz~% Ya y`i` qe` gqay~r` `ca`~lrk` na
0: `z`r`~ly# { ya i`ciqc` c` nay}k`ikùe ay~àen`r gqay~r`c na 0%=% ¼Avky~a yqmkikae~a
a}knaeik` ile ΰ 7 :%:1 z`r` `zl{`r c` ile~aeikùe nac m`orki`e~a<
Ylcqikùe8
0% Zrkgar`gae~a# naoknl ` pqa yar rapqkara na~argke`r yk c` nay}k`ikùe ay~àen`r ay
gaelr l gaely pqa 9 zz~# c` zrqao` na fkzù~ayky `c~are`~k}` ay F08ω? 3 ;% Zlr cl
~`e~l# c` zrqao` na fkzù~ayky eqc` naoa yar Fl8ω? 7 ;
?% C` rabkùe na raif`xl ay ς ? 3 ς ?0'ΰ4e'0 l ya` ς ? 3 ς ?%;14; l ya` ς ? 3 9%99
9% C` ay~`nçy~ki` ` qy`rya ay8 ς ? 7 &e ’ 0+y? , ω?
>% Cly iàciqcly yle8 ς ? 7 ;&0%=+? , ; 7 ?%1=
1% Ae ileicqykùe# ya raif`x` c` fkzù~ayky eqc` { ya nkia pqa yk f`{ yqmkikae~a a}knaeik` z`r` zl{`r c` ile~aeikùe nac m orki`e~a%
=% Ac }`clr na z ya f`ia oqyi`enl ?%1= ae c` ~`oc` na c` nky~rkoqikùe na HK iq`nr`n`
ile η 7 ; { }agly pqa ay &%:?1 3 z 3 %:0+% Qy`enl c` mqeikùe &1'?;+# { yqy~k~q{aenl
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 348/846
Nr% Fëi~lr Pqa}anl Qrç`y
='9:
cly }`clray n`8 &:%:?1 ’ :%:0+,&?%2 ’ ?%:;+ 7 &:%:?1 ’ V+,&?%2 ’ ?%1=+ { c`
zrlo`okckn`n ay z 7 :%:??% C` mkbqr` na `o`hl gqay~r` ay~` yk~q`ikùe%
Mkbqr` =%1% Bràmki` gly~r`enl ac }`clr na c` ay~`nçy~ki` ς ?# c` rabkùe na raif`xl# c`
rabkùe irç~ki` { ac }`clr na c` zrlo`okckn`n z# z`r` ac Ahagzcl )09%%
Ahagzcl )01% Ae qe ay~qnkl na `flrrl na aearbç` acëi~rki` &cl pqa li`ykle`rç` pqa
fqokar` gaely ile~`gke`ikùe nac gankl `gokae~a+ ya loyar}` pqë# c` }`rk`ex`
&zloc`ikle`c+ nac ileyqgl ay na ?6%: dRf% Ya naikna zlear mlily mcqlrayiae~ay {
`z`b`r c`y cqiay iq`enl el ya qyae# z`r` }ar yk f`{ qe` ranqiikùe ae c` }`rk`ikùe
nac ileyqgl% Z`r` ay~l ya y`i` qe` gqay~r` `ca`~lrk` na ?= ileyqgly na aearbç`# {
ya ay~kg` qe` }`rk`ex` gqay~r`c na 0=%: dRf% Qy`r qe ek}ac na ykbekmki`eik` na ΰ 7
:%:1# { zrlo`r pqa c` }`rk`ex` nac ileyqgl na aearbç` ya f` ranqiknl# o`hl c`y
ilenkikleay n`n`y% ^`gokëe f`iar qe` zrqao` na fkzù~ayky el ~r`nkikle`ci`ciqc`enl ac }`clr na c` zrlo`okckn`n z a ke~arzra~`rc`# `ilrnagae~a%
Ylcqikùe8
0% C` zrqao` na fkzù~ayky eqc` ay Fl8ω? 7 ?6%:% C` zrqao` na fkzù~ayky `c~are`~k}` ay
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 349/846
Nr% Fëi~lr Pqa}anl Qrç`y
='90
F08ω? 3 ?6%:%
?% C` rabkùe irç~ki` ya i`ciqc` oqyi`enl ac eqgarl na br`nly na ckoar~`n η 7 ?= ae
c` ~`oc` na c` nky~rkoqikùe na c` HK iq`nr`n` ile ς ?%:14?= { n` 01%92; &zrlo`okckn`nayna c` ilc` kemarklr l kxpqkarn`+%
9% Qy`enl c` ay~`nçy~ki` na c` nky~rkoqikùe na c` HK iq`nr`n` ς ? z`r` c` zrqao` na
fkzù~ayky &='01+# ay~l ay# ς ? 7 &e ' 0+y? , ω?# { yqy~k~q{aenl cly }`clray n`8
ς ?i`ci% 7 &?= ’ 0+&0=%:+ , ?6%: 7 0>%?;
>% @flr`# ilgz`r`enl ac }`clr na ς ?i`ci% 7 0>%?; ile c` rabkùe irç~ki` kxpqkarn` na
01%96# ay naikr# ς ?i`ci%
7 0>%?; 3 ς ?%:14?=
7 01%96# ya raif`x` c` fkzù~ayky Fl8ω? 7 ?6%:
{ ya keicke` zlr c` fkzù~ayky `c~are`~k}`%
1% Ac }`clr na z ya f`ia oqyi`enl 0>%?; ae c` ~`oc` na c` HK iq`nr`n` ile η 7 ?= {
}agly pqa ay~à ae~ra δ? 7 %:1 ile ς ?? 7 01%92; { δ0 7 %:?1 ile ς ?0 7 09%6>>%
Yqy~k~q{aenl cly }`clray ae c` mqeikùe &1'?;+ { raylc}kaenl zlr V n`8
&δ? ’ δ0+,&ς ?? ’ ς ?0+ 7 &δ? ’ V+,&ς ?? ’ ς ?i`ci%+
&%:1 ' %:?1+,&01%> ’ 09%6+ 7 &%:1 ’ V+,&01%> ’ 0>%?;+
C` zrlo`okckn`n ay na z 7 %:?;% Zlr cl ~`e~l# yk c` fkzù~ayky eqc` ay }arn`nar`#
ayzar`rç`gly na ~aear qe }`clr na ς ? gày br`ena pqa# l kbq`c pqa 0>%?;# ile qe`
zrlo`okckn`n na %:9%
@zcki`ikùe na c` HK iq`nr`n`# ς ?
ae iq`e~l ` c` zrqao` na olen`n na `hqy~a
ilgz`r`enl c`y mraiqaeik`y loyar}`n`y &cl zrài~kil l cly rayqc~`nly na
c`olr`~lrkl+ { c`y mraiqaeik`y ~aùrki`y &cl ayzar`nl+
C` zrqao` na olen`n na `hqy~a ya qy` z`r` zrlo`r c` fkzù~ayky na pqa qe`mraiqaeik` loyar}`n` ay~à na `iqarnl ile `cbqe`y nky~rkoqikleay ~aùrki`y# l pqa
f`{ ileyky~aeik` ae~ra qe` nky~rkoqikùe fkzl~ë~ki` &ilgl c` nky~rkoqikùe elrg`c# c`
okelgk`c# a~i%+ pqë aei`ha ile qe` nky~rkoqikùe agzçrki` l gqay~r`c%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 350/846
Nr% Fëi~lr Pqa}anl Qrç`y
='9?
@pqç ya el~` pqë# c`y zrqao`y na fkzù~ayky eqc`y yle ykagzra qekc`~ar`cay
naraif`y% ^`gokëe# ay na el~`rya pqa ya zqana qy`r c` zrqao` na Dlcglblrl}'
Ygkrel} ae c`y zrqao`y na olen`n na `hqy~a%Nayirkzikùe na c` ay~`nçy~ki` na ς
?qy`n` z`r` c` zrqao` na olen`n na `hqy~a
ς ? 7 &l0 ’ a0+? , a0 $ &l? ’ a?+
? , a? $%%%$ &ld ’ ad + , ad &='0=+
d
ς ? 7 Ω &l h ’ a h+? , a h &='02+
h70
Nlena8
ς ? 7 ay~`nçy~ki` qy`n` z`r` c` zrqao` na olen`n na `hqy~a
l 7 mraiqaeik`y loyar}`n`
a 7 mraiqaeik` ayzar`n`
d 7 eøgarl na i`~ablrç`y nkmarae~ay na qe rayqc~`nl
e 7 eøgarl ~l~`c na i`yly l ~`g`ðl na c` gqay~r`
η 7 d ’ 0 7 eøgarl na br`nly na ckoar~`n
El~`8 Ae `cbqe`y li`ykleay# yk ya }`e ` `ilgln`r cly n`~ly zlr nky~rkoqikleay
~aùrki`y# ilgl c` okelgk`c# ya qy` c` rac`ikùe η 7 d ’ 0 ’ g &Yzkabac# 0;=0+%
Iq`enl ya qy` c` zrqao` na olen`n na `hqy~a# ac irk~arkl z`r` raif`x`r l
ra~aear c` fkzù~ayky eqc` ay pqa# yk ς ? 7 :# ae~leiay# c`y loyar}`ikleay ~aùrki`y { c`y
loyar}`n`y yle kbq`cay% Zarl# yk ς ? 5 :# ae~leiay# c`y mraiqaeik`y ~aùrki`y { c`y
loyar}`n`y el yle kbq`cay% Ay~l pqkara naikr pqa# yk ac }`clr na c` ay~`nçy~ki` ς ?i`ci% 5
ς ?~`o%# ae~leiay# ya raif`x` c` fkzù~ayky eqc`4 na l~r` g`ear`# ya ra~kaea Fl8%
Yqzlykikleay z`r` f`iar c`y zrqao`y na olen`n na `hqy~a
0% Cly n`~ly gqay~r`cay ileyky~ae na ile~aly na mraiqaeik` na nkmarae~ay i`~ablrç`y# d
na gqay~r`y `ca`~lrk y%
?% Z`r` i`n` qe` na c`y i`~ablrç`y d# c` mraiqaeik` ayzar`n` ay na iq`enl gaely 1%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 351/846
Nr% Fëi~lr Pqa}anl Qrç`y
='99
m&ς ?+
: ς
?
ΰ Mkbqr` =%=% _abc` na naikykùe ay~`nçy~ki` gly~r`enl c` rabkùe irç~ki` { c` rabkùe na
`iaz~`ikùe# z`r` c` zrqao` na olen`n na `hqy~a# ay naikr# qy`enl c` nky~rkoqikùe na
HK iq`nr`n`% &Ac`olr`ikùe zrlzk`+
Ahagzcl )0=% Ae qe ay~qnkl na yabqrkn`n gqekikz`c# ya `e`ckx` ac eøgarl na
`iiknae~ay zlr nç`y na c` yag`e`% Zrlo`r c` fkzù~ayky eqc` na pqa cly `iiknae~ay
liqrrae ile kbq`cay mraiqaeik`y ae cly 1 nç`y na c` yag`e`% Z`r` ay~l qy`r qe ek}ac
na ykbekmki`eik` na ΰ 7 :%:1 { i`ciqc`r ac }`clr na z% Cly n`~ly ya n`e ae c` ^@OC@
=%9 na `o`hl%
^@OC@ =%9% Mraiqaeik`y loyar}`n`y { ayzar`n`y% &Ac`olr`ikùe zrlzk`+
Nç` na c` yag`e` s Cqeay G`r~ay Gkërilcay Hqa}ay ]kareay
@iiknae~ay loyar}`nlys 90 >? 06 ?1 90
@iiknae~ay ayzar`nly s ?;%> ?;%> ?;%> ?;%> ?;%>
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 352/846
Nr% Fëi~lr Pqa}anl Qrç`y
='9>
Ylcqikùe8
Cly n`~ly i`ciqc`nly { cly rayqc~`nly na c` zrqao` na fkzù~ayky ya n`e `o`hl%
^@OC@ =%1% Iàciqcly z`r` c` zrqao` na olen`n na `hqy~a% &Ac`olr`ikùe zrlzk`+
I`~ablrç` Mraiqaeik` Mraiqaeik` &l ’ a+ &l ’ a+? &l ’ a+?,aloyar}`n` ayzar`n`
Cqeay 90 ?;%> 0%= ?%1= :%:620
G`r~ay >? ?;%> 0?%= 016%2= 1%>:::
Gkërilcay 06 ?;%> '00%> 0?;%1= >%>?:>
Hqa}ay ?1 ?;%> '>%> 0;%9= :%=161
]kareay 90 ?;%> 0%= ?%1= :%:620
1
ς ? 7 Ω&l ’ a+? , a 7 &:%:620+ $ &1%>::+ $ &>%>?:>+ $ &:%=161+ $ &:%6200+ 7 0:%=1 h70
C` zrqao` na fkzù~ayky eqc` nkia pqa el f`{ nkmaraeik`y ae~ra c`y mraiqaeik`y
loyar}`n`y { c`y ayzar`n`y &cly `iiknae~ay yk liqrrae ile c` gkyg` mraiqaeik`+% C`
rabkùe irç~ki` ay nac av~ragl naraifl# ile η 7 d ’ 0 7 1 ’ 0 7 > br`nly na ckoar~`n%
C` ay~`nçy~ki` ~`oqc`n` ay na ς ?ΰ4 η 7 ς ?%:14> 7 ;%>;% Ae ileicqykùe# naoknl ` pqa ac
}`clr na ς ? 7 0:%=1 5 ς ?~`o% 7 ;%>;# ya raif`x` c` fkzù~ayky eqc`# { ya nkia pqa yk f`{
nkmaraeik`y ae~ra c`y mraiqaeik`y loyar}`n`y { c`y ayzar`n`y% @flr` qy`enl c`
mùrgqc` na ke~arzlc`ikùe z`r` c` HK iq`nr`n`# ile δ? 7 %:?1# δ0 7 %:1# ς ?? 7 00%0>#
ς ?0 7 ;%>66 { ς ?i`ci%7 0:%=1 { yqy~k~q{aenl ~lnly cly }`clray n`8
&:%:?1 ’ :%:1+,&00%0> ’ ;%>66+ 7 &:%:?1 ’ V+,&00%0> ’ 0:%=1+_aylc}kaenl zlr ac }`clr ` ke~arzlc`rya n` V 7 :%:01 7 z 7 :%:01%
Ahagzcly ile c` ~ na Ay~qnk`e~a qy`enl ac zrlbr`g` Gkek~`o
Z`r` qy`r ac zrlbr`g` Gkek~`o ae c`y zrqao`y na fkzù~ayky ile c` nky~rkoqikùe na ~ ya
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 353/846
Nr% Fëi~lr Pqa}anl Qrç`y
='91
zrliana ilgl8
Y~`~ 5 O`yki Y~`~ky~kiy 5 0'y`gzca ~…
Zrliankgkae~l8Ae c` }ae~`e` na –]`rk`ocay‗ zlear cly n`~ly nac zrlocag` ae c` ilcqge` I0% Ae c`
}ae~`e` na –^ay~ ga`e‗ zlear ac zrlgankl zrlo`nl% Ae c` }ae~`e` na –Lz~kley‗ ae c`
}ae~`ekcc` na –@c~are`~k}a‗ zlear c` fkzù~ayky `c~are`~k}` naya`n` { cqabl zraykle`r c`
~aic` na –LD‗%
Ahagzcl )02% Ay~a zrlocag` ay~à rac`ikle`nl ile ac ahagzcl na c` yaiikùe na cly
gai`ekygly qy`nly z`r` i`ciqc`r ac }`clr na z% Qy`enl cly n`~ly na aya ahagzcl )>
ilrrayzlenkae~ay ` ay` yaiikùe { `zcki`enl c` mqeikùe na `rrko` nac zrlbr`g` Gkek~`o#
zrlo`r8
&`+ Fl8± 7 9>%1 }y% F08± ≪ 9>%1
&o+ Fl8± ≡ 9>%1 }y% F08± 3 9>%1
&i+ Fl8± ≮ 99%? }y% F08± 5 99%?
Nayzqëy na yqy~k~qkr ~lnly cly }`clray# ac zrlbr`g` Gkek~`o n` cly rayqc~`nly
gly~r`nly ae c` ~`oc` na `o`hl%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 354/846
Nr% Fëi~lr Pqa}anl Qrç`y
='9=
^@OC@ =%>% ^`oc` gly~r`enl cly iàciqcly faifly zlr ac Gkek~`o% &Ac`olr`ikùe zrlzk`+% \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
Ahagzcly na zrlocag`y na c` ~ na Ay~qnk`e~a qy`enl ac zrlbr`g` na Aviac
Ay~a ay qe ahagzcl rac`ikle`nl ile qe zrlocag` na loyar}`ikleay z`ray% ^`gokëe ya
n`e c`y key~rqiikleay z`r` zrlo`r c`y nkmaraeik`y ae~ra nly zrlgankly zloc`ikle`cay#iq`enl c`y }`rk`ex`y yle kbq`cay% Mke`cgae~a ya n`e c`y key~rqiikleay z`r` raylc}ar
zrlocag`y iq`enl c`y }`rk ex`y yle naykbq`cay%
Zlr ahagzcl# z`r` ac qyl na c` mqeikùe na ~ na Ay~qnk`e~a qy`n` z`r`
agz`rah`gkae~ly nackoar`nly ya zrliana na c` ykbqkae~a g`ear`8
^llcy 5 N`~` `e`c{yky 5 ~'^ay~ Z`kran ^l Ykgzcay mlr Ga`ey
Ahagzcl )02% Ay~a ay qe ahagzcl y`i`nl nac ~av~l na Zrlo`okckn`n { Ay~`nçy~ki` na
R`czlca a~ `c% &0;;;+% Ay~` ke}ay~kb`ikùe ay~à rac`ikle`n` ile ac nay`rrlccl na cl
cc`g`nl ai~lg{ilrrfkx`c# qe` rac`ikùe ykgokù~ki` ae~ra c`y r`çiay na cly àrolcay { qe
flebl ae c` pqa ya ~r`eymkarae gkear`cay nac flebl ` cly àrolcay { `xøi`ray na cly
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 355/846
Nr% Fëi~lr Pqa}anl Qrç`y
='92
àrolcay ` cly flebly% Ay~a avzarkgae~l ileyky~kù ae `zcki`r ek~rùbael ` c` gk~`n na cly
àrolcay { ` c` l~r` gk~`n l ya` ac brqzl na ile~rlc `c iq`c el ya ca `zckiù ac ek~rùbael%
Cly zayly na cly àrolcay ya rabky~r`rle ae br`gly `c mke`c nac avzarkgae~l% Zrlo`r pqael f`{ nkmaraeik`y ae~ra cly zayly na c`y nly zloc`ikleay na àrolcay% @yqgkr qe
z`ra`gkae~l ae ay~a zrlocag % @yqgkr ΰ 7 :%:1% Cly n`~ly ya n`e ae c` ~`oc` na `o`hl%
^@OC@ =%1% ^`oc` gly~r`enl cly n`~ly nac zrlocag`% &R`czlca a~ `c% 0;;;+
Yke ek~rùbael s :%9? :%19 :%?6 :%92 :%>2 :%>9 :%9= :%>? :%96 :%>9
Ile ek~rùbael s :%?= :%>9 :%>2 :%>; :%1? :%21 :%2; :%6= :%=? :%>=
Ylcqikùe8Ac zrlbr`g` Aviac n` cly rayqc~`nly ae c` ~`oc` na `o`hl%
^@OC@ =%=% ^`oc` gly~r`enl cly rayqc~`nly nac zrlbr`g` Gkek~`o% &Ac`olr`ikùe
zrlzk`+%
Ilgl ya }a ae c` ^@OC@ =%=# ac }`clr na c` ay~`nçy~ki` ~ ay na '?%2>% C`y rabkleay
irç~ki`y z`r` qe` { nly ilc`y yle na 0%69 { ?%?=# rayzai~k}`gae~a% @nagày# cly }`clray
na c` zrlo`okckn`n z yle na :%:0 { na :%:? z`r` qe` { nly ilc`y# rayzai~k}`gae~a% Ae
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 356/846
Nr% Fëi~lr Pqa}anl Qrç`y
='96
ileicqykùe# c` fkzù~ayky eqc` na el nkmaraeik`y ya raif`x` ae `goly i`yly { ya
ileicq{a pqa yk f`{ nkmaraeik`y ae~ra cly zayly na c`y nly gqay~r`y na àrolcay%
Zrlo`enl c`y nkmaraeik`y ae~ra nly zrlgankly zloc`ikle`cay# iq`enl c`y
}`rk`ex`y yle kbq`cay8
^llcy 5 N`~` @e`c{yky 5 ~'^ay~ ^l Ykgzcay @yyqgkeb Apq`c ]`rk`eiay
Ykgkc`rgae~a# z`r` c` mqeikùe na ~ na Ay~qnk`e~a iq`enl c`y }`rk`ex`y yle
naykbq`cay ya zrliana ilgl8
^llcy 5 N`~` @e`c{yky 5 ~'^ay~ ^l Ykgzca @yyqgkeb Qeapq`c ]`rk`eiay
Ya railgkaen` `c cai~lr qy`r ay~`y nly øc~kg`y mqeikleay na c` ~ na Ay~qnk`e~a ile ac
zrlbr`g` Gkek~`o%
Mqeikùe zrlo`okcçy~ki` na naeykn`n na c` nky~rkoqikùe M { yq `zcki`ikùe ae c`
ilgz`r`ikùe na }`rk`ex`y gqay~r`cay
C` nky~rkoqikùe M ~kaea gqif` `zcki`ikùe ae c` ilgz`r`ikùe na }`rk`ex`y
gqay~r`cay% Ay~` nky~rkoqikùe M ya aeiqae~r` ae zrlocag`y pqa ke}lcqir`e nly `
gày gqay~r`y% Naoknl ` pqa# c` ay~`nçy~ki` M ya namkea ilgl qe` rac`ikùe# c`
nky~rkoqikùe M na zrlo`okckn`n ~kaea nly z`ràga~rly razrayae~`nly zlr η0 { η?#
nlena ay~ly }`clray yle ae~arly zlyk~k}ly% Ac z`ràga~rl η0 ya cc`g` eøgarl na
br`nly na ckoar~`n nac eqgar`nlr { η? ya cc`g` ac eøgarl na br`nly na ckoar~`n nac
naelgke`nlr% Z`r` ay~kg`r cly br`nly na ckoar~`n η0 { η? ya qy` c` ~`oc` na c`
nky~rkoqikùe M n`n` ae ac `zëenkia na ay~a ckorl%
C` nky~rkoqikùe na M ay ykgkc`r ` c` nky~rkoqikùe na ~ na Ay~qnk`e~a { na HK
iq`nr`n` &ς ?
+# zlrpqa ay qe` m`gkck` na nky~rkoqikleay% I`n` z`r na }`clray na η0 { η? ayzaikmki`e qe` nky~rkoqikùe na M nkmarae~a% L~rlyç# M ay qe` }`rk`oca `ca`~lrk`
ile~keq` pqa }`rç` na iarl f`y~` kemkek~l% Naoknl ` pqa c`y }`rk`ex`y ae `goly# ac
eqgar`nlr { naelgke`nlr na c` rac`ikùe M# ay~àe aca}`n`y `c iq`nr`nl# ac }`clr na
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 357/846
Nr% Fëi~lr Pqa}anl Qrç`y
='9;
M ay ykagzra zlyk~k}l% C` mlrg` na c` iqr}` M ay `ykgë~rki` { yayb`n` f`ik` c`
naraif`% Yke ago`rbl# c` nky~rkoqikùe M ~kaena f`ik` c` ykga~rç`# ` gankn` pqa η0 {
η? `qgae~`e% El loy~`e~a# c` zrqao` na M ay av~rag`n`gae~a yaeykoca `nky~rkoqikleay pqa el yle elrg`cay { ay~` m`c~` na rloqy~ax el ya gahlr` ile
gqay~r`y br`enay &Glra a~ `c% 0;;9+% C` Mkbqr` =%2 gqay~r` }`rk`y iqr}`y na
naeykn`n na c` nky~rkoqikùe na M z`r` nkmarae~ay br`nly na ckoar~`n%
C` nky~rkoqikùe na M ya qy` ae yk~q`ikleay ile nly gqay~r`y z`r` y`i`r
kemaraeik`y `iari` na gày na nly }`rk`ex`y zloc`ikle`cay# ilgl ae ac i`yl na
zrlocag`y na `eàckyky na }`rk`ex`% Zlr ahagzcl# yk y?
0{ y?
?yle c`y }`rk`ex`y na
gqay~r`y `ca`~lrk`y kenazaenkae~ay na ~`g`ðl e0 { e? ~lg`n`y na zloc`ikleay
elrg`cay ile }`rk`ex`y zloc`ikle`cay ω?0 { ω?
?# rayzai~k}`gae~a# ae~leiay c`
rac`ikùe na `o`hl8
M 7 y?0,ω
?0 , y
??,ω
?? 7 ω
?? y?
0 , ω?0 y?
? &='06+
~kaea qe` nky~rkoqikùe na M ile η0 7 e0 ’ 0 { η? 7 e? ’ 0 br`nly na ckoar~`n
C` mqeikùe &='06+ ay `gzck`gae~a qy`n` z`r` f`iar zrqao`y na fkzù~ayky#
z`r` }ar yk c`y }`rk`ex`y yle kbq`cay l naykbq`cay% Qe` `zcki`ikùe na c` mqeikùe &='
06+ ay~à aemli`n` ae ac qyl# zlr ahagzcl# na c`y mqeikleay &='2+ l &='6+# ay naikr#
z`r` naiknkr yk c`y }`rk`ex`y yle kbq`cay l naykbq`cay%
Z`r` zrlo`r zlr }`rk`ex`y kbq`cay zloc`ikle`cay ya qy` ac ykbqkae~a irk~arkl
z`r` zrqao`y qekc`~ar`cay { zrqao`y okc`~ar`cay% Ay~` kemlrg`ikùe ya n` ae c` ~`oc`
na `o`hl &GiIc`}a a~ `c% 0;6?+%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 358/846
Nr% Fëi~lr Pqa}anl Qrç`y
='>:
^`oc` =%2% Nk`br`g` gly~r`enl cly irk~arkly pqa ya ykbqae z`r` zrqao`y nafkzù~ayky ile c` nky~rkoqikùe M%
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
Zrqao` qekc`~ar`c Zrqao` okc`~ar`cFl8 ω
?0 7 ω?
? Fl8 ω?
0 7 ω??
F`8 ω?
0 3 ω?? F`8 ω
?0 ≪ ω?
?
&l F`8 ω?0 5 ω?
?+
Zrqao` ay~`nçy~ki`8 Zrqao` ay~`nçy~ki`8
M 7 y??,y
?0 M 7 ]`rk`ex` gqay~r`c br`ena,}`rk`ex` gqay~r`c zapqað`
&l M 7 y?0,y
?? iq`enl F`8 ω
?0 5 ω?
?+ 7 y?0,y
?? iq`enl y?
0 5 y??
&l y??,y
?0 iq`enl y?
? 5 y?0+
_abkùe na raif`xl8 _abkùe na raif`xl8
Mi`ci% 5 M~`o% Mi`ci% 5 Mΰ,? iq`enl y?0 5 y?
?
nlena M~`o% ay~à o`y`n` ae η0 7 e? '0 nlena Mΰ,? ya o`y` ae η0 7 e? '0
{ η? 7 e0 ’ 0 br`nly na ckoar~`n% { η? 7 e0 ’ 0 br`nly na ckoar~`n
&l Mi`ci% 5 M~`o% nlena F`8 ω?0 5 ω?
? &l Mi`ci% 5 Mΰ,? iq`enl y?0 5 y?
?
nlena M~`o% ya o`y` ae η0 7 e0 ’ 0 nlena Mΰ,? ya o`y` η0 7 e0 ’ 0
{ η? 7 e? ’ 0 br`nly na ckoar~`n+ { η? 7 e? ’ 0 br`nly na ckoar~`n+
Mqae~a8 GiIc`}a a~ `c% &0;6?+
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 359/846
Nr% Fëi~lr Pqa}anl Qrç`y
='>0
% Mraiqaeik` rac`~k}`
Mkbqr` =%2% Bràmki` gly~r`enl qe` m`gkck` na nky~rkoqikleay na M ile nkmarae~ay
br`nly na ckoar~`n% Eù~aya pqa z`r` c` iqr}` ile η0 7 9: { η? 7 9: br`nly na
ckoar~`n# c` rabkùe irç~ki` ay kbq`c ` >%?6%
Mkbqr` =%6% Mkbqr` gly~r`enl c` nky~rkoqikùe M# ile ac }`clr irç~kil na M kbq`c `
>%?=# ile ΰ 7 :%:1%
Ahagzcl )06% Ay~a ahagzcl ay~à aei`gke`nl ` aeile~r`r cly }`clray irç~kily qy`enl
c` nky~rkoqikùe M% Ykaenl `yç# aeile~r`r8
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 360/846
Nr% Fëi~lr Pqa}anl Qrç`y
='>?
&`+ M:%:1 ile η0 7 = { η? 7 0:
Ylcqikùe8
Cly br`nly na ckoar~`n nac eqgar`nlr yle η0 7 = { cly br`nly na ckoar~`n nacnaelgke`nlr yle η? 7 0:% Ile qe }`clr na ykbekmki`eik` na ΰ 7 :%:1 na c` ~`oc` ya
caa 9%??% Zlr cl ~`e~l# M:%:14=#0: 7 9%??
&o+ M:%:0 ile η0 7 = { η? 7 0:
Ylcqikùe8
Eqa}`gae~a ya oqyi` ΰ 7 :%:0 ae c` ~`oc` na M ile η0 7 = { η? 7 0: { n` M:%:04=#0: 7
1%9;
&i+ Yk ac ~`g`ðl na qe` gqay~r` ay na e0 7 9 { ac ~`g`ðl na l~r` gqay~r` ay na e? 7
0:# aeile~r`r c` rabkùe irç~ki` ile ΰ 7 :%:1 { :%:0% Nkoqh`r qe` bràmki` yað`c`enl c`
rabkùe irç~ki` iq`enl ΰ 7 :%:1%
Ylcqikùe8
M:%:14?#; 7 >%?= { M:%:04?#; 7 6%:?
C` Mkbqr` =%6 gqay~r` c` rabkùe irç~ki` { yq }`clr ilrrayzlenkae~a ile qe ek}ac na
ykbekmki`eik` na 1%
Ahagzcl )0;% Ay~a zrlocag` ay~à aei`gke`nl ` ay~kg`r ac }`clr na c` zrlo`okckn`n
z z`r` zrqao`y na M% Zlr ahagzcl# ile ΰ 7 :%:1# z`r` qe` zrqao` na fkzù~ayky ile e0
7 1 { e? 7 2 { ile qe }`clr na Mi`ci% 7 1%2: c` rabkùe irç~ki` ay M:%:14>#= 7 >%19%
Ae~leiay# `c ilgz`r`r ac }`clr na Mi`ci% 7 1%2: ile M:%:14>#= 7 >%19 ya raif`x` c`
fkzù~ayky% Yke ago`rbl# ay~` zrqao` na fkzù~ayky ~r`nkikle`c el nkia# pqë ~`e~`
mknackn`n ya ca zqana n`r ` ac rayqc~`nl lo~aeknl% Z`r` ay~l# ya f`ia qe` zrqao` nafkzù~ayky el ~r`nkikle`c qy`enl ac }`clr na c` zrlo`okckn`n z% Ykaenl `yç# ya oqyi`
ae c` ~`oc` na c` nky~rkoqikùe M ac }`clr na Mi`ci% 7 1%2:# ile > { = br`nly na ckoar~`n
{ ile ΰ 7 :%:1# zarl }agly pqa el ay~à avzcçik~`gae~a gly~r`nl% Yke ago`rbl#
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 361/846
Nr% Fëi~lr Pqa}anl Qrç`y
='>9
}agly pqa ay~à ae~ra >%19 { ;%01 ile yqy }`clray rayzai~k}ly na ΰ 7 :%1: { %:0:#
zlr cl ~`e~l c` zrlo`okckn`n ay %:0 3 z 3 %:1% @flr`# z`r` lo~aear qe }`clr na z gày
ayzaikmkil ya qy` c` mùrgqc` na ke~arzlc`ikùe &1'9:+ 8&δ? ’ δ0+,&M? ’ M0+ 7 &δ? ’ V+,&M? ’ Mi`ci%+
Nlena δ? 7 }`clr zlriae~q`c gày `c~l pqa ac }`clr na Mi`ci%# δ0 7 }`clr zlriae~q`c
gày o`hl pqa Mi`ci%# M? 7 }`clr na c` nky~rkoqikùe M ilrrayzlenkae~a ` δ?# M0 7 }`clr
na c` nky~rkoqikùe M ilrrayzlenkae~a ` δ0# V }`clr pqa ya pqkara ke~arzlc`r { Mi`ci% 7
}`clr i`ciqc`nl%
@flr` ile δ?
7 :%:1# δ0
7 :%:0# M?
7 >%19# M0
7 ;%01 { Mi`ci%
7 1%2: { yqy~k~q{aenl {
raylc}kaenl zlr V n`8
&:%:1 ’ :%:0+,&>%19 ’ ;%01+ 7 &:%:1 ’ V+,&>%19 ’ 1%2:+
V 7 z 7 :%:>
Ahagzcl )?:% Yqzùeb`ya pqa qe kebaekarl `gokae~`c y`i` nly gqay~r`y `ca`~lrk`y
na nly yk~kly nkmarae~ay ` cl c`rbl na qe` ilrrkae~a na `bq` { gkna c`y
ileiae~r`ikleay na NOL1% Z`r` c` zrqao` na fkzù~ayky ac kebaekarl pqkara qy`r ΰ 7
%0:% C` zrkgar` gqay~r` ileyky~a na e0 7 ?1 ileiae~r`ikleay na NOL1# iq{l
zrlgankl ay na V 0 7 ?1 gb,C ile qe` nay}k`ikùe ay~àen`r na y0 7 21 gb,C%
Ykgkc`rgae~a# c` yabqen` gqay~r` ileyky~a na e? 7 ?1# V ? 7 0?1 gb,C ile y? 7 >=%
Z`r` ay~l# ya ~kaea pqa f`iar qe` naikykùe yk ya }` ` qy`r c` nky~rkoqikùe &='2+ na ~
na Ay~qnk`e~a pqa rapqkara na }`rk`ex`y kbq`cay {,l c` nky~rkoqikùe &='6+ pqa el
rapqkara na }`rk`ex`y kbq`cay% Z`r` raylc}ar ay~a zrlocag` f`iar cl ykbqkae~a8
&`+ Zrlo`r c` fkzù~ayky eqc` na pqa c`y }`rk`ex`y na c`y nly gqay~r`y yle kbq`cay%&o+ @nagày# i`ciqc`r ac }`clr na z%
Ylcqikùe8
0% Naoknl ` pqa ya pqkara na~ai~`r qe` nkmaraeik` ae c`y }`rk`ex`y zloc`ikle`cay#
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 362/846
Nr% Fëi~lr Pqa}anl Qrç`y
='>>
~aenragly pqa ay~kg`r# {` ya` ω?0 5 ω?
?# l okae# ω?? 5 ω?
0%
?% Zlr cl ~`e~l# c` fkzù~ayky `c~are`~k}` ay F`8ω?0 ≪ ω?
?%
9% C` zrqao` ay okc`~ar`c# ay naikr8 Fl8ω?0,ω?? 7 0 { F`8ω?0,ω??%>% C` zrqao` ay~`nçy~ki` ay8
M 7 }`rk`ex` gqay~r`c br`ena,}`rk`ex` gqay~r` zapqað` 7 y?0,y
??
1% C`y yqzlykikleay yle na pqa c`y gqay~r`y ~kaeae mraiqaeik`y rac`~k}`y pqa yle
`zrlvkg`n`gae~a elrg`cay% @nagày# ya yqzlea pqa c`y gqay~r`y yle `ca`~lrk`y a
kenazaenkae~ay%
=% C` naikykùe ay~`nçy~ki` ya o`y`rà ae ilgz`r`r c` rabkùe irç~ki` na 0%;6# ile ac
}`clr ay~`nçy~kil# ay~l ay8 Mi`ci% 5 M~`o% 7 M%:14?>#?> 7 0%;6
Nlena η0 7 e0 ’ 0 7 ?> { η? 7 e? ’ 0 7 ?> br`nly na ckoar~`n
2% @flr` ya i`ciqc` c` zrqao` ay~`nçy~ki` &='06+ { ya yqy~k~q{ae cly }`clray8
M 7 y?0,y
?? 7 &2=+?,&>=+? 7 ?%29
6% Naoknl ` pqa# ?%29 5 0%;6# zlr cl pqa ya raif`x` Fl8 na }`rk`ex`y kbq`cay%
;% Qy`enl ΰ 7 %0: ay~l nkia pqë# ylc`gae~a qe` }ax ae nkax# ay~` zrqao` ay~`nçy~ki`
ely cca}`rç` ` ileicqkr arrùea`gae~a pqa c`y }`rk`ex`y ω?0,ω?
? mqar`e nkmarae~ay#
iq`enl na faifl mqar`e kbq`cay%
0:% Z`r` i`ciqc`r z ya oqyi` ac }`clr na ?%29 ae c` ~`oc` M ile η0 7 ?> { η? 7 ?> {
ay~à ae~ra %0:: { %:1:% Ay~l ay8 %:1: 3 z 3 %0::%
El loy~`e~a# yk ya naya`r` gày zraikykùe ya zqana qy`r c` mùrgqc` na ke~arzlc`ikùe
&1'9:+# ile δ? 7 %0::# δ0 7 %:1:# ile M? 7 0%;6# M0 7 ?%>0 { Mi`ci% 7 ?%29
Yqy~k~q{aenl ~lnly cly }`clray ae &1'9:+ n`8 z 7 :%:09&?+ 7 %:?% &Eù~aya pqa `pqçya gqc~kzcki` zlr ? zlrpqa c` zrqao` ay okc`~ar`c+%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 363/846
Nr% Fëi~lr Pqa}anl Qrç`y
='>1
Aharikikly I`zç~qcl =
=%0% Aeile~r`r cly }`clray irç~kily na ~ zlr cly iq`cay ac àra` nac av~ragl naraifl na c`
nky~rkoqikùe na ~ ay na ΰ 7 :%:1# { na ΰ 7 :%:0# yk8
&`+ η 7 0= &~Tΰ4ηW 7 ~T%;140=W 7 0%21# ~T%;;40=W 7 ?%169+
o+ e 7 ?6 &~Tΰ4ηW 7 ~T%;140=W 7 0%2:# ~T%;;4?6W 7 0%2:0
&i+ η 7 ∕ &~Tΰ4ηW 7 ~T%;14∕W 7 ?%99# ~T;;4∕W 7 ?%99+
=%?% F`iar ac zrlocag` =%0# zarl okc`~ar`cgae~a%
=%9% Z`r` g`e~aear ac ile~rlc na c` i`ckn`n kenqy~rk`c# qe m`orki`e~a na yky~ag`y na
ile~rlc na z`r~çiqc`y &ikicleay+# yqzlea pqa c` zrlnqiikùe na ay~ly yky~ag`y z`r` acile~rlc na z`r~çiqc`y 3 0: gkir`y# ~kaeae qe amkikaeik` zrlgankl na 9?% Z`r` zrlo`r
ay~` `ya}ar`ikùe ya ~lgù qe` gqay~r` na 6 ikicleay { ya gknkarle c`y amkikaeik`y na
i`n` qel z`r` aya ~`g`ðl na z`r~çiqc`y% C`y amkikaeik`y &+ mqarle8 ?;%># 9:%6# 9:%=#
90%1# 9?%0# 90%2# 9:%9# { 9:%6# rayzai~k}`gae~a% F`iar c`y ykbqkae~ay ay~kg`ikleay8
&`+ Ay~`ocaiar qe ke~ar}`cl na ilemk`ex` z`r` ±# ile ΰ 7 :%:1% &9:%06 3 ± 3 90%=?+
&o+ F`iar qe` zrqao` na fkzù~ayky okc`~ar`c `c ;1% &~ 7 '9%=?+
&i+ I`ciqc`r ac }`clr na c` zrlo`okckn`n# z% &:%::;+
=%>% Ae qe` zrqao` z`r` gankr c` `iqgqc`ikùe na zclgl `~glymërkil &Zo+ ae c`
y`ebra# ya ra`ckxù qe avzarkgae~l ile 01 }lcqe~`rkly% C` zrqao` ileyky~kù ae avzlear
cly yqha~ly ae qe yk~kl `can`ðl ` qe` zc`e~` na mqenkikùe na ga~`cay { na av`c~`r ac
ga~`olckygl# ay~l ay# ilrrkaenl% Nayzqëy na pqa cly yqha~ly ~argke`rle na ilrrar# ya
cay y`iù y`ebra { ya gankù c` ileiae~r`ikùe na Zo# ay naikr# `e~ay na ilrrar { nayzqëy
na ilrrar% Z`r` ay~l qy`r c` ay~`nçy~ki` na ~ gày `zrlzk`n` z`r` raylc}ar ay~a zrlocag`{ y`i`r c`y ileicqykleay `zrlzk`n`y% C` ~`oc` na `o`hl gqay~r` c` kemlrg`ikùe
rapqarkn` z`r` ay~a avzarkgae~l%
^`oc` gly~r`enl cly n`~ly nac zrlocag`% &Ac`olr`ikùe zrlzk`+
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 364/846
Nr% Fëi~lr Pqa}anl Qrç`y
='>=
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ El% yqha~l Ileiae~r`ikùe na Zo `e~ay na ilrrar Ileiae~r`ikùe na Zo nayzqëy na ilrrar
0 ?%2= 2%:?
? 1%06 9%0:9 ?%=6 1%>>> 9%:1 9%;;1 >%0: 1%?0= 2%:1 0:%?=2 =%=: 09%;06 >%2; 06%19; 2%9; 2%;00: 2%9: >%61
00 00%26 00%0:0? 9%;: 9%2>09 ?=%:: ;>%:90> =2%>6 ;>%:901 02%:> >0%2:
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
=%1% Ae qe` zrqao` z`r` nkyað`r qe apqkzl na ile~rlc z`r` z`r~çiqc`y agk~kn`y zlr qe`
mqae~a kenqy~rk`c# ya fkikarle nly zrqao`y z`r` y`oar iq`c na cly nly yky~ag`y naile~rlc ar`e gày amkikae~ay% C` zrkgar` zrqao` ileyky~kù ae key~`c`r qe mkc~rl na }knrkl
&o`bflqya+% C` l~r` zrqao` ileyky~kù ae `brab`r `c yky~ag` na ile~rlc nac o`bflqya# qe
ikicùe% Zrlo`r c` fkzù~ayky# `c ;1 na ek}ac na ilemk`ex` na pqë# ile ac apqkzl
`nkikle`c# el fqol nkmaraeik` ae c`y ranqiikleay na ile~`gke`e~ay% I`ciqc`r ac }`clr
na c` zrlo`okckn`n# z% C` ~`oc` na `o`hl gqay~r` cly rayqc~`nly na cly nly apqkzly na
ile~rlc% @yøg`ya pqa ac gqay~ral na yacaiikùe mqa ilgzca~`gae~a `c `x`r# yke
agz`rah`gkae~l { `yqgkr pqa c`y zloc`ikleay yle elrg`cay% &~ 7 9%1># z 7 :%:?6+
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 365/846
Nr% Fëi~lr Pqa}anl Qrç`y
='>2
^`oc` gly~r`enl c`y ileiae~r`ikleay na z`r~çiqc`y z`r` `go`y yk~q`ikleay%&Ac`olr`ikùe zrlzk`+
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
Ileiae~r`ikùe na z`r~çiqc`y ile Ileiae~r`ikùe na z`r~çiqc`y ile ac yky~ag`ac yky~ag` na ile~rlc `brab`nl na ile~rlc# `c iq`c ya ca `brabù ac ikicùe
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ Gkirlbr`gly,g9 Gkirlbr`gly,g9
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ >?0 ?:2>=? 02>:: >0?926 2>>09 00=
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
Loyar}`ikleay { iàciqcly yqbarknly8
@e~ay na ilgaex`r# ~aeagly pqa f`iar qe` naikykùe ylora iq`c zrqao` na ~ ay c` gày
`zrlzk`n`% Zqnkër`gly qy`r c` }arykùe na ~ z`r` loyar}`ikùe z`ray nlena f`{ qe
`z`rah`gkae~l nackoar`nl# ay~l ay qy`enl c` mqeikùe &='=+% ^`c }ax zqnkër`gly qy`r c`
mqeikùe na ~ pqa `yqga pqa c`y }`rk`ex`y na c`y zloc`ikleay yle kbq`cay { ile
gqay~r`y nac gkygl ~`g`ðl &mqeikùe &='2++% C` ~ariar` lzikùe# yarç` qy`r c` }arykùe na
~ z`r` }`rk`ex`y naykbq`cay { qy`enl c` mqeikùe &='6+% Yke ago`rbl# yk `yqgkgly pqa
ya qyù ac gë~lnl na yacaiikùe ilgzca~`gae~a `ca`~lrkl# yke agz`rah`gkae~l# { yk
`e`ckx`gly ` ykgzca }ky~` cly n`~ly na c` ~`oc` na `rrko`# zlnagly }ar pqa f`{ gqif`
}`rk`ikùe ae c`y loyar}`ikleay &~`gokëe ya zqana f`iar qe` zrqao` na fkzù~ayky ile ac
loha~l na }ar yk c`y }`rk`ex`y yle kbq`cay+# cl pqa ely keicke`rç` ` qy`r c` ~ariar`
lzikùe# ay~l ay# c` mqeikùe &='6+% El loy~`e~a# `e~ay na naiknkrya zlr ac qyl na ay~`mqeikùe ay ile}aekae~a f`iar qe` zrqao` ile c` mqeikùe ay~`nçy~ki` M 7 y?
0,y??%
=%=% Z`r` y`oar yk qe` nrlb` avzarkgae~`c zqana iqr`r cly yçe~lg`y na c` caqiagk`
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 366/846
Nr% Fëi~lr Pqa}anl Qrç`y
='>6
&zlrpqa c` cc`g`n` caqiagk` el ay qe` aemargan`n ae z`r~kiqc`r na c` y`ebra# ykel qe
yçe~lg` pqa `iqy` pqa ~lnl ac iqarzl ay~à aemargl# el øeki`gae~a c` y`ebra% Na el
zaey rya `yç# ae~leiay# ya nkrç pqa c` y`ebra ay qe` z`r~a kenazaenkae~a nac iqarzl+#0: yqha~ly ile ac yçe~lg` `}`ex`nl# mqarle ylga~knly ` qe` zrqao`% Ikeil na accly
raikokarle ac ~r`~`gkae~l avzarkgae~`c { ikeil na accly el% Ac ~kagzl na
yqzar}k}aeik`# ae `ðly# ya gknkù ae i`n` qel na cly yqha~ly% Zrlo`r ile ΰ 7 :%:1 pqa
ay~` nrlb` avzarkgae~`c mqa amai~k}`% @yqgkr pqa c`y nly nky~rkoqikleay yle elrg`cay
{ ile }`rk`ex`y kbq`cay% Cly n`~ly ya n`e `o`hl%
^`oc` gly~r`enl cly n`~ly% &Ac`olr`ikùe zrlzk`+ \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
Yqzar}k}aeik` ae `ðly \\\\\\\\\\\\\\\\\\\\\\\
Yqha~ly ~r`~`nly ?%0 1%9 0%> >%= ?%; \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ Yqha~ly yke ~r`~`gkae~l 0%; 0%1 ?%6 9%0 ?%:
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
=%2% Ae qe ay~qnkl na kebaekarç` nac `bq` na `eàckyky na lvçbael nkyqac~l &LN+ }`rkly
c`olr`~lrkly ya `}li`rle ` f`iar ay~ly `eàckyky qy`enl ac gë~lnl na Rkedcar &GR+
&~k~qc`ikùe+ { ac gë~lnl na acai~rlnly &GA+% Qy`r qe` ~ ay~`nçy~ki` na gqay~r`y
z`ra`n`y { zrlo`r pqa el f`{ nkmaraeik`y ae~ra cly nly gë~lnly% Qy`r ΰ 7 %:1%
I`ciqc`r ac }`clr na z% Cly n`~ly ya n`e ae c` ~`oc` na `o`hl% C` ~`oc` na `o`hl gqay~r`
cly n`~ly na lvçbael nkyqac~l &LN+ na }`rkly c`olr`~lrkly qy`enl ac gë~lnl na
Rkedcar { ac gë~lnl na acai~rlnly% C`y ileiae~r`ikleay nac lvçbael nkyqac~l &LN+# ya
avzray`e ae gb,C yle ae gb,C% Yqbaraeik`8 Qy`r ac zrlbr`g` na ilgzq~`nlr`Gkek~`o l Aviac% &~ 7 '?%>;# z 7 %:0+
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 367/846
Nr% Fëi~lr Pqa}anl Qrç`y
='>;
^`oc` gly~r`enl cly n`~ly nac zrlocag`% &Ac`olr`ikùe zrlzk`+
Gë~lnl na s 0%? 0%> 0%> 0%9 0%? 0%9 0%> ?%: 0%; 0%0 0%6 0%: 0%0 0%>Rkedcar
Gë~lnl na s 0%= 0%> 0%; ?%9 0%2 0%9 ?%? 0%> 0%9 0%2 0%; 0%6 0%6 0%6Acai~rlnly
=%6% Ay~a ay qe avzarkgae~l rac`ikle`nl ile fkbkaea kenqy~rk`c { yabqrkn`n# z`r`
ranqikr ac eøgarl na flgora'flr`y zarnkn`y# ilgl rayqc~`nl na cly `iiknae~ay
kenqy~rk`cay% Z`r` ay~l ya key~`cù qe eqa}l apqkzl na yabqrkn`n% Ae qe` zrqao` z`r`
gankr c` amkikaeik` nac apqkzl na yabqrkn`n key~`c`nl# ya av`gkeù qe` gqay~r`
`ca`~lrk` ae }`rkly naz`r~`gae~ly na ay~` kenqy~rk`% Ac eøgarl na flr`y'flgora
zarnknl ae ac gay `e~ay na c` key~ c`ikùe nac apqkzl { ac ykbqkae~a gay nayzqëy na
key~`c`r ac apqkzl# ac eøgarl na flr`y zarnkn`y zlr `iiknae~ay kenqy~rk`cay ya rabky~rù%
C` ~`oc` na `o`hl gqay~r` cly n`~ly na c` gqay~r` `ca`~lrk` pqa ya y`iù%
^`oc` gly~r`enl cly n`~ly na flr`y'flgora zarnkn`y `e~ay { nayzqëy na key~`c`r acapqkzl na yabqrkn`n% \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
Flr`y zarnkn`y zlr naz`r~`gae~l
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ Gay 0 ? 9 > 1 =
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ @e~ay na key~`c`r ac apqkzl 06 ?= >9 02 ?; 9:Nayzqëy na key~`c`r ac apqkzl 01 ?: 90 02 ?1 ?2
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
F`iar cly ykbqkae~ay iàciqcly8
&`+ ¼_a`cgae~a }`ckù c` zae` c` ke}arykùe ae c` key~`c`ikùe nac apqkzl na yabqrkn`n<
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 368/846
Nr% Fëi~lr Pqa}anl Qrç`y
='1:
=%;% Ya ilcaiikle`rle cly ykbqkae~ay n`~ly na qe` gqay~r` `ca`~lrk` na ùvknly na
`xqmra &YL?+# ae zzg# zrl}aekae~ay na qe` mqenkikùe% @yqgkr pqa cly n`~ly zrl}kaeae
na qe` zloc`ikùe elrg`c na ùvknly na `xqmra% Qy`r ΰ 7 :%:1% C` ~`oc` na `o`hl n` c`kemlrg`ikùe%
^`oc` gly~r`enl cly n`~ly nac zrlocag`% &Ac`olr`ikùe zrlzk`+Ùvknly na `xqmra &zzg+ s 1= 16 16 1; 12 12 1= 12 16
F`iar cly ykbqkae~ay iàciqcly8
&`+ Ay~kg`r ac ke~ar} cl na ilemk`ex` nac ;1% Keicqkr ~lnly cly z`yly eaiay`rkly a
ke~arzra~`rcl `ilrnagae~a% &12%99 ° :%2=6+
&o+ Zrlo`r c` fkzù~ayky eqc` na pqa ac zrlgankl zloc`ikle`c ay na 16%1 zzg%
Ay~`ocaiar ~lnly cly z`yly pqa rapqkara ay~a zrlocag`% F`iar qe` bràmki` gly~r`enl
c`y rabkleay na raif`xl { `iaz~`ikùe%
&i+ F`iar qe` zrqao` na fkzù~ayky el ~r`nkikle`c a ke~arzra~`rc` `ilrnagae~a% F`iar
qe` bràmki` ile c`y zrlo`okckn`nay% &~ 7 '9%0;+
&n+ Yk el ya zqnkar` raif`x`r c` fkzù~ayky eqc`# gaeikle`r ~ray m`i~lray pqa zqnkar`ef`oar `mai~`nl ac rayqc~`nl na ay~a avzarkgae~l%
=%0:% F`iar ac gkygl zrlocag` =%; na cly ùvknly na `xqmra { zrlo`r c` fkzù~ayky eqc`
na pqa ac }`clr nac zrlgankl zloc`ikle`c ay na el gày na 1=%0 zzg% Qy`r qe ek}ac na
ykbekmki`eik` na :%:1% @nagày# ay~kg`r c` fkzù~ayky el ~r`nkikle`c &ac }`clr na z+ a
ke~arzra~`rc` `ilrnagae~a% F`iar qe` bràmki` gly~r`enl c` zrlo`okckn`n%
=%00% F`iar ac gkygl zrlocag` =%0: na cly ùvknly na `xqmra { zrlianar na c`ykbqkae~a g`ear`8
&`+ Zrlo`r c` fkzù~ayky na pqa ± ay na iq`enl gaely 16%1 qy`enl ac ek}ac na
ykbekmki`eik` na :%:1% &~ 7 '9%10# ya raif`x` Fl8+
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 369/846
Nr% Fëi~lr Pqa}anl Qrç`y
='10
&o+ I`ciqc`r ac }`clr na z% &z 7 :%:::?+
&i+ Br`mki`r cly rayqc~`nly%
=%0?% Qe m`orki`e~a na mqykocay `mkrg` pqa ile qe` ylorai`rb` na ?1# cly mqykocay yamqenkràe ae 0>%:: gkeq~ly# ae zrlgankl% Z`r` zrlo`r ay~` `mkrg`ikùe# ya ~lgl qe`
gqay~r` `ca`~lrk` na ?: mqykocay { ya ylga~kù ` qe` i`rb` na ?: { cly ~kagzly pqa
~`rn`rle ae mqenkrya ~q}karle qe zrlgankl na 0:%=9 gkeq~ly# ile qe` nay}k`ikùe
ay~àen`r na ?%>6 gkeq~ly% @yqgkaenl pqa c` zloc`ikùe gqay~ra`n` ay elrg`c# f`iar
qe` zrqao` na fkzù~ayky z`r` ramraen`r l raif`x`r c` `mkrg`ikùe nac m`orki`e~a na
mqykocay% @yqgkr ΰ 7 %:1% ^`gokëe# i`ciqc`r ac }`clr na z%
=%09% Ae qe ay~qnkl na yabqrkn`n ae cly i`gkely i`rra~arly# faifl z`r` a}k~`r cly
`iiknae~ay# c` zlckiç` manar`c na i`gkely iraa pqa c` }aclikn`n zrlgankl na cly
gl~lrky~`y# pqa g`eah`e ylora ikar~` xle` i`rra~ar`# avianae ac cçgk~a na }aclikn`n na
00: dkcùga~rly zlr flr`% Z`r` ay~l# ya ~lgù qe` gqay~r` `ca`~lrk` na ?: }afçiqcly ile
yqy rayzai~k}`y }aclikn`nay# ae dkcùga~rly zlr flr` rabky~r`n` zlr ac r`n`r% Cly
rayqc~`nly ae dkcùga~rly zlr flr` na i`n` qel na cly ?: }afçiqcly mqarle8 009%=#
001%:# 002%:# 006%:# 001%;# 6>%:# 62%:# ;:%:# 00:%:# ;1%:# ;6%:# ;;%:# 006%:# 0?:%:#
0?0%:# 00;%:# 006%:# 000%:# 00?%:# 00?%=% Qy`r ΰ 7 :%:1% F`iar c`y ykbqkae~ay
ay~kg`ikleay8
&`+ ¼Zrl}aae ay~ly n`~ly yqmkikae~a a}knaeik` z`r` `zl{`r c` `ya}ar`ikùe na c` zlckiç`
manar`c na i`gkely na pqa cly gl~lrky~`y ay~àe }klc`enl ac rabc`gae~l nac cçgk~a na
}aclikn`n na 00: dkcùga~rly zlr flr`< &El f`{ a}knaeik` na pqa ya ay~ë }klc`enl `
cçgk~a na }aclikn`n na 00: dkcùga~rly zlr flr`+& o+ Ay~kg`r ac ke~ar}`cl na ilemk`ex` ile ΰ 7 :%:1 { ile ΰ 7 :%0 z`r` ac zrlgankl
zloc ikle`c na }aclikn`n% &0:9%0> 3 ± 3 009%06+
=%0>% Ae qe ay~qnkl na kebaekarç` na g`eqm`i~qr`# ae qe aymqarxl zlr ay~`ocaiar ac
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 370/846
Nr% Fëi~lr Pqa}anl Qrç`y
='1?
~kagzl ay~àen`r z`r` ra`ckx`r na~argke`n` ~`ra` ae ac aey`goca na z`r~ay na
i`roqr`nlray z`r` `q~lgù}kcay# ac kebaekarl na zrlnqiikùe yacaiikle`# `ca`~lrk`gae~a#
` 0= ~r`o`h`nlray avzarkgae~`nly z`r` ra`ckx`r ay~` m`ae`% Ac ~kagzl zrlganklrapqarknl zlr cly 0= ~r`o`h`nlray mqa na 09 gkeq~ly ile qe` }`rk`ex` na ; gkeq~ly% Ac
kebaekarl na zrlnqiikùe naya` iley~rqkr qe ke~ar}`cl na ilemk`ex` na ;; z`r` c`
clebk~qn na ~kagzl nac }arn`narl zrlgankl ± rapqarknl z`r` ra`ckx`r c` m`ae`% F`iar
qe ke~ar}`cl na ilemk`ex` ile qe ek}ac na ykbekmki`eik` na :%:1% ¼Ilgl ya ilgz`r`e
cly nly ke~ar}`cly< ¼Iqàc ay gày `gzckl { zlrpqë<
=%01% Ae qe ay~qnkl fkzl~ë~kil na ileyqgl na b`ylcke`# ac dkclga~r`ha na b`ylcke`
n`nl zlr cly `q~ly na ikar~ly glnacly ay na 0:%> dkcùga~rly zlr ck~rl ile qe`
nay}k`ikùe ay~àen`r na 0%= dkcùga~rly zlr ck~rl% Ya i`ciqc` ac zrlgankl na raenkgkae~l
ae dkcùga~rly zlr ck~rl z`r` gqay~r`y na ay~a ~kzl na glnacly na `q~ly% ¼Iqàc ay c`
zrlo`okckn`n na pqa ac zrlgankl na raenkgkae~l na b ylcke` ya` na 0? dkcùga~rly zlr
ck~rl# yk ya y`i` qe` gqay~r` `ca`~lrk` na ?: `q~ly< @yqgkr qe ek}ac ykbekmki`e~a na ΰ 7
:%:0% &z ≆ %:::;6+
=%0=% Ac kebrayl zrlgankl gaeyq`c# na ikar~l brqzl na zrlmaykleky~`y ay ilgl ykbqa8
9:#:::# 9?#:::# 90#:::# ?;#:::# ?;#1::# 99#:::# 90#1::# 9:#1::# ?;#6::# ?;#;::%
¼Iqàc ay c` zrlo`okckn`n na pqa ac }arn`narl zrlgankl ya` na 90#?:: zayly< ¼Iqàcay
yarç`e cly m`i~lray pqa zqnkar`e `mai~`r ac }`clr na c` zrlo`okckn`n z<
=%02% Ae qe ay~qnkl ay~`nçy~kil# z`r` nagly~r`r pqa c` zrqao` na ~ na ay~qnk`e~a ay
kenazaenkae~a na c`y qekn`nay na gankikùe# ya y`i`rle gqay~r`y na c`y ~agzar`~qr`y
na flrely na c`nrkccar`y gankn`y ae br`nly Iacykqy &l
I+ { ae br`nly M`fraefak~ &l
M+%C` fkzù~ayky ay pqa ac zrlgankl na c` ~agzar`~qr` nac flrel ay na 1: lI% F`iar c`
gkyg` zrqao`# zarl `flr` ile ac zrlgankl ae lM% C` ~`oc` na `o`hl gqay~r` cly
rayqc~`nly na c`y ~agzar`~qr`y na cly flrely ae lI% Ile}ar~kr ay~`y ~agzar`~qr`y ` lM
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 371/846
Nr% Fëi~lr Pqa}anl Qrç`y
='19
ilgzca~`r c` ~`oc` na `o`hl { ilgz`r`r cly rayqc~`nly na c`y nly zrqao`y na fkzù~ayky%
¼Yle cly rayqc~`nly na c` ~ na ay~qnk`e~a { na c` zrlo`okckn`n z# kbq`cay l nkmarae~ay<
^`oc` gly~r`enl c`y ~agzar`~qr y% &Ac`olr`ikùe zrlzk`+%^agzar`~qr`y lI s >2 11 =6 11 10 1: >; >1 19 >2 >6 10
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ ^agzar`~qr`y lM s
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
=%06% Aeile~r`r cly }`clray irç~kily na ς ?# zlr cly iq`cay ac àra` na c` ilc` naraif`
na c` nky~rkoqikùe ay na :%:1 &ς ?%;1+# yk cly br`nly na ckoar~`n yle na8
&`+ η 7 01
&o+ η ?0
&i+ η 7 1:%
=%0;% Z`r` ay~a zrlocag`# ya n`e cly ykbqkae~ay n`~ly lo~aeknly na qe` gqay~r` na
ileiae~r`ikleay &ae gb,C+ na ek~r`~ly &EL9'+ ~lg`nly nac amcqae~a na qe` zc`e~` na
~r`~`gkae~l na `bq`y rayknq`cay kenqy~rk`cay% Iley~rqkr qe ke~ar}`cl na ilemk`ex`
z`r` ac }arn`narl }`clr na c` }`rk`ex`# ay naikr# c` }`rk`ex` zloc`ikle`c ω?# qy`enlqe ek}ac ykbekmki`e~a na ΰ 7 :%:0% &:%?0 3 ω? 3 0%90+
^`oc` gly~r`enl cly n`~ly nac zrlocag`% &Ac`olr`ikùe zrlzk`+
92%=0 96%=0 92%=; 92%2? 9=%21 96%=0
96%66 96%0; 92%66 96%:: 92%?: 92%?:
92%19 96%?0 96%00 92%>: 92%>: 9;%9;
=%?:% Yk qe` gqay~r` na z`r~çiqc`y na i`ngkl `~glymërkil na qe ~`g`ðl na 02 gkir`y
~kaea qe` }`rk`ex` na y? 7 0;=%96# aeile~r`r ac ke~ar}`cl na ilemk`ex` z`r` c` }`rk`ex`
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 372/846
Nr% Fëi~lr Pqa}anl Qrç`y
='1>
zloc ikle`c# yk cly ek}acay ykbekmki`e~ay yle na8
&`+ :%:1
&o+ :%:0%=%?0% Yk ~aeagly qe ~`g`ðl na gqay~r` na e 7 ?: { qe ek}ac na ΰ 7 %:1# ae~leiay#
aeile~r`r cly }`clray irç~kily na c` nky~rkoqikùe na c` HK iq`nr`n` yk8
&`+ C` zrqao` ay qekc`~ar`c kxpqkarn` &0:%002+
&o+ Yk c` zrqao` ay okc`~ar`c &6%;:2# 90%>0+
&i+ Yk ya `yqga qe` zrqao` na olen`n na `hqy~a &9:%0>+
=%??% Ae qe ay~qnkl na `flrrl na aearbç` acëi~rki` &pqa ile~rkoqkrç` ` gaely
ile~`gke`ikùe `gokae~`c% Zlrpqa<+ ya aecky~ù ac ileyqgl na aearbç` acëi~rki` &ae
dRf+ nqr`e~a 2 `ðly nkmarae~ay% Qy`enl qe ek}ac na ilemk`ex` na ;1 zrlo`r c`
`mkrg`ikùe na pqa c` nay}k`ikùe ay~àen`r z`r` ~lnly ayly `ðly ay na 0#:::#:::%
^`oc` gly~r`enl cly n`~ly nac zrlocag`% &Ac`olr`ikùe zrlzk`+
00#;>9 00#>=9 0:#26; ;;:2 ;:0? ;;>? 00#019
=%?9% Ac ckorl Acagae~`r{ Y~`~ky~kiy nac `q~l G`rkl ^rklc` &0;;1+ n` qe ahagzcl na
qe r`nk`nlr na qe `q~l pqa ile~kaea 9261 gC na `e~kilebac`e~a% @yqgkaenl pqa
c`y mcqi~q`ikleay yle kea}k~`ocay# ac g`eah`nlr na ile~rlc na i`ckn`n pqkara ay~`r
yabqrl na pqa c` nay}k`ikùe ay~àen`r ya` gaely pqa 9: gC% Na l~r` g`ear`#
`cbqely r`nk`nlray ya narr`g`rç`e# gkae~r`y pqa l~rly# pqa el ~aenrç`e yqmkikae~a
`e~kilebac`e~a# el% Z`r` ay~l ya yacaiikle` qe` gqay~r` `ca`~lrk` iq{ly rayqc~`nly
ya n`e `o`hl% Qy`r ay~ly n`~ly z`r` iley~rqkr qe ke~ar}`cl na ilemk`ex` nac ;; z`r` ac }arn`narl }`clr na ω?% ¼Yqbkara ay~a ke~ar}`cl na ilemk`ex` pqa c`y
mcqi~q`ikleay ay~àe ae qe ek}ac `iaz~`oca< @yøg`ya pqa c`y nky~rkoqikleay na cly
ccae`nly na cly r`nk`nlray ile ac `e~kilebac`e~a ay~àe elrg`cgae~a nky~rkoqkn`y%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 373/846
Nr% Fëi~lr Pqa}anl Qrç`y
='11
&96%? 3 ω 3 ;1%2+
^`oc` aecky~`enl c` gqay~r` na cly ccae`nly na `e~kilebac`e~a%
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ 92=0 96=0 92=; 922? 9=21 96=09666 960; 9266 96:: 92?: 92>69219 96?0 9600 92>: 92>; 969;
&Mqae~a8 ^rklc`# 0;;1+
=%?>% Qe` gqay~r` `ca`~lrk` na 2:: ~r`o`h`nlray na c` kenqy~rk` z`r~kikzù ae qe`
zrqao` z`r` na~argke`r# iqàe~l ~kagzl eaiayk~`o`e z`r` yq zrl~aiikùe zaryle`c
f`ikaenl na~argke`n` m`ae`% Ay~l ya fkxl nayzqëy na ~lg`r qe iqryl naae~rae`gkae~l na fkbkaea kenqy~rk`c { yabqrkn`n% @yøg`ya qe` zrqao` na olen`n na
`hqy~a% @yqgkr e 7 6% F`iar cl ykbqkae~a
&`+ Zrlo`r c` fkzù~ayky eqc` na Fl8 { ra}ky`r yk c` zloc`ikùe gqay~ra`n` ay elrg`c l
`zrlvkg`n`gae~a elrg`c% &ς ? 7 ?:%9=# ς ?irç~ki` 7 01%10# ya raif`x` Fl8+
&o+ I`ciqc`r ac }`clr na z% &%:1 3 z 3 %::1+
=%?1% Qe kebaekarl `gokae~`c gkna c` i`e~kn`n na NOL1 zrlianae~ay na 01 cqb`ray `
cl c`rbl na qe` ilrrkae~a# c` iq`c ay~à ile~`gke`n` zlr ` qe` nayi`rb` kenqy~rk`c% Ac
kebaekarl razlr~` c`y ileiae~r`ikleay ae gb,C% Ilgl kemlrg`ikùe kekik`c ya y`oa pqa
c` yqg` na cly iq`nr`nly ay kbq`c ` 1:6%0 gb,C% Iley~rqkr qe ke~ar}`cl na ilemk`ex`
nac ;1 z`r` c` }`rk`ex` zloc`ikle`c% &;:%? 5 ω? 5 0;%>=+
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 374/846
Nr% Fëi~lr Pqa}anl Qrç`y
2'0
I@ZK^QCL 2
@eàckyky na ]`rk`ex`
Nkyaðly na `eàckyky na }`rk`ex` ilgzca~`gae~a `ca`~lrkx`nly%' Gë~lnl na
ilgz`r`ikleay gøc~kzcay z`r` y`oar iq`cay zloc`ikleay yle kbq`cay { iq`cay
yle naykbq`cay%' @eàckyky na }`rk`ex` na nkyaðl na oclpqay `ca`~lrkx`nly%'
Yqzlykikleay nac glnacl na oclpqay `ca`~lrkly ilgzca~ly%' @eàckyky na
}`rk`ex` ae nly yae~knly%' Ke~ar`iikùe ile @EL]@ na nly m`i~lray%' @eàckyky
na }`rk`ex` na ~ray yae~knly8 nkyaðl ilgzca~`gae~a `ca`~lrkl%' Ke~ar`iikùe ile
@EL]@ na nkyaðly m`i~lrk`cay na ~ray ic`ykmki`ikleay%' Ahagzcly na `eàckyky na
}`rk`ex` qy`enl ac zrlbr`g` Gkek~`o%'
Ac gë~lnl z`r` ilgz`r`r }`rkly zrlgankly ya cc`g` `eàckyky na }`rk`ex` l
ykgzcagae~a @EL]@% Ae yq gày ykgzca mlrg`# ac `eàckyky na }`rk`ex` ilgz`r`
}`rkly ~r`~`gkae~ly z`r` na~argke`r c` kbq`cn`n na cly zrlgankly% Ae ile~r`y~a ile
c` zrqao` na ~ na ay~qnk`e~a# pqa ay~qnk` c` kbq`cn`n na nly zloc`ikleay &Fl8 ±0 7
±?+# ac `eàckyky na }`rk`ex` ay~qnk` gày na ? nky~rkoqikleay# { qy` c` ay~`nçy~ki` M%
Ayzaiçmki`gae~a# ac glnacl @EL]@ ykgzca ay~qnk` c`y kbq`cn`nay na gày na ?
zrlgankly# ay~l ykbekmki` pqa ay~qnk` cly amai~ly na gày na nly /~r`~`gkae~ly#/ ay
naikr# na c` fkzù~ayky eqc` Fl8 ±0 7 ±? 7 ±9 7 %%%%% 7 ±e# ay~l ay# na pqa c`y }`rk`ex`y
na cly zrlgankly yle kbq`c ` iarl &ω?± 7 :+% @ zay`r na pqa ay~a `eàckyky na }`rk`ex`
ay~qnk` cly zrlgankly# `e`ckx`# na faifl# c` }`rk`ex` na c`y zloc`ikleay%
C`y zrlzkan`nay { yqzlykikleay ae ac `eàckyky na }`rk`ex` &@EL]@+ yle8
0% Z`r` c`y zrqao`y nac `eàckyky na }`rk`ex` ya qy` c` nky~rkoqikùe na M% Ay~`
nky~rkoqikùe M el ay ykgë~rki`# ykel yayb`n`# ay naikr# lockiq` f`ik` c` naraif`%
?% Cly }`clray na M zqanae yar na iarl l zlyk~k}ly# zarl el
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 375/846
Nr% Fëi~lr Pqa}anl Qrç`y
2'?
zqanae yar eab`~k}ly%
9% C` zrqao` na fkzù~ayky ay ykagzra qekc`~ar`c naraif`%
>% F`{ qe` nky~rkoqikùe na M nkmarae~a z`r` i`n` z`r na br`nly na ckoar~`n# &b%c%+%
C` Mkbqr` 2%0 gqay~r` ay~` yk~q`ikùe% Z`r` nael~`r cly br`nly na ckoar~`n z`r` ac
eqgar`nlr ya qy` c` `el~`ikùe# η0 { z`r` cly br`nly na ckoar~`n ac naelgke`nlr ya
qy` c` `el~`ikùe# η?%
1% C`y zloc`ikleay ~kaeae nky~rkoqikleay elrg`cay%
=% C`y zloc`ikleay ~kaeae c` gkyg` }`rk`ex` l nay}k`ikùe ay~àen`r% Yk ay~`
ilenkikùe el zqana yar iqgzckn`# c` zrqao` na M el ay }àckn`% Ae ay~a i`yl ya naoana qy`r qe` zrqao` na fkzù~ayky nkmarae~a%
2% C`y gqay~r`y yle `ca`~lrk`y a kenazaenkae~ay qe` na c` l~r`%
El~`8 Iq`enl el ya zqanae iqgzckr c`y ilenkikleay na elrg`ckn`n l na
kenazaenaeik` na cly n`~ly# qel ya ~kaea pqa ragk~kr ` c` zrqao`y el z`r`gë~rki`y#
pqa el rapqkarae na ay~`y yqzlykikleay%
Mkbqr` 2%0% Bràmki` gly~r`enl c` nky~rkoqikùe M% F`{ qe` nky~rkoqikùe nkmarae~a na
M z`r` i`n` z`r na br`nly na ckoar~`n nac eqgar`nlr# η0 { nac naelgke`nlr# η?%
&Ac`olr`ikùe zrlzk`+%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 376/846
Nr% Fëi~lr Pqa}anl Qrç`y
2'9
Nkyaðly na `eàckyky na }`rk`ex` ilgzca~`gae~a `ca`~lrkx`nly
Avky~ae nly ~kzly oàykily na `eàckyky na }`rk`ex`8 ac nkyaðl ilgzca~`gae~a
`ca`~lrkx`nl { ac nkyaðl na oclpqa ilgzca~`gae~a `ca`~lrkx`nl%
Ae ac i`yl nac nkyaðl ilgzca~`gae~a `ca`~lrkx`nl# ileliknl zlr `eàckyky na
}`rk`ex` ae qe yae~knl &@EL]@ na qe` ic`ykmki`ikùe+# ya `ykbe`e cly ~r`~`gkae~ly
`ca`~lrk`gae~a ` c`y qekn`nay avzarkgae~`cay% Ae ay~a nkyaðl ya y`i`e c`y gqay~r`y
kenazaenkae~agae~a# zlr cl ~`e~l# c` yacaiikùe na qe` gqay~r` el `mai~` c`
yacaiikùe na iq`cpqkar l~r` gqay~r`% Z`r` i`n` gqay~r` ya zqana i`ciqc`r ac
zrlgankl# V h { c` }`rk`ex` y
?
h% Zlr ahagzcl# yqzùeb`ya pqa ya pqkar`e zrlo`r iq`~rl g`ri`y na eaqgà~kily# 0# ?# 9 { ># z`r` na~argke`r yk f`{ nkmaraeik`y ile
rayzai~l ` c` nqr`ikùe% Z`r` ay~l ya zqanae `ykbe`r# `ca`~lrk`gae~a# qe` gqay~r` na
0: eaqgà~kily na i`n` g`ri`# ` nkb`gly ?1 }afçiqcly { zrlo`r yq nayb`y~`gkae~l%
Qe` }ax zrlo`n`y c`y g`ri`y na cly eaqgà~kily# ya qy` ac `eàckyky na }`rk`ex`# z`r`
}ar yk c`y g`ri`y nkmkarae ile rayzai~l ` yq nqr`ikùe%
Zlr l~r` z`r~a# ae ac i`yl na @EL]@ na nkyaðl na oclpqay ilgzca~`gae~a
`ca`~lrkly# ay~a aemlpqa ya qy` iq`enl ac arrlr avzarkgae~`c ay br`ena# cl pqa
ilecca}` `c el raif`xl na fkzù~ayky naoknl ` pqa f`{ gqif` }`rk`ikùe% Na g`ear`
pqa# `c –oclpqa`r‗ c`y loyar}`ikleay ya ranqia c` }`rk`ikùe% Ac ~ërgkel –oclpqa‗ ya
nark}` na nkyaðly avzarkgae~`cay `zcki`nly ` c` `brkiqc~qr`# ae cly iq`cay c`y
z`riac`y na ~karr`y na iqc~k}ly ya ramkarae ilgl –oclpqay‗% Zlr ahagzcl# ae ac i`yl
nac nkyaðl na oclpqa `ca`~lrkl# cly ~r`~`gkae~ly &ilgl mar~kckx`e~ay+ ya `ykbe`e
`ca`~lrk`gae~a ` qekn`nay nae~rl na i`n` oclpqa# ay naikr# na z`riac`y pqa ~aeb`e
yqacly z`raiknly%
Qe` yqzlykikùe kgzlr~`e~a nac glnacl z`r` qe nkyaðl na oclpqay ilgzca~ly
`ca`~lrkx`nly ay pqa cly amai~ly na ~r`~`gkae~l { na oclpqal ya `yqga pqa yle
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 377/846
Nr% Fëi~lr Pqa}anl Qrç`y
2'>
`nk~k}ly% Zlr ahagzcl# z`r` kcqy~r`r ay~` yk~q`ikùe# yk ya br`mki`e cly zrlgankly
zloc`ikle`cay }aryqy ~r`~`gkae~ly# nkb`gly na oclpqa 0 { ? {# yk c`y bràmki`y yle
z`r`cac`y# ya nkia pqa cly amai~ly na ~r`~`gkae~l { na oclpqay yle `nk~k}ly l pqa el
ke~ar`i~ø`e% Yke ago`rbl# yk c`y cçea`y ya irqx`e ae~ra yk# ya nkia pqa f`{
ke~ar`iikle l el `nk~k}kn`n%
Ac mlrg`~l na c` ~`oc` na @EL]@ na qe yae~knl ilgzca~`gae~a `ca`~lrkx`nl
ya n` `o`hl% C` ^@OC@ 2%0 n` qe` nayirkzikùe na ~lnly cly ilgzleae~ay na
ic`ykmki`ikleay qekc`~ar`cay l na nkyaðly ilgzca~`gae~a `ca`~lrkx`nly%
^@OC@ 2%0% @eàckyky na }`rk`ex` na qe yae~knl na nkyaðly ilgzca~`gae~a`ca`~lrkx`nly%
Mqae~a na c` Yqg` na &YY+ Br`nly na Iq`nr`nl &GY`+ Mi`ci% M~`o% ]`clr
}`rk`ikùe cly iq`nr`nly ckoar~`n gankl na z
^r`~`gkae~ly YY` ` ’ 0 GY` 7 YY`,&` ’ 0+ M0 7 GY`,y? MT0'ΰ4`'0#`&e'0+W
Arrlr YYa `&e ’ 0+ y?a 7 YYa,T`&e ’ 0+W
^l~`c YY~ `e ’ 0
Nlena8
`
YY` 7 e Ω & { k% ' { %% +?
&2'0+k70
` e
YYa 7 Ω Ω &{kh ’ { k%+?
7 YY~ ’ YY` &2'?+k70 h70
` e
YY~ 7 Ω Ω &{kh ’ { %%+?
&2'9+k70 h70
` 7 eøgarl na ~r`~`gkae~ly
e 7 ~`g`ðl na c` gqay~r`
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 378/846
Nr% Fëi~lr Pqa}anl Qrç`y
2'1
Z`r` nael~`r cly ykgolckygly qy`nly ae c` ^@OC@ 2%0# ay~ly ya n`e ae c` ~`oc` na
`o`hl%
^@OC@ 2%?% ^`oc` gly~r`enl cly ykgolckygly qy`nly ae c` ^@OC@ 2%0% &R`czlca
a~ `c% 0;;;+
^r`~`gkae~l8 0 ? …… k …… d
{00 {00 …… {k0 …… {d0
{0? {?? …… {k? …… {d?
% % …… % …… %
% % % %
{0e {?e …% {ke …%% {de
^l~`c ^0% ^?% …% ^k% …% ^d% ^%%
Zrlgankl { 0% { ?% …% { k% …% { d% { %%
Nlena8 {kh 7 h'ëykg` loyar}`ikùe nac k'ëykgl ~r`~`gkae~l
{ k% 7 zrlgankl na ~ln`y c`y loyar}`ikleay z`r` ac k'ëykgl ~r`~`gkae~l
{ %% 7 zrlgankl na ~ln`y c`y `e loyar}`ikleay l zrlgankl na cly
zrlgankly
^k% 7 ^l~`c na ~lnly cly zrlgankly
Ahagzcl )0% Ay~a ay qe ahagzcl rac`ikle`nl ile ac qyl na @EL]@ qekc`~ar`c l na
nkyaðl ilgzca~`gae~a `ca`~lrkx`nl% Z`r` ay~l ya ilcaiikle`rle c`y ileiae~r`ikleay
`~glymërki`y na YL? &ae zzg+ zrl}aekae~ay na 1 gqay~ra`nlray cli`ckx`nly `
nkmarae~ay nky~`eik`y &`ca`~lrk`gae~a `ykbe`n`y+# na qe` mqae~a kenqy~rk`c agkylr`%
Zrlo`r c` fkzù~ayky eqc` na pqa c`y 1 zloc`ikleay na YL? yle kbq`cay# ay naikr# Fl8
±0 7 ±? 7 ±9 7 ±> 7 ±1% I`ciqc`r ac }`clr na z% Cly n`~ly ya n`e ae c` ~`oc` na `o`hl%
Qy`r qe z`pqa~a na ilgzq~`nlr` z`r` zrliay`r cly n`~ly%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 379/846
Nr% Fëi~lr Pqa}anl Qrç`y
2'=
^@OC@ 2%9% ^`oc` gly~r`enl cly n`~ly nac zrlocag`%
Eøgarl na gqay~ra`nlr s 0 ? 9 > 1
1:: 11: =>6 2?: 6;:
10: 1>: =9: 2:: ;::
>;: 1:: =?: 20: ;?:
19: 1?: =:: 29= 66:
Ylcqikùe8
Yk ya qy` ac zrlbr`g` Aviac krya `8 @EL]@ ↟ Ykebca m`i~lr%
Qy`enl ay~a zrlbr`g`# cly rayqc~`nly ya n`e `o`hl8
^@OC@ 2%>% ^`oc` gly~r`enl cly rayqc~`nly na ay~a zrlocag qy`enl ac zrlbr`g`
na Aviac%
@EL]@8 Qe ylcl m`i~lr
_AYQGAE
Brqzly Ile~al Yqg` Zrlgankl ]`rk`ex`
Ilcqge` 0 > ?:9: 1:2%1 ?;0%===2Ilcqge` ? > ?00: 1?2%1 >;0%===2
Ilcqge` 9 > ?>;6 =?>%1 >:0
Ilcqge` > > ?6== 20=%1 ?91%===2
Ilcqge` 1 > 91;: 6;2%1 ?;0%===2
^`oc` na @EL]@
Mqae~a na]`rk`ikùe YY bc GY M i`ci% ]`clr'z M irk~%
Ae~ra cly brqzly >:=0?9%? > 0:019:%6 ?;=%16>= >%>A'0> 9%:111=6Nae~rl na cly
brqzly 1091 01 9>?%9999
^l~`c >00?16%? 0;
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 380/846
Nr% Fëi~lr Pqa}anl Qrç`y
2'2
Yk ya qy` ac zrlbr`g` Gkek~`o krya `8 Y~`~ ↟ @EL]@ ↟ Lea `{ &qey~`idan+%
Cly rayqc~`nly na ay~a zrlocag` qy`enl ac Gkek~`o ya n`e ae c` ~`oc` na `o`hl%
^@OC@ 2%1% ^`oc` gly~r`enl cly rayqc~`nly qy`enl ac Gkek~`o%
Lea'`{ @EL]@8 Gqay~ra`nlr 0# Gqay~ra`nlr ?# Gqay~ra`nlr 9# Gqay~ra`nlr >#Gqay~ra`nlr 1
Ylqria NM YY GY M Z
M`i~lr > >:=0?9 0:0190 ?;=%16 :%:::
Arrlr 01 1091 9>?
^l~`c 0; >00?16
y 7 06%1: _'Yp 7 ;6%21! _'Yp&`nh+ 7 ;6%>?!
Eù~aya pqa i`n` qel na ay~ly z`pqa~ay na ilgzq~`nlr` ~kaea yqy }ae~`h`y {
nay}ae~`h`y% Na iq`cpqkar g`ear`# `c hqxb`r zlr ac }`clr na M 7 ?;=%16 5555 Mirç~ki`
7 9%:=# c` fkzù~ayky eqc` na kbq`cn`n na zloc`ikleay na YL? ya raif`x` na qe`
g`ear` gqifl gq{ ykbekmki`e~a% Ay~` naikykùe ay ile~qenae~agae~a `zl{`n` zlr
ac }`clr ~`e zapqaðl na z 7 >%>v0:'0>
%
Ahagzcl )?% Ya n` c` ykbqkae~a kemlrg`ikùe ae c` ~`oc` na `o`hl rac`ikle`n` ile
ikar~l ay~qnkl ailcùbkil% @yøg`ya qe nkyaðl ilgzca~`gae~a `ca`~lrkx`nl% Y`i`r c`y
ileicqykleay `naiq`n`y%
^@OC@ 2%=% ^`oc` gly~r`enl cly n`~ly nac zrlocag`% &Ac`olr`ikùe zrlzk`+
=
^r`~`gkae~l Loyar}`ikleay ∐ {kh { k% H70
0 ;; >: =0 2? 2= 6> >9? 2?
? ;= 6> 6? 0:> ;; 0:1 12: ;1
9 =9 12 60 1; => 2? 9;= ==
> 2; ;? ;0 62 26 20 >;6 69
Ylcqikùe8
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 381/846
Nr% Fëi~lr Pqa}anl Qrç`y
2'6
Qy`enl qe z`pqa~a na zrlbr`g` na ilgzq~`nlr` n`8
^@OC@ 2%2% ^`oc` na `eàckyky na }`rk`ex`% &Ac`olr`ikùe zrlzk`+
Brqzly Ile~al Yqg` Zrlgankl ]`rk`ex`
^r`~`gkae~l 0 = >9? 2? >:=%6
^r`~`gkae~l ? = 12: ;1 ;2%=
^r`~`gkae~l 9 = 9;= == 6:%6
^r`~`gkae~l > = >;6 69 =;%?
^`oc` na @EL]@
Mqae~a na]`rk`ikle YY bc GY M i`ci% ]`clr'z M irk~%
Ae~ra cly Brqzly ?;>: 9 ;6: 1%;;:?? :%::>962 9%:;69;0Nae~rl na clybrqzly 9?2? ?: 0=9%=
^l~`c =?0? ?9
Ac }`clr na c` zrlo`okckn`n na z ay na :%::>>% Ay~a }`clr ~`gokëe ya zqana i`ciqc`r
g`eq`cgae~a oqyi`enl ac }`clr na c` Mi`ci% 7 1%;; ae c` ~`oc` na c` nky~rkoqikùe M a
ke~arzlc`enl ae~ra ac }`clr gày `c~l { ac gày o`hl qy`enl c` rac`ikùe &2'>+ na `o`hl8
&δ? ’ δ0+ , &M? ’ M0+ 7 &δ? ’ V+ , &M? ’ Mi`ci%+ &2'>+
Nlena8
δ? 7 }`clr zlriae~q`c na M gày `c~l pqa ac }`clr na Mi`ci%
δ0 7 }`clr zlriae~q`c na M gày o`hl pqa ac }`clr na Mi`ci%
M? 7 }`clr na c` nky~rkoqikùe M ilrrayzlenkae~a ` δ?
M0 7 }`clr na c` nky~rkoqikùe M ilrrayzlenkae~a ` δ0
V 7 }`clr pqa ya pqkara ke~arzlc`r
Mi`ci% 7 }`clr i`ciqc`nl qy`enl c` ~`oc` na c` nky~rkoqikùe M%
El~`8 Ac gai`ekygl pqa ya ykbqa z`r` ke~arzlc`r ay oqyi`enl ac }`clr na c` Mi`ci% 7
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 382/846
Nr% Fëi~lr Pqa}anl Qrç`y
2';
1%;; ae c` ~`oc` na c` nky~rkoqikùe M ile η0 7 9 &eqgar`nlr+ { η? 7 ?:
&naelgke`nlr+ { }agly pqa 1%;; ay~à ae~raδ? 7 %::0 ile M? 7 6%0: &}`clr gày `c~l+
{ δ0 7 %:0 ile M0 7 >%;> &}`clr gày o`hl+% Aeyabqkn`# yqy~k~qkgly ac }`clr na c`
Mi`ci% 7 1%;; { cly nagày }`clray ae c` mùrgqc` na ke~arzlc`ikùe &2'>+ z`r` n`r8
&%::0 ' %:0+,&6%0: ’ >%;>+ 7 &%::0 ’ V+,&6%0: ’ 1%;;+
_aylc}kaenl zlr V 7 %::17 z 7 %::1% Ay~a }`clr ay~à gq{ na `iqarnl `c }`clr na
%::>> na c` ^@OC@ 2%2% Ae ileicqykùe# ac }`clr na z 7 %::>> kenki` qe nkyaðl
avzarkgae~`c zraikyl { ileikyl%
Ahagzcl )9% Cly ek~r`~ly &EL
'
9+ razrayae~`e c` m`ya gày lvkn`n` ae ac ikicl nacek~rùbael% Baear`cgae~a# ay~l liqrra ae gq{ zapqað`y i`e~kn`nay ae c`y
yqzarmkikay na cly `cg`iae`gkae~ly na `bq`# zarl zqana avky~kr ae br`enay
i`e~kn`nay ae `cbqe`y `bq`y yqo~arràea`y% Ae i`e~kn`nay aviayk}`y# cly ek~r`~ly
zqanae li`ykle`r qe` aemargan`n kem`e~kc cc`g`n` ga~agabclokeagk`% &Gë~lnly
Ay~àen`ray z`r` ac av`gae nac `bq` { na c`y `bq`y rayknq`cay# 0;20+% Zlr ay~`
r`xùe# ac cçgk~a ay na >1 gb,C z`r` ac `bq` zl~`oca% Z`r` cly `eàckyky na cly ek~r`~ly#
avky~ae }`rkly gë~lnly% Zlr ahagzcl# qe gë~lnl ay ac nac àiknl maelcnkyqcmùekil4
l~rl ay ac gë~lnl na c` ranqiikùe na i`ngkl4 l~rl gày ay ac gë~lnl na àiknl
irlgl~rùzkil {# l~rl gày ay ac gë~lnl na orqike` &`ci`clkna ~ùvkil+% Z`r` ay~l# ya
fkxl qe ay~qnkl ay~`nçy~kil z`r` ilgz`r`r cly rayqc~`nly na cly iq`~rl gë~lnly
gaeikle`nly `rrko` z`r` `e`ckx`r cly ek~r`~ly% Cly ykbqkae~ay n`~ly ya n`e `o`hl%
Z`r` ay~l# cc`gagly ~r`~`gkae~l &0+ `c gë~lnl nac àiknl maelcnkyqcmùekil#
~r`~`gkae~l &?+ `c gë~lnl na c` ranqiikùe nac i`ngkl# ~r`~`gkae~l &9+ `c gë~lnl na
àiknl irlgl~rùzkil# { ~r`~`gkae~l &>+ `c gë~lnl na orqike`% C` ~`oc` na `o`hl n`
cly rayqc~`nly ae gb,C% @yqgkr qe ek}ac na ykbekmki`eik` na :%:1% F`iar cly
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 383/846
Nr% Fëi~lr Pqa}anl Qrç`y
2'0:
ykbqkae~ay iàciqcly8
&`+ Aecky~`r c`y yqzlykikleay kgzcki`n`y zlr ac glnacl na @EL]@%
&o+ F`iar qe` ~`oc` na `eàckyky na }`rk`ex` { zrlo`r pqa el f`{ nkmaraeik`y ae~ra
cly > gë~lnly%
&i+ Ay~kg`r ac }`clr na c` zrlo`okckn`n z { y`i`r c`y ileicqykleay `zrlzk`n`y%
^@OC@ 2%6% ^`oc` ile cly n`~ly% &Ac`olr`ikùe zrlzk`+
^r`~`gkae~l _ayqc~`nly na cly yaky `eàckyky ae gb,C
&0+ ;; >: =0 2? 2= 6>
&?+ ;= 6> 6? 0:> ;; 0:1
&9+ =9 12 60 1; => 2?
&>+ 2; ;? ;0 62 26 20
Ylcqikùe8
&`+ C`y yqzlykikleay kgzcki`n`y zlr ac glnacl na `eàckyky na }`rk`ex` na qe` ylc`
ic`ykmki`ikùe yle8
0% C`y iq`~rl zloc`ikleay na cly ek~r`~ly ay~àe elrg`cgae~a nky~rkoqkn`y%
?% C`y }`rk`ex`y na c`y iq`~rl zloc`ikleay na ek~r`~ly yle kbq`cay%
9% C`y ?> loyar}`ikleay &`eàckyky+ yle kenazaenkae~ay# ay naikr# pqa c`y gqay~r`y
mqarle yacaiikle`n`y `ca`~lrk`gae~a%
&o+ Qy`enl ac zrlbr`g` Gkek~`o krya `8
Y~`~ ↟ @EL]@ ↟ Lea `{ &qey~`idan+ n` cly kbqkae~ay rayqc~`nly gly~r`nly ae
c` ^`oc` 2%;%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 384/846
Nr% Fëi~lr Pqa}anl Qrç`y
2'00
^@OC@ 2%;% ^`oc` na @EL]@ z`r` cly rayqc~`nly na ahagzcl na `rrko` qy`enl ac
Gkek~`o% &Ac`olr`ikùe zrlzk`+
Lea'`{ @EL]@8 ^r`~`gkae~l 0# ^r`~`gkae~l ?# ^r`~`gkae~l 9# ^r`~`gkae~l >
Ylqria NM YY GY M Z
M`i~lr 9 ?;>: ;6: 1%;; :%::>
Arrlr ?: 9?2? 0=>
^l~`c ?9 =?0?
∘GY 7 ∘0=> 7 y 7 0?%60 _'Yp 7 >2%99! _'Yp&`nh+ 7 9;%>9!
Zlr l~r` z`r~a# qe gë~lnl ilr~l z`r` f`iar `eàckyky na }`rk`ex` na qe yae~knl# ay
naikr# g`eq`cgae~a# ya n` qy`enl ac mlrg`~l na c` ~`oc` na `o`hl%
^@OC@ 2%0:% ^`oc` na `eàckyky na }`rk`ex` &@EL]@+ z`r` qe` ic`ykmki`ikùe# ile
gqay~r`y na ~`g`ðly kbq`cay qy`enl ac gë~lnl `ora}k`nl% &Ac`olr`ikùe zrlzk`+%
Mqae~a na Yqg` na cly b%c% Iq`nr`nl nac Mi`ci% M~`o% ]`clr
]`rk`ikùe iq`nr`nly zrlgankl na z
Naoknl `c YY` 7 ∐^?,e ’ B
?,`e `’0 GY` 7 YY`,&`'0+ GY`,y
?a MT0'ΰ4`'0#`&e'0+W Ay~kg`nl
~r`~`gkae~l
_ayknql YYr 7 ∐V
?
' ∐ ^
?
,e `&e'0+ y
?
a 7 YYr ,`&e'0+
^l~`c YY~ 7 ∐V?
’ B?,`e e`'0
Nlena8
^?
7 iq`nr`nl na cly ~l~`cay
b%c% 7 η 7 br`nly na ckoar~`n
e 7 ~`g`ðl na c` gqay~r`
B 7 br`e ~l~`c
` 7 eøgarl na gqay~r`y
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 385/846
Nr% Fëi~lr Pqa}anl Qrç`y
2'0?
Ahagzcl )>% C` ~`oc` na `o`hl gqay~r` cly n`~ly na cly `eàckyky na nag`en`
pqçgki` na lvçbael &NPL+ faifly zlr 9 c`olr`~lrkly nkmarae~ay% Ya ~lg`rle 9
gqay~r`y na 1 loyar}`ikleay i`n` qe`% @yqgkr pqa c`y 9 gqay~r`y }kaeae na
zloc`ikleay elrg`cay `ca`~lrk`y { pqa ~kaeae c` gkyg` }`rk`ex`% @yqgkr qe ek}ac
na ykbekmki`eik` na ΰ 7 :%:1% F`iar cl ykbqkae~a8
&`+ Qe` ~`oc` ile qe `eàckyky na }`rk`ex` z`r` ac NPL%
&o+ Ay~`ocaiar c` rabkùe irç~ki`%
&i+ Zrlo`r c` fkzù~ayky eqc` na Fl8 ±0 7 ±? 7 ±9# l ya` pqa ω?± 7 :# ay naikr# pqa cly
zrlgankly na c`y ~ray zloc`ikleay na NPL yle kbq`cay% @nagày# ay~`ocaiar c`fkzù~ayky `c~are`~k}` `zrlzk`n`%
&n+ Yk ya raif`x` Fl8 i`ciqc`r ac }`clr na c` zrlo`okckn`n z%
Ya n` c` ~`oc` na `o`hl ile `cbqely iàciqcly zrackgke`ray8
^@OC@ 2%00% ^`oc` gly~r`enl cly iàciqcly zrackgke`ray% &Ac`olr`ikùe zrlzk`+
Eøgarl na gqay~r` &0+ &?+ &9+ Ilgoke`ikùe
Loyar}`ikùe 9 ; 0
2 0? ?
2 00 =
= 6 >
? 1 2
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
^l~`cay ?1 >1 ?: B 7 ;:
Zrlgankl V 1 ; > V 7 =
Ylcqikùe8
Qy`enl c`y ay~`nçy~ki`y na c` ^@OC@ 2%0:# cly iàciqcly yle8
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 386/846
Nr% Fëi~lr Pqa}anl Qrç`y
2'09
B 7 Ω^ 7 ΩV 7 ^0 $ ^? $ ^9 $%%%$ ^d 7 ?1 $ >1 $ ?: 7 ;:# `e 7 &9+&1+ 7 01
Zrlgankl baear`c l zrlgankl na cly zrlgankly 7 V 7 B , `e 7 ;: , 01 7 =
^`gokëe# V 7 & V 0 $ V ? $ V 9+ , ` 7 &1 $ ; $ >+ , 9 7 =
ΩV?
7 =66# e 7 1# ` 7 9# Ω^?
, e 7 9#:1: , 1 7 =0:
YY&ae~ra c`y gqay~r`y+ 7 &Ω^?
, e+ ' &Ω^+?
, `e 7 Ω^?
, e ' B?,`e
7 &?1?
$ >1?
$ ?:?+,1 ' T&?1 $ >1 $ ?:+
?W , T&9+&1+W 7 2:%:
El~` 08 c` yqg` na cly iq`nr`nly YY 7 YY&ae~ra c`y gqay~r`y+ gkna c` }`rk`ikùe ae~ra cly
zrlgankly gqay~r`cay `%
YY&nae~rl na c`y gqay~r`y+ 7 Ω V ? ' Ω^?,e 7 Ω& V ' V +? 7 =66 ' =0: 7 26
El~` ?8 YYr 7 YY&nae~rl na c`y gqay~r`y+ gkna c` }`rk`ikùe na c`y loyar}`ikleay nae~rl na
cly zrlgankly gqay~r`cay%
YY&~l~`c+ 7 YY&ae~ra c`y gqay~r`y+ $ YY&nae~rl na c`y gqay~r`y+
7 Ω V ? ' B?,`e 7 Ω& V ' V +
?
El~` 9% YY&~l~`c+ gkna c` }`rk`ikùe ~l~`c na c`y loyar}`ikleay `e%
C` }`rk`ex` na cly zrlgankly gqay~r`cay ay8y
? v 7 iq`nr`nl nac zrlgankl na YY&ae~ra c`y gqay~r`y+ 7 Ω& V ' V +
?, `'0
7 T&1 ' =+?
$ &; ' =+?
$ &> ' =+?W,9 ' 0
7 &'0?
$ 9?
' ??+,?
7 2%:
y?a 7 iq`nr`nl nac zrlgankl na YY&nae~rl na c`y gqay~r`y+
7 Ω& V ' V +?
, ` &e ' 0+
7 YY&nae~rl na c`y gqay~r`y+ , ` &e ' 0+
7 26 , 9&1 ' 0+
7 =%1
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 387/846
Nr% Fëi~lr Pqa}anl Qrç`y
2'0>
Gë~lnl na ilgz`r`ikleay gøc~kzcay z`r` y`oar iq`cay zloc`ikleay yle kbq`cay
{ iq`cay yle naykbq`cay
Qe` }ax pqa ya zrqao`e c`y fkzù~ayky na pqa cly zrlgankly yle kbq`cay# l
naykbq`cay# ae~leiay# eaiayk~`gly y`oar iq`cay zrlgankly yle naykbq`cay { iq`cay
yle kbq`cay% Z`r` ay~l# ya qy` cl pqa ya cc`g`e ilgz`r`ikleay gøc~kzcay avzcki`nly
zlr R`czlca a~ `c% 0;;9%
Ac `eàckyky na }`rk`ex` ay qe zrliankgkae~l zlnarlyl z`r` zrlo`r c`
flglbaeakn`n na qe brqzl na zrlgankly% Yke ago`rbl# yk raif`x`gly c` fkzù~ayky
na kbq`cn`n &Fl8±0 7 ±? 7 ±9 7 ±e+# { ely keicke`gly zlr c` fkzù~ayky `c~are`~k}` napqë# iq`enl gaely nly na cly zrlgankly yle kbq`cay# ~ln`}ç` el y`oagly iq`cay
na cly zrlgankly yle kbq`cay { iq`cay yle naykbq`cay% Ac qyl nac gë~lnl na
ilgz`r`ikleay gøc~kzcay kgzcki` f`iar }`rk`y ilgz`r`ikleay agz`rah`n`y ae~ra
cly ~r`~`gkae~ly l zrlgankly%
Zlr ahagzcl# c`y ilgz`r`ikleay agz`rah`n`y yle zrqao`y ilgl c` na `o`hl
c`y iq`cay nkiae pqa yle kbq`cay l pqa el f`{ nkmaraeik`8
Fl8±k ' ± h 7 : &2'1+
F08±k ' ± h≪ : &2'1`+
Z`r` f`iar ay~`y zrqao`y agz`rah`n`y qy`gly c` }arykùe na ~ na Ay~qnk`e~a na c`
mlrg` na8
V k ' V h~ 7 \„„„„„„„„„ ∟ &2'=+
y ?,e
Nlena8
V k 7 qely na cly zrlgankly pqa ya pqkar` ilgz`r`r
V h 7 l~rl na cly zrlgankly pqa ya pqkar` ilgz`r`r
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 388/846
Nr% Fëi~lr Pqa}anl Qrç`y
2'01
y 7 nay}k`ikùe ay~àen`r ilgoke`n` l c` r`çx iq`nr`n` nac iq`nrà~kil
zrlgankl nac arrlr GY
e 7 ~`g`ðl na c` gqay~r` na i`n` ~r`~`gkae~l
Ahagzcl )1% Ac ckorl Zrlo`okckn`n { Ay~`nçy~kyi` na R`czlca a~ `c% &0;;9+ n` qe
ahagzcl nac qyl na c`y ilgz`r`ikleay gøc~kzcay% C` ~`oc` na `o`hl n` cly n`~ly
rac`ikle`nly ile ay~a zrlocag`% @yqgkr qe ek}ac na ykbekmki`eik` na ΰ 7 :%:1%
Ay~kg`r na }`clr na c` zrlo`okckn`n z%
^@OC@ 2%0?% ^`oc` gly~r`enl cly n`~ly nac zrlocag`%
Eøgarl na @brab`nly
0 ? 9 > 1
110 1;1 =9; >02 1=9
>12 16: =01 >>; =90
>1: 1:6 100 102 1??
290 169 129 >96 =09
>;; =99 =>6 >01 =1=
=9? 102 =22 111 =2;
_aylc}ar cly ykbqkae~ay aeqeik`nly8
&`+ Ilrrar qe `eàckyky na }`rk`ex` qy`enl ae z`pqa~a na ilgzq~`nlr`%
&o+ Zrlo`r c` fkzù~ayky eqc` na pqa c` zloc`ikùe nac `brab`nl 0 ay kbq`c ` c`
zloc`ikùe nac `brab`nl 1# ay naikr# Fl8±0 7 ±1 ile~r` c` fkzù~ayky `c~are`~k}` na
F08±0 ≪ ±1%
&i+ Zrlo`r c` fkzù~ayky eqc` na pqa c` zloc`ikùe nac `brab`nl > ay kbq`c ` c`
zloc`ikùe nac `brab`nl 1# ay naikr# Fl8±> ’ ±1 7 :# ile~r` F08±> ' ±1 ≪ :%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 389/846
Nr% Fëi~lr Pqa}anl Qrç`y
2'0=
Ylcqikùe8
Qy`enl qe zrlbr`g` na ilgzq~`nlr` ilgl AVIAC n` cly ykbqkae~ay rayqc~`nly%
&`+ C` ~`oc` na `o`hl gqay~r` cly rayqc~`nly lo~aeknly qy`enl ac z`pqa~a na
ilgzq~`nlr`%
^@OC@ 2%09% _ayqc~`nly qy`enl `eàckyky na }`rk`ex` na qe ylcl m`i~lr%
@eàckyky na }`rk`ex` na qe ylcl m`i~lr
_ayqgae
Brqzly Ile~al Yqg` Zrlgankly ]`rk`ex`y
@brab`nl 0 = 99?: 119%9999 0?099%62
@brab`nl ? = 9>0= 1=;%9999 ?9:?%==2
@brab`nl 9 = 9==9 =0:%1 91;9%1 @brab`nl > = ?2;0 >=1%0==2 9906%1=2
@brab`nl 1 = 9==> =0:%===2 9>11%>=2
@EL]@
Mqae~a na ]`rk`ikùe YY bc GY M i`ci% ]`clr'z M irk~%
Ae~ra cly brqzly 6191=%>2 > ?099;%0? >%9:019= :%::621? ?%21620
Nae~rl na cly brqzly 0?>:?:%9 ?1 >;=:%609
^l~`c ?:;92=%6 ?;
@c hqxb`r zlr cly rayqc~`nly lo~aeknly ya raif`x` cly fkzù~ayky na kbq`cn`n na
zrlgankly# ay naikr# Fl8±0 7 ±? 7 ±9 7 ±> 7 ±1# ile qe` zrlo`okckn`n na z 7 :%::;%
& o+ @flr` okae# z`r` zrlo`r c` fkzù~ayky na pqa c` zloc`ikùe nac `brab`nl 0 ay kbq`c
` c` zloc`ikùe nac `brab`nl 1# ya qy` c` rac`ikùe8 Fl8±0 7 ±1 { F08 ±0 ≪ ±1% Qy`enl
c` mqeikùe &2'=+ { yqy~k~q{aenl cly }`clray na ±0 7 119%99# ±1 7 =0:%=2# nay}k`ikùe
ay~`en`r ilgoke`n` 7 y 7 ∘ >;=:%609 7 2:%>9 { e 7 = n`8 V k ' V h
~ 7 ┊┊┊┊┊┊┊┊ y ?,e
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 390/846
Nr% Fëi~lr Pqa}anl Qrç`y
2'02
~ 7 &119%99 ’ =0:%=2+ , T&2:%>9+ &∘ ?,=+W 7 '0%>0
Z`r` i`ciqc`r ac }`clr na c` zrlo`okckn`n z ya oqyi` ac }`clr `oylcq~l# s'0%>0s ae c`
~`oc` na c` nky~rkoqikùe na ~ na Ay~qnk`e~a { ay~à ae~ra :%:1 { :%0: { zlr
ke~arzlc`ikùe n` z 7 :%02% Ay~a }`clr el ay ykbekmki`e~a {# zlr cl ~`e~l# ya nkia pqa
–~`c }ax‗ ±0 7 ±1%
&i+ @pqç ya pqkara zrlo`r c` fkzù~ayky eqc` na pqa el f`{ nkmaraeik`y ae~ra c`y
zloc`ikleay na cly `brab`nly > { 1# ay~l ay# Fl8±> 7 ±1% Z`r` ay~l# ya zrliana ae
mlrg` `eàclb` `c keikyl &o+ qy`enl cly }`clray na ±> 7 >=1%02# ±1 7 =0:%=2# y 7
2:%>9 { ~`g`ðl na gqay~r` na e 7 =% Yqy~k~q{aenl ~lnly ay~ly }`clray ae c` mqeikùe&2'=+ n`8
~ 7 &>=1%02 ’ =0:%=2+ , T&2:%>9+&∘ ?,=+ 7 '9%16
Z`r` i`ciqc`r ac }`clr na c` zrlo`okckn`n z ya ileyqc~` c` ~`oc` na c`
nky~rkoqikùe na ~ ile ?1 br`nly na ckoar~`n { }agly pqa ac }`clr z ilrrayzlenkae~a
` 9%16 ay~à ae~ra %:::1 3 z 3 :%::0% Zlr ke~arzlc`ikùe# ac }`clr i`ciqc`nl na z ay
kbq`c ` :%:::6% Ay~a }`clr `zl{`# namkek~k}`gae~a# c` fkzù~ayky `c~are`~k}` na F08±>
≪ ±1%
@eàckyky na }`rk`ex` na nkyaðl na oclpqay ilgzca~`gae~a `ca`~lrkx`nly
Ilgl ya nkhl `e~arklrgae~a# ac nkyaðl na oclpqay ilgzca~`gae~a `ca`~lrkly ya qy`
z`r` ranqikr ac arrlr avzarkgae~`c# {` ya` naoknl ` gqay~r`y zapqað`y l naoknl `
}`rk`ikùe kefarae~a na c`y loyar}`ikleay% Ile ay~a ~kzl na nkyaðl zlr oclpqay
ilgzca~ly ay zlykoca ile~rlc`r c` }`rk`ikùe nae~rl na c`y gqay~r`y &rayknq`c+
baear`n` zlr `cbqe m`i~lr kenaya`oca% Na g`ear` pqë# `c oclpqa`r c`y
loyar}`ikleay# ya ranqia c` }`rk`ikùe# pqa ~`c }ax el ya zqan` ile~rlc`r iq`enl ya
qy`e nkyaðly ilgzca~`gae~a `ca`~lrkx`nly%
Ac nkyaðl na oclpqay `ca`~lrkx`nly ~`gokëe ya ramkara ilgl @EL]@ ile
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 391/846
Nr% Fëi~lr Pqa}anl Qrç`y
2'06
nly m`i~lray# ae ac yae~knl na pqa ya qy` K z`r` razrayae~`r ac eøgarl na ek}acay nac
zrkgar m`i~lr @ { H z`r` razrayae~`r ac eøgarl na ek}acay nac yabqenl m`i~lr O
&oclpqay+% Ykaenl `yç# f`{ KH zlykocay ilgoke`ikleay pqa iley~`e na qe ek}ac na
m`i~lr @ { l~rl na m`i~lr O% I`n` qe` na ay~`y ilgoke`ikleay ya cc`g` ~r`~`gkae~l#
zlr cl pqa f`{ KH nkmarae~ay ~r`~`gkae~ly% @pqç# ae ac nkyaðl na oclpqay# ac eøgarl
na loyar}`ikleay faif`y ae ac ~r`~`gkae~l KH ya razrayae~`e ile D kh 7 0# ac iq`c ay
qe i`yl ayzaik`c nac nkyaðl na oclpqay `ca`~lrkx`nly# nlena qe ylcl m`i~lr @ ay na
ke~arëy zrkeikz`c# { ac l~rl m`i~lr &O+ oclpqay ay keicqknl z`r` ranqikr ac arrlr
avzarkgae~`c% Ae c` ykbqkae~a nkyiqykùe na @EL]@ na nly m`i~lray# elyiae~r`ragly ae ac i`yl na D kh 7 D 5 0# z`r` nkmaraeik`rcl nac nkyaðl na oclpqay
`ca`~lrkly ile D kh 7 0%
Na iq`cpqkar g`ear`# ac ~ërgkel –oclpqa‗ ya nark}` na nkyaðly
avzarkgae~`cay `brçilc`y# ae cly iq`cay c`y z`riac`y na ~karr`y na iqc~k}ly ya ramkarae
ilgl –oclpqay‗% Zlr ahagzcl# ae ac i`yl nac nkyaðl na oclpqay `ca`~lrkly# cly
~r`~`gkae~ly ya `ykbe`e `ca`~lrk`gae~a ` qekn`nay nae~rl na i`n` oclpqa ile
i`r`i~arçy~ki`y na yqacly yagah`e~ay% Na el yar `yç# c`y z`riac`y ` c`y pqa ya ca
`zcki` mar~kckx`e~a# el ~ln`y zqnkar`e ~aear ac gkygl ~kzl na ~karr`# eq~rkae~ay l
fqgan`n# &cl pqa zqankar` i`qy`r }`rk`ikleay ae cly raenkgkae~ly `brçilc`y+% @c
`brqz`r c`y z`riac`y zlr i`r`i~arçy~ki`y ykgkc`ray na yqacly# gkear`cay# eq~rkae~ay#
fqgan`n# a~i%# ac arrlr avzarkgae~`c ya ranqia%
L~rl ahagzcl# ay ac rac`ikle`nl ile avzarkgae~ly gënkily% Zlr ahagzcl# yk
cly ~r`~`gkae~ly yle 9 nrlb`y { f`{ ?> z`ikae~ay# qy`enl ac nkyaðl ilgzca~`gae~a
`ca`~lrkx`nl# 6 z`ikae~ay yle `ykbe`nly `ca`~lrk`gae~a ` i`n` qel na cly
~r`~`gkae~ly% Zarl zqana liqrrkr pqa ac fky~lrk`c icçekil na cly ?> z`ikae~ay el ya`
ac gkygl# cl iq`c zqana `mai~`r yq ilgzlr~`gkae~l ` c`y nrlb`y &cl pqa zqana
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 392/846
Nr% Fëi~lr Pqa}anl Qrç`y
2'0;
i`qy`r qe arrlr l rayknq`c br`ena+% Yke ago`rbl# `brqz`enl cly z`ikae~ay zlr
fky~lrk`cay icçekily ykgkc`ray# an`nay# yavl# zayly# mqg`nlray# ~lg`nlray#
lrkae~`ikleay yavq`cay# a~i%# ya ile~rlc` ay~` }`rk`ikùe%
Ae ac i`yl na c` kebaekarç` `gokae~`c# qy`enl glnacly na ile~`gke`ikùe
`~glymërki`# ya ayzar`rç` pqa c`y ileiae~r`ikleay na cly ile~`gke`e~ay
nkygkeq{ar`e ae mqeikùe na c` nky~`eik` &ykagzra { iq`enl c`y `c~qr`y na cly
gqay~ra`nlray mqar`e kbq`cay# c`y ilenkikleay ga~aralcùbki`y mqar`e qekmlrgay {
ac ~kzl ~arrael zlr nlena ay~à z`y`enl c` zcqg` mqar` ykgkc`r+% @c ile~rlc`r ay~ly
m`i~lray# c`y ileiae~r`ikleay na cly ile~`gke`e~ay nkygkeq{ae avzleaeik`cgae~a#ae mqeikùe na c` nky~`eik` na c` mqae~a agkylr`# yke zrlnqikr gqif` }`rk`ikùe%
C` ~`oc` na `o`hl n` ac @EL]@ z`r` ac nkyaðl na oclpqay ilgzca~ly%
^@OC@ 2% 0>% @EL]@ na qe nkyaðl `ca`~lrkx`nl zlr oclpqay ilgzca~ly%
Mqae~a na Yqg` na cly Br`nly na Iq`nr`nl Mi`ci% M~`o% ]`clr na
}`rk`ikùe iq`nr`nly ckoar~`n gankl z
Naoknl ` cly YY` ` ’ 0 GY` 7 YY`,&` ’ 0+ GY`,y?0 MT0'ΰ4`'0#&`'0+&o'0+W I`ciqc`n`
~r`~`gkae~ly
Naoknl ` cly YY o o ’ 0 GY o 7 YY o,&o ’ 0+ GY o,y?? MT0'ΰ4o'0#&`'0+&o'0+W
oclpqay
_ayknq`c &Arrlr+ YYa &` ’ 0+&o ’ 0+ GYa 7 YYa,T&` ’ 0+&o ' 0+W
^l~`c YY~ `o ’ 0
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
Nlena8
`
YY` 7 o Ω & { k% ’ { %% +?
Yqg` na iq`nr`nly na ~r`~`gkae~ly &2'2+k70
o
YY o 7 ` Ω & { %h ’ { %% +?
Yqg` na iq`nr`nly na oclpqay &2'6+H70
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 393/846
Nr% Fëi~lr Pqa}anl Qrç`y
2'?:
` o
YYa 7 Ω Ω &{kh ’ { k% ’ { %h $ { %% +?
Yqg` na iq`nr`nly nac arrlr &2';+k70 h70
` o
YY~ 7 Ω Ω &{kh ’ { %%+?
Yqg` ~l~`c na cly iq`nr`nly &2'0:+k70 h70
Nlena8
{ k% 7 zrlgankl na c`y loyar}`ikleay z`r` ac k'ëykgl ~r`~`gkae~l
{ %h 7 zrlgankl na c`y loyar}`ikleay z`r` ac h'ëykgl oclpqa
{ %% 7 zrlgankl na ~ln`y c`y o` loyar}`ikleay l ac zrlgankl na cly zrlgankly
{kh 7 h'ëykg` loyar}`ikùe nac k'ëykgl ~r`~`gkae~l
Yqzlykikleay nac glnacl na oclpqay `ca`~lrkly ilgzca~ly
Ac glnacl l nkyaðl na oclpqay `ca`~lrkly ilgzca~ly `yqga iq`~rl yqzlykikleay
&Nqee a~ `c% 0;2>+ 8
0% C` rayzqay~` `c k'ëykgl ~r`~`gkae~l ae ac h'ëykgl oclpqa zrl}kaea na qe`
nky~rkoqikùe elrg`c%
?% Cly zrlgankly na c`y nky~rkoqikleay elrg`cay `o zqanae avzray`rya ae c` mlrg`
na ± $ ΰ $ ο% Ay~` zrlzkan`n qyq`cgae~a ya cc`g` `nk~k}kn`n l el ke~ar`iikùe%
9% C`y }`rk`ex`y na c`y zloc`ikleay `o yle ~ln`y kbq`cay% Ay~l ya cc`g`
flglyian`y~kikn`n% Ay~a ~ërgkel ya nkyiq~krà# eqa}`gae~a# ae ac i`zç~ql na
rabraykùe { ilrrac`ikùe%
>% C`y na}k`ikleay na cly zrlgankly αkh yle kenazaenkae~ay% Zlr ahagzcl# yk ya y`oa
pqa α00 ay br`ena# el ya zqana ayzar`r pqa α0? ya` zapqað` l br`ena%
Qe` yqzlykikùe kgzlr~`e~a nac glnacl z`r` qe nkyaðl na oclpqay ilgzca~ly
`ca`~lrkx`nly ay pqa cly amai~ly na ~r`~`gkae~l { na oclpqal ya `yqgae pqa yle
`nk~k}ly% Zlr ahagzcl# z`r` kcqy~r`r ay~` yk~q`ikùe# yk ya br`mki`e cly zrlgankly
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 394/846
Nr% Fëi~lr Pqa}anl Qrç`y
2'?0
zloc`ikle`cay }aryqy ~r`~`gkae~ly# nkb`gly na cly oclpqay 0 { ? {# yk c`y bràmki`y
yle z`r`cac`y# ya nkia pqa cly amai~ly na ~r`~`gkae~l { na oclpqay yle `nk~k}ly l
pqa el ke~ar`i~ø`e% Yke ago`rbl# yk c`y cçea`y na c` bràmki` ya irqx`e ae~ra yk# ya
nkia pqa f`{ ke~ar`iikùe% Ae ay~a raebcùe# yk el ya iqgzca c` ilenkikùe na
`nk~k}kn`n# ay~l ilenqia ` ileicqykleay arrùea`y%
Ac nkyaðl ilgzca~`gae~a `ca`~lrkl# ~kaea gqif`y `zcki`ikleay ae c`
zrlnqiikùe kenqy~rk`c { ae ` glnacly anqi`~k}ly% Z`r` ay~l }`gly ` qy`r qe
ahagzcl z`r` kcqy~r`r ay~` yk~q`ikùe%
Ahagzcl )=% Yqzùeb`ya pqa > nkmarae~ay gàpqke`y yle g`eah`n`y zlr > lzar`nlraynkmarae~ay% Ya pqkara y`oar yk cly lzar`nlray nkmkarae ile rayzai~l ` c`
zrlnqi~k}kn`n na ~kagzl# iq`enl yle `ykbe`nly ` }`rk`nly ~kzly na g`pqke`rk`y%
@zcki`r ac `eàckyky na }`rk`ex` gày `zrlzk`nl z`r` ay~a zrlocag`# qy`enl ΰ 7 :%:1%
Y`i`r ileicqykleay `c rayzai~l qy`enl qe z`pqa~a na ilgzq~`nlr`% Cly n`~ly ya
n`e ae c` ~`oc` na `o`hl8
^@OC@ 2%01% ^`oc` gly~r`enl c` zrlnqi~k}kn`n zlr ~kagzl na cly nkmarae~ay
lzar`nlray `ykbe`nly `ca`~lrk`gae~a ` > ~kzly na gàpqke`y nkmarae~ay%
Lzar`nlray
Gàpqke`y 0 ? 9 >
@ =6%1 2;%? 69%6 62%1
O 2?%? 6:%= 6;%9 ;1%9
I 29%9 6:%? 66%: ;>%0
N 60%0 66%6 ;1%? 0::%1
Ylcqikùe8
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 395/846
Nr% Fëi~lr Pqa}anl Qrç`y
2'??
Qy`enl qe z`pqa~a na ilgzq~`nlr` ilgl Aviac# ya zrliana ilgl8
^llcy ↟ N`~` @e`c{yky ↟ @e`c{yky ^llcy ↟ @el}`8 ^l M`i~lry Rk~flq~
_azcki`~kle &@pqç# yke ago`rbl# f`{ pqa key~`c`r ac gùnqcl na N`~` @e`c{yky+
C`y zrqao`y na fkzù~ayky z`r` cly lzar`nlray { c`y gàpqke`y ya ay~`ocaiae na c`
ykbqkae~a g`ear`8
Fl8 Cly lzar`nlray el nkmkarae ile rayzai~l `c zrlgankl na zrlnqi~k}kn`n zlr
~kagzl# ile~r` F08 Cly lzar`nlray yk nkmkarae ile rayzai~l `c zrlgankl na
zrlnqi~k}kn`n zlr ~kagzl
Fl8 C`y gàpqke`y el nkmkarae ile rayzai~l `c zrlgankl na zrlnqi~k}kn`n zlr ~kagzl# ile~r` F08 C`y gàpqke`y yk nkmkarae ile rayzai~l ` c` zrlnqi~k}kn`n
Qy`enl ac zrlbr`g` Aviac n` cly ykbqkae~ay rayqc~`nly gly~r`nly `o`hl%
^@OC@ 2%0=% ^`oc` gly~r`enl ac `eàckyky na }`rk`ex` ile nly m`i~lray%
@el}` ile nly m`i~lray
_AYQGAE Ile~al Yqg` Zrlgankl ]`rk`ex`
Gàpqke` @ > 90; 2;%21 =2%22==2
Gàpqke` O > 992%> 6>%91 0:?%:9
Gàpqke` I > 991%= 69%; 6?%9Gàpqke` N > 9=1%= ;0%> 2:%:9999
Lzar`nlr 0 > ?;1%0 29%221 ?6%:=?1
Lzar`nlr ? > 9?6%6 6?%? 0;%2:==2
Lzar`nlr 9 > 91=%9 6;%:21 ??%06?1
Lzar`nlr > > 922%> ;>%91 ?6%12
^`oc` na @EL]@
Mqae~a na }`rk`ikùe YY bc GY M i`ci% ]`clr na z M irç~ki`
G`pqke`rk` ?6:%?= 9 ;9%>? 1>%;9>;; %:::::>0> 9%6=?1>6
Lzar`nlray ;10%001 9 902%:969 06=%>9?? %:::::::? 9%6=?1>6
Arrlr 01%9:1 ; 0%2::11=
^l~`c 0?>=%=6 01
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 396/846
Nr% Fëi~lr Pqa}anl Qrç`y
2'?9
Ileicqykùe8
Ile rayzai~l ` c` g`pqke`rk`# naoknl ` pqa ac }`clr na c` Mi`ci% 7 1>%; 555 Mirç~ki` ya
raif`x` Fl8 Ay~` naikykùe ay gqifl gq{ ykbekmki`~k}`# `c hqxb`r zlr ac }`clr na z 7
%:::::>% C`y gàpqke`y yç nkmkarae gq{ ykbekmki`e~agae~a# ile rayzai~l ` c`
zrlnqi~k}kn`n% Ile rayzai~l ` cly lzar`nlray# naoknl ` pqa ac }`clr na Mi`ci% 7 06=%>
5555 Mirç~ki` ya raif`x` Fl8 Ay~` naikykùe ay gqifl# gqifl gq{ ykbekmki`e~a `c
hqxb`r zlr ac }`clr na z 7 ?v0:'6
% Cly lzar`nlray yç nkmkarae gq{
ykbekmki`e~agae~a# ile rayzai~l ` c` zrlnqi~k}kn`n na ~kagzl% Ay~l ay `zl{`nl#
gq{ ile~qenae~agae~a# zlr ac }`clr ~`e o`hl na c` zrlo`okckn`n z%Ahagzcl )2% Ay~a ay qe ahagzcl rac`ikle`nl ile qe avzarkgae~l na oclpqay
`ca`~lrkly ilgzca~ly z`r` na~argke`r cly amai~ly ilrrlyk}ly na iq`~rl yqy~`eik`y
pqçgki`y nkmarae~ay# }%b%# FIc# F?YL># FEL9 { FM% Ay naikr# àiknly b`yalyly pqa
ae~r`e ae ac mcqhl na `kra &mcqhl ~r`eyzlr~`nlr pqa ae~r` `c apqkzl na ile~rlc# ac iq`c
ya baear` na qe zrliay`gkae~l kenqy~rk`c+# pqa z`y`e zlr cly mkc~rly# ay naikr# ae c`y
~ac`y qy`n`y ae cly mkc~rly l o`bflqyay &faif`y na mkor` na }knrkl# `yoay~ly# n`irle#
ekcùe# zlcka~kcael+# z`r` ile~rlc`r c` ile~`gke`ikùe nac `kra% Z`r` ~`cay mkeay ya
yacaiikle`e ikeil gqay~r`y na ~ac`y { ya `zcki` qe nkyaðl `ca`~lrkl zlr oclpqay
ilgzca~ly# zlr gankl na zrlo`r i`n` yqy~`eik` pqçgki`# ae qe lrnae `ca`~lrkl#
ylora i`n` qe` na c`y gqay~r`y na c`y ~ac`y% Y`i`r c`y ileicqykleay naokn`y% Cly
n`~ly ya n`e ae c` ~`oc` na `o`hl% F`iar cl ykbqkae~a8
&`+ Zrlo`r c` fkzù~ayky eqc` na kbq`cn`n na zrlgankly
&o+ F`iar qe` ~`oc` na `eàckyky na }`rk`ex` na nkyaðl `ca`~lrkx`nl zlr oclpqay
ilgzca~ly% Y`i`r c`y ileicqykleay `zrlzk`n`y
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 397/846
Nr% Fëi~lr Pqa}anl Qrç`y
2'?>
^@OC@ 2% 02% C` rayzqay~` na cly çenkiay na ilrrlyk}kn`n na c`y iq`~rl yqy~`eik`y
pqçgki`y ae c`y gqay~r`y na ~ac`y% &Ac`olr`ikùe zrlzk`+
^kzly na ~ac`y
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ Yqy~`eik`y pqçgki`y ]knrkl `yoay~ly n`irle ekcùe zlcka~kcael
FIc 0%6 ?%0 0%0 0%2 0%=
F?YL> ?%2 ?%; :%6 ?%1 ?%1
FEL9 ?%9 ?%9 0%0 ?%: 0%6
FM >%> >%6 ?%1 >%> 9%;
Cly rayqc~`nly qy`enl ac z`pqa~a na Aviac ya n`e `o`hl%
^@OC@ 2%06% _ayqc~`nly na c`y rayky~aeik`y ` c` ilrrlykùe na c`y ~ac`y qy`enl qenkyaðl `ca`~lrkx`nl na oclpqay ilgzca~ly%
@el}` na nly m`i~lray yke razcki`ikleay
_ayqgae Ile~al Yqg` Zrlgankl ]`rk`ex`
FIc 1 6%> 0%=6 :%012
F?YL> 1 00%> ?%?6 :%20?
FEL9 1 ;%1 0%; :%?>1
FM 1 0; 9%6 :%=:1
]knrkl > 00%? ?%6 0%?29999
@yoay~ly > 00%? ?%6 :%1>
N`irle > 1%1 0%921 :%16?1
Ekcùe > 0:%= ?%=1 0%>2
Zlcka~kcael > ;%6 ?%>1 0%:69999
^`oc` na @EL]@
Mqae~a na }`rk`ikùe YY bc GY M i`ci% ]`clr na
z M irk~ki`
Naoknl cly àiknly 09%2:;1 9 >%1=;699 >6%066:1 1%21A':2 9%>;:?;1
Naoknl c`y ~ac`y 1%296 > 0%>9>1 01%0?=1> :%:::0?9 9%?1;0=2
Arrlr 0%096 0? :%:;>699
^l~`c ?:%1611 0;
Naoknl ` pqa ac }`clr na c` Mi`ci% 7 >6%0; 5 M:%:1#9#0? 7 9%>; ya raif`x` c` fkzù~ayky
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 398/846
Nr% Fëi~lr Pqa}anl Qrç`y
2'?1
eqc` na kbq`cn`n na ~r`~`gkae~ly# { ya nkia pqa f`{ qe` nkmaraeik` gq{
ykbekmki`~k}` ae c` `iikùe na cly àiknly# ae iq`e~l ac amai~l pqa ~kaeae ylora c`
rayky~aeik` zrlgankl na c`y ~ac`y% Ay~` ile~aeikùe ay~à gq{ okae yqy~ae~`n` zlr ac
}`clr ~`e zapqaðl na z 7 1%21v0:'2
%
Zlr l~r` z`r~a# ae iq`e~l ` glnacly ay~`nçy~kily z`r` ile~rlc`r c` }`rk`ikùe#
avky~a l~rl ~kzl na nkyaðl z`r` ranqikr ac arrlr avzarkgae~`c cc`g`nl iq`nr`nly
c`~kely% @øe# iq`enl ac nkyaðl ae oclpqay `ca`~lrkx`nly ay gq{ amai~k}l z`r`
ranqikr ac arrlr avzarkgae~`c &rayknq`c+# `c ackgke`r qe` mqae~a na }`rk`ikùe# cly
iq`nr`nly c`~kely yle gq{ ø~kcay z`r` ranqikr nly mqae~ay na }`rk`ikùe# gkae~r`y yaranqia ac eøgarl na ilgoke`ikleay% Ay~a nkyaðl# yke ago`rbl# el ya nkyiq~krà ae
ay~a ~av~l%
Ic`ykmki`ikleay irqx`n`y8 @eàckyky na }`rk`ex` ae nly yae~knly
Ac `eàckyky na }`rk`ex` ae nly nkraiikleay l na nly ic`ykmki`ikleay l na nly
yae~knly ay ø~kc z`r` ay~qnk`r nly ~kzly nkmarae~ay na ~r`~`gkae~ly% C` i`r`i~arçy~ki`
nac nkyaņl m`i~lrk`c ae nly yae~knly ay pqa# i`n` ek}ac na qe m`i~lr# ya qy` ae
ilgoke`ikùe ile i`n` ek}ac nac l~rl m`i~lr% Zlr ahagzcl# ileyknëraya ac i`yl na e
rëzcki`y na c`y ilgoke`ikleay nac ~r`~`gkae~l pqa ya na~argke`e zlr ek}acay nac
m`i~lr @ { o ek}acay nac m`i~lr O% Ae ay~a `yzai~l# c`y loyar}`ikleay ya ay~rqi~qr`e
zlr gankl na qe `rrabcl rai~`ebqc`r# nlena cly raebcleay razrayae~`e cly ek}acay
nac m`i~lr @ { c`y ilcqge`y razrayae~`e cly ek}acay nac m`i~lr O% Ykaenl `yç# f`{ `o
iacn`y# i`n` qe` na c`y iq`cay ile~aekaeae e loyar}`ikleay &~`g`ņl na c` gqay~r`+%
Zlr ahagzcl# yk qe kebaekarl `brùelgl ke}ay~kb` ac ilgzlr~`gkae~l na nly
~kzly na yagkcc`y# zlr gankl na }`rk`r ac ek}ac nac mar~kckx`e~a# nkb`gly# ` ~ray
ek}acay# `c~l# gank`el { o`hl# qe m`i~lr yarç` ac ~kzl na yagkcc` { ac yabqenl m`i~lr
yarç` ac ek}ac na mar~kckx`e~a% Ay~a yarç` qe ahagzcl m`i~lrk`c ile nly m`i~lray# ac
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 399/846
Nr% Fëi~lr Pqa}anl Qrç`y
2'?=
iq`c ileyky~krç` ae qy`r yaky ~r`~`gkae~ly mlrg`nly zlr gankl na qy`r i`n` ~kzl na
yagkcc` ile i`n` ek}ac na mar~kckx`e~a%
L~rl ahagzcl# na @EL]@ na nly m`i~lray ay~à rac`ikle`nl ile c` gankikùe
na c`y ileiae~r`ikleay na ile~`gke`~ay nac `kra agk~knly zlr qe` mqae~a kenqy~rk`c%
@pqç z`r` qe m`i~lr ya zqanae yacaiikle`r nkmarae~ay ek}acay nky~`eik`y na c` mqae~a
agkylr` {# z`r` ac l~rl m`i~lr# ya zqanae yacaiikle`r nkmarae~ay `c~qr`y nlena ay~àe
yk~q`nly cly gqay~ra`nl~ay &zlrpqa c` `c~qr` `mai~` c`y ileiae~r`ikleay+%
Ke~ar`iikùe ile @EL]@ na nly m`i~lray
Iq`enl ya ay~qnk`e avzarkgae~ly m`i~lrk`cay ay kgzlr~`e~a na~argke`r yk clym`i~lray zrkeikz`cay ~kaeae qe` kemcqaeik` ae c` rayzqay~`# ykel ~`gokëe `e`ckx`r cl
pqa ya cc`g` ke~ar`iikùe &el `nk~k}kn`n+ ae~ra cly m`i~lray% Ac ~av~l na Nqee a~ `c%
&0;2>+ `zcki` qe avzarkgae~l na nly ic`ykmki`ikleay# z`r` avzcki`r ac ileiaz~l na
c` ke~ar`iikùe% Zlr ahagzcl# ae c` Mkbqr` 2%?# ae qe avzarkgae~l pqa ke}lcqir` ~ray
ek}acay na `bq` { ~ray ek}acay na mar~kckx`e~a# c`y cçea`y yle z`r`cac`y# cl pqa kenki`
pqa el f`{ ke~ar`iikùe# l ya` pqa f`{ kenazaenaeik` ae cly n`~ly% Yke ago`rbl# ae
c` Mkbqr` 2%9 ya loyar}` pqë# ae `go`y bràmki`y f`{ qe` rayzqay~` zrlgankl ile
ke~ar`iikùe# ay naikr# pqa f`{ nazaenaeik`% Zlr ahagzcl# ae c` zrkgar` bràmki` qe
ek}ac `c~l na mar~kckx`e~a ke~ar`iikle` zlyk~k}`gae~a ile qe ek}ac `c~l na `bq`4
gkae~r`y pqa ae c` yabqen` bràmki` ek}acay `c~ly na `bq` { mar~kckx`e~a rayqc~`e ae
qe` rayzqay~` o`h`# ae ilgz`r`ikùe ile c` rayzqay~` ` ek}acay o`hly { gank`ely na
`bq`% Ae ~ërgkely ykgzcay# ya nkia pqa f`{ ke~ar`iikùe ae~ra nly m`i~lray &nkb`gly
@ { O+# yk ac i`gokl ae qel na cly m`i~lray &nkb`gly m`i~lr O+ zrlnqia qe i`gokl
ae rayzqay~` ` qe ek}ac &nkb`gly ek}ac 0+ nac l~rl m`i~lr &nkb`gly @+ nkmarae~a na
`pqëc zrlnqiknl ae cly l~rly ek}acay &nkb`gly ek}ac ?+ na ay~a yabqenl m`i~lr @#
nlena qe ek}ac ay qel na cly ~r`~`gkae~ly nae~rl na qe m`i~lr%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 400/846
Nr% Fëi~lr Pqa}anl Qrç`y
2'?2
Mkbqr` 2%?% Bràmki` kenki`enl qe` rayzqay~` zrlgankl yke ke~ar`iikùe &`nk~k}kn`n+#
l ya` pqa f`{ kenazaenaeik` ae cly n`~ly% &Nqee a~ `c% 0;2>+%
Mkbqr` 2%9% Bràmki`y kenki`enl qe` rayzqay~` zrlgankl ile ke~ar`iikùe &el
`nk~k}kn`n+ l ya` pqa f`{ nazaenaeik` ae~ra cly n`~ly% &Nqee a~ `c% 0;2>+%
Iq`enl liqrra qe` ke~ar`iikùe ae `cbøe avzarkgae~l ay kgzlr~`e~a
ke}ay~kb`r zlrpqa liqrrkù% Zlr ahagzcl# iq`enl ya ay~`ocaia c` ~`oc` na `eàckyky na
}`rk`ex`# ya ay~qnk`e cly ilgzlr~`gkae~ly na cly amai~ly zrkeikz`cay { ~`gokëe# c`
zlykoca ke~ar`iikùe ae~ra cly nly m`i~lray o`hl ay~qnkl% Ae ~ërgkely ay~`nçy~kily# yk
c` M i`ciqc`n` ay g`{lr pqa c` M irç~ki` ayl kenki` pqa cly m`i~lray ay~àe
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 401/846
Nr% Fëi~lr Pqa}anl Qrç`y
2'?6
ke~ar`i~q`enl% El loy~`e~a# c` ke~ar`iikùe zqana liqrrkr zlr gar` i`yq`ckn`n% Zarl
~`gokëe c` ke~ar`iikùe zqana liqrrkr# i`qy`cgae~a# naoknl ` `cbøe }`clr av~ragl l
` `cbøe m`i~lr pqa el ya f` zlnknl ile~rlc`r% C` ke~ar`iikùe# ~`gokëe ya zqana
naoar ` `cbøe zrlocag` ae cly n`~ly l ` qe` rayzqay~` arrùea`% Na iq`cpqkar
g`ear` iq`enl cly n`~ly lo~aeknly kenki`e pqa avky~a qe` ke~ar`iikùe br`ena# cly
amai~ly zrkeikz`cay ilrrayzlenkae~ay yaràe na zli` q~kckn`n%
Na ay~` g`ear`# ae ac ahagzcl )2 na `o`hl# f`{ ke~ar`iikleay ae~ra c`y
`c~qr`y { c`y nky~`eik`y% Iq`enl ya glnac`e c`y agkykleay na ile~`gke`e~ay
`~glymërkily# f`{ gqif`y }`rk`ocay pqa zqanae `mai~`r cly rayqc~`nly% Ae ay~aahagzcl# ~`c }ax fqol i`gokly ga~aralcùbkily kgzra}ky~ly# agkykleay mqbk~k}`y l
nkmaraeik`y ae cly ~kzly na ~arrael zlr nlena z`y` c` zcqg` na c` ifkgaea`% Ay~l
zqnl ile~rkoqkr ` c` ke~ar`iikùe na cly nly m`i~lray ay~qnk`nly ae aya ahagzcl%
Yk~q`ikleay ykgkc`ray zqanae liqrrkr ae ay~qnkly na `brkiqc~qr`% Zlr ahagzcl#
yk ac kebaekarl `brùelgl naya` ay~qnk`r cly raenkgkae~ly `brçilc`y qy`enl nly
m`i~lray# ilgl ac ~kzl na yagkcc` { c` i`e~kn`n na mar~kckx`e~a `zcki`nl# ~kaea pqa
`e`ckx`r yk fqol ke~ar`iikùe ae~ra cly m`i~lray yagkcc`'mar~kckx`e~a% Yk f`{
ke~ar`iikùe ae~ra ay~ly nly m`i~lray# ay~l zqnl naoarya ` pqa# ae c`y z`riac`y
yacaiikle`n`y z`r` cly iqc~k}ly avzarkgae~`cay# el f`oç` qekmlrgkn`n na }`rk`ocay
ilgl fqgan`n# ~kzly na yqacly# l na i`e~kn`n na eq~rkae~ay% Z`r` ragank`r ay~`
yk~q`ikùe ya ~aenrç` pqa f`iar qe avzarkgae~l zlr oclpqay `ca`~lrkx`nly# ay naikr#
~aekaenl iqkn`nl na pqa c`y z`riac`y `brçilc`y mqar`e ~ln`y qekmlrgay ae c`y
}`rk`ocay `e~arklrgae~a nayirk~`y%
Na iq`cpqkar g`ear`# c` ~`oc` na `o`hl gqay~r` ac mlrg`~l pqa ya qy` z`r`
avzarkgae~ly m`i~lrk`cay ae nly yae~knly l ile nly ~r`~`gkae~ly%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 402/846
Nr% Fëi~lr Pqa}anl Qrç`y
2'?;
C` ^@OC@ 2%0; na `o`hl gqay~r` ac mlrg`~l pqa ya ykbqa z`r` cly `eàckyky na
}`rk`ex` ae nly yae~knly%
^@OC@ 2% 0;% ^`oc` na `eàckyky na }`rk`ex` ae nly yae~knly% &Ac`olr`ikùe zrlzk`+Mqae~a na YY b%c% GY Mi`ci% M~`o%
}`rk`ikùe
Amai~l zrkeikz`c
Naoknl ` @ YY` ` ' 0 GY`7 YY`,&`'0+ M0 7 GY`,y?a MT0'ΰ4`'0#`o&e'0+W
Naoknl ` O YY o o ' 0 GY o 7 YY o,&o'0+ M? 7 GY o,y?a MT0'ΰ4o'0#`o&e'0+W
Ke~ar`iikùe na
nly m`i~lray
Naoknl ` @O YY`o &`'0+&o'0+ GY`o 7 YY`o,&`'0+&o'0+ M9 7 GY`o,y?
a MT0'ΰ4&`'0+&o'0+#`o&e'0+W
_ayknq`c YYa `o&e'0+ y?a7YYa,T`o&e'0+W
^l~`c YY~ `oe'0
Nlena8
`YY` 7 oe Ω & { k%% ' { …+
?&2'00+
k70
o
YY o 7 `e Ω & { %h% ' { %%% +?
&2'0?+H70
` o
YY`o 7 e Ω Ω & { kh% ' { k%% ' { %h% $ { …+?
&2'09+k70 h70
` o e
YYa 7 Ω Ω Ω &{khd ' { kh%+? &2'0>+k70 h70 d70
` o e
YY~ 7 Ω Ω Ω &{khd ’ { …+?
&2'01+k70 h70 d70
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 403/846
Nr% Fëi~lr Pqa}anl Qrç`y
2'9:
@ 7 }`rk`ikùe naoknl `c zrkgar m`i~lr @
O 7 }`rk`ikùe naoknl `c yabqenl m`i~lr O
@O 7 ke~ar`iikùe ae~ra ac m`i~lr @ { O &ke~ar`iikùe pqa liqrra iq`enl el f`{
`nk~k}kn`n+
y?0# y
??# y
?9 { y
?a yle c` mlrg`ikùe na cly iq`nr`nly gankly { ya lo~kaeae
nk}knkëenlcly ae~ra yqy ilrrayzlenkae~ay br`nly na ckoar~`n
{khd 7 yqg` na c`y loyar}`ikleay ae c` &kh+'ëykg` iacn`
{ k%% 7 zrlgankl na c`y loyar}`ikleay z`r` ac k'ëykgl ek}ac nac m`i~lr @
{ … 7 zrlgankl na ~ln`y c`y `oe loyar}`ikleay
{ %h% 7 zrlgankl na c`y loyar}`ikleay z`r` ac h'ëykgl ek}ac nac m`i~lr O
{ kh% 7 zrlgankl na c`y loyar}`ikleay ae c` &kh+'ëykg` iacn`
{khd 7 d 'ëykg` loyar}`ikùe ae ac k'ëykgl ek}ac nac m`i~lr @ { ac ek}ac h'ëykgl ek}ac
nac m`i~lr O
` 7 eøgarl na gqay~r`y nac zrkgar m`i~lr
o 7 eøgarl na gqay~r`y nac yabqenl m`i~lr
e 7 eøgarl ~l~`c na i`yly
Ae ac `eàcyky na }`rk`ex` na nly yae~knly# z`r` ac glnacl o`hl ay~qnkl ya
nk}kna i`n` loyar}`ikùe {khd ae iq`~rl z`r~ay { c` pqke~` ae c`y nay}k`ikleay na c`y
loyar}`ikleay nac zrlgankl zloc`ikle`c &Nqee a~ `c% 0;2>+% Ay~l ay8
{khd 7 ± $ ΰk $ο h $ &ΰο+kh $ αkhd z`r` k 7 0#…% # `4 h 7 0#…%# o4 d 7 0#%%%# e# &2'0=+
Nlena8` o ` o
Ω ΰk 7 Ω ο h 7 Ω &ΰο+kh 7 Ω &ΰο+kh 7 : &2'02+k70 h70 k70 h70
[ nlena8
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 404/846
Nr% Fëi~lr Pqa}anl Qrç`y
2'90
± 7 rayzqay~` zrlgankl nac ilehqe~l l c` rayzqay~` zrlgankl na c`y zloc`ikleay
`o4 ΰk 7 ac amai~l nac k ’ëykgl ek}ac nac m`i~lr @ zrlgank`nl ylora o ek}acay nac
m`i~lr O4 ο h 7 amai~l h ’ëykgl ek}ac nac m`i~lr O4 &ΰο+kh 7 ke~ar`iikùe ae~ra ac k ’ëykgl
ek}ac nac m`i~lr @ { ac h ’ëykgl ek}ac nac m`i~lr O {4 αkhd 7 nay}k`ikùe na c`y
loyar}`ikleay {khd na c` rayzqay~` nac zrlgankl zloc`ikle`c z`r` c` kh'ëykg`
zloc`ikùe%
@pqç# ay kgzlr~`e~a rai`zk~qc`r c`y yqzlykikleay nac glnac na @EL]@ ae
nly nkraiikleay# ay naikr8
0% Cly arrlray αkhd naoae yar kenazaenkae~ay?% Cly rayknq`cay αkhd naoae ay~`r elrg`cgae~a nky~rkoqknly
9% Cly rayknq`cay αkhd naoae na }aekr na qe` zloc`ikùe ile c` gkyg` }`rk`ex`
Na el iqgzckrya ile ay~`y yqzlykikleay# ac nkyaðl yarà keikar~l%
Ahagzcl )6% Z`r` ay~qnk`r cly amai~ly na c` `c~qr` { c` nky~`eik` ae c`y
ileiae~r`ikleay na ile~`gke`e~ay `~glymërkily &YL?+ agk~knly zlr qe` ifkgae`
kenqy~rk`c ya key~`c`rle ~ray gqay~ra`nlray# ` ~ray `c~qr`y nkmarae~ay &9 ek}acay na @+
{# ` iq`~rl nky~`eik`y nkmarae~ay &> ek}acay na O+ }kae~l `o`hl na c` mqae~a agkylr`%
Z`r` ay~l ya n`e cly ykbqkae~ay `}`eiay kemlrg`~k}ly8 YY` 7 2%::# YY o 7 ?:%::# YYa
7 2%: { YY~ 7 >1%::% @yqgkaenl qe ek}ac na ykbekmki`eik` na ΰ 7 :%:1# raylc}ar cly
ykbqkae~ay aeqeik`nly8
&`+ Ay~`ocaiar qe` ~`oc` na `eàckyky na }`rk`ex`%
&o+ F`iar zrqao`y na M z`r` nagly~r`r pqa ekebqel na cly }`clray na M z`r`
ke~ar`iikleay na c` `c~qr` { c nky~`eik` ay ykbekmki`~k}l% Zrlo`r c` fkzù~ayky eqc`
F‛l8 na pqa el f`{ nkmaraeik`y ae c`y ileiae~r`ikleay zrlgankl na YL? ae c`y
nky~`eik`y# iq`enl ya qy`e ~ray `c~qr`y nkmarae~ay# ae c`y iq`cay mqarle yk~q`nly cly
gqay~ranlray pqa ay~àe gknkaenl c`y ileiae~r`ikleay nac okùvknl na `xqmra%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 405/846
Nr% Fëi~lr Pqa}anl Qrç`y
2'9?
@nagày# zrlo`r c` fkzù~ayky eqc` F‛‛l8 na pqa el f`{ nkmaraeik` ae c`y
ileiae~r`ikleay zrlgankl ae c`y iq`~rl nky~`eik`y ` c`y pqa ya yk~q`rle cly
yaeylray% Mke`cgae~a# zrlo`r c` fkzù~ayky eqc` F‛‛‛l8 na pqa el f`{ ke~ar`iikle
ae~ra c`y nkmarae~ay `c~qr`y { c`y nkmarae~ay nky~`eik`y na cly yaeylray%
&i+ ]ar iq`cay amai~ly zrkeikz`cay yle ykbekmki`~k}ly%
&n+ I`ciqc`r cly }`clray na z%
Ylcqikùe8
&`+ C` ~`oc` na @EL]@ ile cly }`clray yqy~k~qknly ya n` `o`hl%
^@OC@ 2%?:% ^`oc` na @EL]@ z`r` ac zrlocag` na cly amai~ly na c` `c~qr` { c`nky~`eik` ae c`y ileiae~r`ikleay na ile~`gke`e~ay nac `kra%
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ Mqae~a na Yqg` na cly b%c% Iq`nr`nl gankl Mi`ci% M~`o% ]`clr z
}`rk`ikùe iq`nr`nly &YY+ &GY+
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
Naoknl ` c` 2%:: ? 9%1: =%:9 9%6; %::0 3 z 3 %:0
`c~qr` &@+
Naoknl ` c` ?:%:: 9 =%=2 00%1: 9%>; z 33 %::0
nky~`eik` &O+
Naoknl ` c` 00%:: = 0%69 9%0= 9%:: %:0 3 z 3 %:1
ke~ar`iikùe na @O
Naoknl `c arrlr 2%:: 0? :%16
^l~`c >1%:: ?9
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
&o+ C`y ~ray zrqao`y na fkzù~ayky eqc`y ya ay~`ocaiae na c` ykbqkae~a g`ear`8
F‛l8ΰ0 7 ΰ? 7 ΰ9 7 : &el f`{ nkmaraeik`y ae c`y ileiae~r`ikleay zrlgankl na YL?
iq`enl ya yk~ø`e cly yaeylray ` c`y nkmarae~ay `c~qr`y+
F‛‛l8 ο0 7ο? 7 ο9 7 ο> 7 : &el f`{ nkmaraeik`y ae c`y ileiae~r`ikleay na YL?# ae c`y
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 406/846
Nr% Fëi~lr Pqa}anl Qrç`y
2'99
iq`~rl nky~`eik`y na c` mqae~a agkylr`+
F‛‛‛l8 &ΰο+00 7 &ΰο+0? 7 …%% 7 &ΰο+?> 7 : &el f`{ ke~ar`iikùe ae~ra c`y nkmarae~ay
`c~qr`y { c`y nkmarae~ay nky~`eik`y
C`y zrqao`y na fkzù~ayky `c~are`~k}`y yle8
F‛08 Iq`enl gaely qe` na c`y ileiae~r`ikleayΰk &zlr c` `c~qr`+ nkmkara na iarl
F‛‛08 Iq`enl gaely qe` na c`y ileiae~r`ikleayοk &zlr c` nky~`eik`+ nkmkara na iarl
F‛‛‛08 Iq`enl gaely qe` na c`y &ΰο+kh &ke~ar`iikùe `c~qr`'nky~`eik`+ nkmkara na iarl
Ileicqykùe8 Ya raif`x` F‛l8 { ya ileicq{a pqa c`y ileiae~r`ikleay na YL? zlr ac
amai~l na c` `c~qr` yle nkmarae~ay# ` `pqëcc`y naokn`y `c amai~l na c` nky~`eik`% Ay~`ile~aeikùe ay `zl{`n` zlr ac }`clr na z 7 %:??% @eàclb`gae~a# F‛‛l8 ~`gokëe ya
raif`x` ile qe }`clr gq{ ykbekmki`~k}l na z 33 %::0% C` ke~ar`iikùe ae~ra c` `c~qr`
{ c` nky~`eik`# ay naikr# Mi`ci% 7 9%0= 5 M~`o% 7 9%:: ay~à ae ac qgor`c na c`
ke~ar`iikùe# ile qe }`clr na z 7 %:1% Ay~l kenki` pqa c` ke~ar`iikùe naoa na
ileyknar`rya% Ay~l pqkara naikr pqa ~kaeae pqa ileyknar`rya }`rk`ocay ilgl ac ~kzl
na ~arrael# i`gokly kgzra}ky~ly ae c`y ilenkikleay ga~alrlcùbki`y# yaeylray g`c
yk~q`nly# g`c mqeiklegkae~l nac apqkzl# agkykleay mqbk~k}`y# iqarzly na `bq`# { `yk
yqiayk}`gae~a%
&i+ { &n+ avzcki`nly zlr ac keikyl &`+% Na `iqarnl ` c` ~`oc` ~lnly cly amai~ly yle
ykbekmki`~k}ly# ayzaik`cgae~a# naoknl ` c` nky~`eik`%
Ahagzcl );% Ac ckorl @zzckan Y~`~ky~kiy8 @eàckyky lm ]`rk`eia `en _abraykùe na
Nqee# a~ `c% &0;2>+ f`ia qe ay~qnkl na @EL]@ na nly ic`ykmki`ikleay rac`ikle`nl
ile ac raenkgkae~l na iao`n`% Ae ay~a avzarkgae~l ya ke}lcqir`e nly ~kzly na
yagkcc`y &0 { ?+# i`n` qel na ay~ly m`i~lray qy`nly ae ~ray ek}acay na mar~kckx`e~ay#
ay naikr# o`hly# gank`ely { `c~ly% C` ~`oc` na `o`hl gqay~r` c` kemlrg`ikùe
rapqarkn` z`r` ay~a zrlocag`%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 407/846
Nr% Fëi~lr Pqa}anl Qrç`y
2'9>
^@OC@ 2%?0% Zrlnqiikùe na iao`n` ae m`eab`y zlr `ira% &Nqee a~ `c% 0;2>+
Ek}ac na mar~kckx`e~a
^kzl na yagkcc` O`hl Gank`el @c~l
0 0>%9 06%0 02%=0>%1 02%= 06%?00%1 02%0 06%;09%= 02%= 06%?
{ 00% 7 09%>21 { 0?% 7 02%=:: { 09% 7 06%??1 { 0%% 7 0=%>99
''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''? 0?%= 0:%1 01%2
00%? 0?%6 02%1
00%: 6%9 0=%20?%0 ;%0 0=%=
{ ?0% 7 00%2?1 { ??% 7 0:%021 { ?9% 7 0=%=?1 { ?%% 7 0?%6>?
{ %0% 7 0?%=:: { %?% 7 09%666 { %9% 7 02%>?1 { … 7 0>%=96
Ile ` 7 ?# o 7 9 { e 7 >
?
YY` 7 oe Ω & { k%% ' { …+?7 &9+&>+T0=%>>9 ' 0>%=96+
?$ &0?%6>? ' 0>%=96+
?W
k70
7 &0?+T9%?16 $ 9%??=W 7 22%6:
9
YY o 7 `e Ω & { %h% ' [ …+?
7 &?+&>+T&0?%=:: ' 0>%=96+?
$ &09%6; ' 0>%=96+?
h70
$ &02%>9 ' 0>%=96+?W
7 &6+T>%019 $ :%11: $ 2%2;1+ 7 ;;%;
? 9
YY`o 7 e Ω Ω &[ kh% ' [ k%% ' [ %h% $ [ %%%+?
k70 h70
7 &>+ tT&09%>21 ' 0=%>99 ' 0?%=:: $ 0>%=96+?$ &02%=:: ' 0=%>99 ' 09%666 $ 0>%=96+
?
$ &06%??1 ' 0=%>99 ' 02%>?1 $ 0>%=96+?W $ T&00%2?1 '0?%6>? ' 0?%=: $ 0>%=96+
?
$ &0:%021 ' 0?%6>? ' 09%666 $ 0>%=96+?
$ &0=%=?1 ' 0?%6>? ' 02%>?1 $ 0>%=96+?Ww
7 &>+&:%6>=> $ 9%2:= $ :%;;: $ :%6>6 $ 9%=21 $ :%;;?+ 7 >>%??;
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 408/846
Nr% Fëi~lr Pqa}anl Qrç`y
2'91
` o e
YYa 7 Ω Ω Ω &{khd ' { kh%+?
k70 h70 d70
7 tT&0>%9 ’ 09%>21+?
$ &0>%1 ’ 09%>21+?
$ &00%1 ’ 09%>21+?
$ &09%= ’ 09%>21+?W
$ T&06%0 ’ 02%=::+?
$ &02%= ’ 02%=::+?
$ &02%0 ’ 02%=::+?
$ &02%= ’ 02%=::+?W
$ T&02%= ’ 06%??1+?
$ &06%? ’ 06%??1+?
$ &06%; ’ 06%??1+?
$ &06%? ’ 06%??1+?W
$ T&0?%= ’ 00%2?1+?
$ &00%? ’ 00%2?1+?
$ &00%: ’ 00%2?1+?
$ &0?%0 ’ 00%2?1+?W
$ T&0:%1 ’ 0:%021+?
$ &0?%6 ’ 0:%021+?
$ &6%9 ' 0:%021+?
$ &;%0 ’ 0:%021+?W
$ T&01%2 ’ 0=%=?1+?
$ &02%1 ’ 0=%=?1+?
$ &0=%2 ’ 0=%=?1+?
$ &0=%= ’ 0=%=?1+?w
7 &9%;:: $ : $ :%1 $ :%0>0 $ 00%==6 $ 0%=?;+ ≆ ??%:
` o e
YY~ 7 Ω Ω Ω &{khd ’ { …+?7 YY` $ YY o $ YY`o $ YYa 7 ?>9%;9
k70 h70 d70
@flr`# ya yqy~k~q{ae ~lnly cly iàciqcly faifly g`eq`cgae~a# z`r` lo~aear c` ~`oc`
na nao`hl na nly ic`ykmki`ikleay irqx`n`y l na nloca yae~knl%
^@OC@ 2%??% ^`oc` na `eàckyky na }`rk`ex` z`r` ac avzarkgae~l `brçilc` na nly
~kzly na yagkcc`y ile ~ray ek}acay nkmarae~ay na mar~kckx`e~ay% &Nqee a~ `c% 0;2>+
Mqae~a na }`rk`ikùe YY b%c% GY Mi`ci% M~`o% ]`clr na z
Naoknl ` c`y yagkcc`y &@+ 22%6: 0 22%6 =>%6 >%>0 3333 %::0
Naoknl ` cly ek}acay &O+ ;;%;: ? >;%; >0%= 9%11 333 %::0nac mar~kckx`e~a
Ke~ar`iikùe na yagkcc` >>%?9 ? ??%0 06%> 9%11 33 %::0{ mar~kckx`e~a &@O+
_ayknq`c ??%: 06 0%?
^l~`c ?>9%;9 ?9
Ae ileicqykùe# naoknl ` pqa c` Mi`ci% ay gqifl g`{lr pqa c` M~`o%# ay naikr# =>%6 55
>%>0 ya raif`x` c` fkzù~ayky na pqa el f`{ nkmaraeik` ae~ra c`y yagkcc`y# { ely
keicke`gly zlr c` fkzù~ayky `c~are`~k}`# ay naikr# F08±0 ≪ ±? ≪ ±9 ≪ ±>% Ay~`
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 409/846
Nr% Fëi~lr Pqa}anl Qrç`y
2'9=
naikykùe ay `zl{`n` zlr qe }`clr na z gq{ ykbkmki`~k}l% Yk~q`ikùe ykgkc`r liqrra
ile cly ek}acay na mar~kckx`e~ay% Yke ago`rbl# ae iq`e~l ` c` ke~ar`iikùe ya }a pqa
cly m`i~lray yagkcc` { ek}acay na mar~kckx`e~a ay~àe ke~ar`i~q`enl# ay~l ay naoknl `
pqa 06%> 5 9%11% Zlr cl ~`e~l# ya ileicq{a pqa f`{ ke~ar`iikùe ae~ra ac ~kzl na
yagkcc` { ac ek}ac na mar~kckx`e~a% Ae ay~a i`yl c` ke~ar`iikùe zqnl liqrrkr zlr gar`
i`yq`ckn`n# zarl ~`gokëe zqnl liqrrkr zlr `cbqe }`clr av~ragl l zlr `cbøe
zrlocag` rac`ikle`nl ile ac nkyaðl avzarkgae~`c%
@eàckyky na }`rk`ex` na ~ray yae~knly8 nkyaðl ilgzca~`gae~a `ca`~lrkl
Zlr l~r` z`r~a# iq`enl ya f`oc` na `eàckyky na }`rk`ex` ile ic`ykmki`ikleay irqx`n`yl nkyaðly m`i~lrk`cay# f`{ ~`gokëe avzarkgae~ly pqa ke}lcqir`e gày na nly
m`i~lray# cl iq`c ely cca}` ` `eàckyky na }`rk`ex` na ic`ykmki`ikleay ae ~ray yae~knly%
@pqç# ay eaiay`rkl naikr# pqa ae ac i`yl na glnacly na @EL]@ m`i~lrk`cay ae ~ray
ic`ykmki`ikleay zqanae yar cly ~ray mkhly# cly ~ray `ca`~lrkly# qel `ca`~lrkx`nl { nly
mkhly# l nly `ca`~lrkx`nly { ac l~rl mkhl% Yke ago`rbl# `pqç ya ileyknar`ràe
qeki`gae~a avzarkgae~ly ile ~ray m`i~lray mkhly @# O { I# ae cly ek}acay `# o { i#
rayzai~k}`gae~a ae nkyaðly avzarkgae~`cay ilgzca~`gae~a `ca`~lrkx`nly%
Cly eøgarly na cly ek}acay na cly ~ray m`i~lray ay~àe razrayae~`nly zlr K # H {
D # rayzai~k}`gae~a# { Ckhd ay kbq`c `c eøgarl na loyar}`ikleay faif`y ile ac m`i~lr
@ `c ek}ac k# m`i~lr O `c ek}ac h { m`i~lr I `c ek}ac d % @pqç# yke ago`rbl# ay eaiay`rkl
`mkrg`r pqa ac `eàckyky m`i~lrk`c ay gq{ ilgzcki`nl iq`enl cly }`clray na Ckhd el
yle ~lnly kbq`cay# zlr cl ~`e~l# ae ay~a ay~qnkl ay~l ya ckgk~`rà ` Ckhd 7 C%
Ae ac avzarkgae~l na c` zrlnqiikùe na iao`n`# ~aeç`gly nly ek}acay# ay
naikr# ac m`i~lr yagkcc` { ac m`i~lr mar~kckx`e~a# zarl yk ay~a avzarkgae~l ya fkikar`
ile qe `eàckyky na }`rk`ex` na ~ray yae~knly# ya ca zqnkar` `brab`r l~rl m`i~lr gày#
ay naikr# nly ek}acay na `bq`% O`hl ay~`y ilenkikleay fqokar` 0? ilgoke`ikleay na
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 410/846
Nr% Fëi~lr Pqa}anl Qrç`y
2'92
~r`~`gkae~ly# { ya `yqgkrç` pqa >6 z`riac`y mqar`e `ykbe`n`y `ca`~lrk`gae~a ` cly
0? ilgoke`ikleay na ~r`~`gkae~ly%
L~rl avzarkgae~l rac`ikle`nl ile c` kebaekarç` `gokae~`c `~glymërki` yarç`
qy`enl ~ray m`i~lray z`r` gankr c`y ileiae~r`ikleay na b`yay { z`r~çiqc`y
ile~`gke`e~ay# ilgl zlr ahagzcl# YL?# EL?# Zo# In# a~i% Ay naikr# z`r` }ar cly
amai~ly pqa ~aenrç`e m`i~lray ilgl nkmarae~ay aca}`ikleay# nkmarae~ay nky~`eik`y {
nkmarae~ay ~kzly na yaeylray# nkmarae~ay ~kzly na ~arraely l ilenkikleay
ga~aralcùbki`y% C`y ic`ykmki`ikleay irqx`n`y ile ~ray m`i~lray# ~r`nkikle`cgae~a# ya
nkyaņ`rle z`r` avzarkgae~ly `brçilc`y# zarl ~`gokëe ~kaeae gqif`y `zcki`ikleayae l~r`y àra`y% C` ^@OC@ 2%?9 gqay~r` ac mlrg`~l qy`nl z`r` avzarkgae~ly
m`i~lrk`cay na ~ray m`i~lray mkhly%
^@OC@ 2%?9% ^`oc` na @EL]@ ile ~ray m`i~lray mkhly% &Ac`olr`ikùe zrlzk`+Mqae~a na YY b%c% Iq`nr`nl Mi`ci% M~`o%
}`rk`ikùe gankl
Amai~ly zrkeikz`cay
@ YY` `’0 GY` 7 YY`,&`'0+ GY`,y?0 M0T0'ΰ4`'0#`oi&e'0+W
O YY o o’0 GY o 7 YY o,&o'0+ GY o,y
?
? M?T0'ΰ4o'0#`oi&e'0+W I YYi i’0 GYi 7 YYi,&i'0+ GYi,y
?9 M9T0'ΰ4i'0#`oi&e'0+W
Ke~ar`iikùe na
nly m`i~lray
@O YY`o &`'0+&o'0+ GY`o 7 YY`o,&`'0+&o'0+ GY`o,y?> M>T0'ΰ4&`'0+&o'0+#`oi&e'0+W
@I YY`i &`'0+&i'0+ GY`i 7 YY`i,&`'0+&i'0+ GY`i,y?1 M1T0'ΰ4&`'0+&i'0+#`oi&e'0+W
OI YY oi &o'0+&i'0+ GY oi 7 YY oi,&o'0+&i'0+ GY oi,y?= M=T0'ΰ4&o'0+&i'0+#`oi&e'0+W
Ke~ar`iikùe na
~ray m`i~lray@OI YY`oi &`'0+&o'0+&i'0+ GY`oi 7 YY`oi,T&`'0+&o'0+&i'0+W GY`oi,y
?2 M2T0'ΰ4&`'0+&o'0+&i'0+#`oi&e'0+W
_ayknq`c YYa `oi&e'0+ y?a 7 YYa,T`oi&e'0+W
^l~`c YY~ `oie'0
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 411/846
Nr% Fëi~lr Pqa}anl Qrç`y
2'96
Nlena8
`
YY` 7 oie Ω & {
k%%% ’ {
%%%%+
?
&2'06+k70
o
YY o 7 `ie Ω & { %h%% ’ { …%+?
&2'0;+ h70
i
YYi 7 `oe Ω & { %%d% ’ { %%%%+?
&2'?:+d70
` o
YY`o 7 ie Ω Ω & { kh%% ’ { k… ' { %h%% $ { …%+?
&2'?0+k70 h70
` i
YY`i 7 oe Ω Ω & { k%d % ’ { k… ' { %%d% $ { …%+?
&2'??+k70 d70
o i
YY oi 7 `e Ω Ω & { %hd% ’ { %h%% ’ { %%d% $ { …%+?
&2'?9+ h70 d70
` o i
YY`oi 7 e Ω Ω Ω & { khd% ’ { kh%% ’ { k%d% ’ { %hd% $ { k… $ { %h%% $ { %%d% ’ { …%+?
&2'?>+k70 h70 d70
` o i e
YYa 7 Ω Ω Ω Ω &{khdc ’ { khd%+ &2'?1+k70 h70 d70 c70
` o i e
YY~ 7 Ω Ω Ω Ω &{khdc ’ { %%%%+ &2'?=+k70 h70 d70 c70
C` ykgolclbç` qy`n` ae c`y mùrgqc`y `e~arklray ya namkea na c` ykbqkae~a g`ear`8
{ k… 7 zrlgankl na c`y loyar}`ikleay z`r` ac k'ëykgl ek}ac nac m`i~lr @
{ %%%% 7 zrlgankl na ~ln`y c`y `oie loyar}`ikleay
{ %h%% 7 zrlgankl na c`y loyar}`ikleay z`r` ac h'ëykgl ek}ac nac ~r`~`gkae~l O
{ %%d% 7 zrlgankl na c`y loyar}`ikleay z`r` ac d 'ëykgl ek}ac nac ~r`~`gkae~l I
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 412/846
Nr% Fëi~lr Pqa}anl Qrç`y
2'9;
{ kh%% 7 zrlgankl na cly i`yly z`r` ac k'ëykgl ek}ac nac m`i~lr @ { ac h'ëykgl ek}ac nac
m`i~lr O
{khdc 7 nael~` c` c 'ëykg` loyar}`ikùe na c` ilgoke`ikùe nac ~r`~`gkae~l khd 'ëykgl
Cly ke}ay~kb`nlray ay~`nçy~kily Nqee a~ `c% &0;2>+ zrlzlrikle`e ac glnacl
z`r` ac `eàckyky na }`rk`ex` ae ~ray yae~knly# ay~l ay8
{khdc 7 ± $ ΰk $ ο h $ &ΰο+kh $ &ΰι+kd $ &οι+ hd $ &ΰοι+khd $ αkhdc &2'?2+
Nlena8
± 7 zrlgankl ~l~`c na cly ~ray ~r`~`gkae~ly `oi
ΰk 7 amai~l zrlgankl nac ek}ac k'ëykgl nac m`i~lr @
ο h 7 amai~l zrlgankl nac ek}ac h'ëykgl nac m`i~lr O
ιd 7 amai~l zrlgankl nac ek}ac d 'ëykgl nac m`i~lr I
&ΰο+kh 7 ke~ar`iikùe na cly m`i~lray @ { O# ay naikr# nac ek}ac k'ëykgl nac m`i~lr @ ile
ac ek}ac h'ëykgl nac m`i~lr O
&ΰι+kd 7 ke~ar`iikùe na cly m`i~lray @ { I# ay naikr# nac ek}ac k'ëykgl nac m`i~lr @
ile ac ek}ac d 'ëykgl nac m`i~lr I
&οι+ hd 7 ke~ar`iikùe na cly m`i~lray O { I# ay naikr# nac ek}ac h'ëykgl nac m`i~lr O
ile ac ek}ac d 'ëykgl nac m`i~lr I
&ΰοι+khd 7 ke~ar`iikùe na cly m`i~lray @# O { I# ay naikr# c`y ke~ar`iikleay ae~ra ac
ek}ac k'ëykgl nac m`i~lr @ ile ac ek}ac h'ëykgl nac m`i~lr O { ile ac ek}ac d 'ëykgl
nac m`i~lr I
Ke~ar`iikùe ile @EL]@ na nkyaðly m`i~lrk`cay na ~ray ic`ykmki`ikleay
Ae iq`e~l `c kgz`i~l na ke~ar`iikleay# iq`enl ya nkyað`e `eàckyky na }`rk`ex` ae
~ray yae~knly# ay kgzlr~`e~a ay~`r ileyikae~a na ay~` yk~q`ikùe# zlrpqa c` ke~ar`iikle
zqana kgz`i~`r c` ke~arzra~`ikùe pqa ya f`ia ile rayzai~l ` cly amai~ly zrkeikz`cay%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 413/846
Nr% Fëi~lr Pqa}anl Qrç`y
2'>:
@nagày# c` zrayaeik` na ke~ar`iikùe zqana nayiqorkr yk~q`ikleay kgzlr~`e~ay pqa
zqanae `{qn`r ` glnkmki`r ac nkyaðl avzarkgae~`c lrkbke`c# z`r` f`iarcl gày
razrayae~`~k}l% C`y ke~ar`iikleay qyq`cgae~a liqrrae iq`enl cly amai~ly
zrkeikz`cay yle gq{ br`enay# zarl zqanae nay`z`raiar iq`enl ac ke}ay~kb`lr
ay~`nçy~kil `gkelr` c`y nkmaraeik`y ae~ra cly ek}acay na qe ~r`~`gkae~l# f`ikaenl#
ile ay~l# pqa cly amai~ly zrkeikz`cay ya`e gaely zrleqeik`nly &Nqee a~ `c% 0;2=+
Ile rac`ikùe ` c`y gankikleay na c` ile~`gke`ikùe nac `kra qy`enl glnacly
na nkmqykùe `~glymërki`# ay naikr# z`r` }`ckn`r ay~qnkly na nkmqykùe `~glymërki`# l
z`r` f`iar ay~qnkly na kgz`i~l `gokae~`c# qe` `zcki`ikùe yarç` gankr c`yileiae~r`ikleay pqa liqrrae ` cl c`rbl na c` zcqg`% Z`r` qe nkyaðl m`i~lrk`c ile
~ray ~r`~`gkae~ly# ya zqana `brab`r l~rl m`i~lr gày `c ahagzcl na c` nkmqykùe
`~glymërki` ile nly ~r`~`gkae~ly# avzcki`nl `e~arklrgae~a% Ae ay~a i`yl# `nagày
na cly m`i~lray nky~`eik` { `c~qr`# ca zlnagly `brab`r qe ~ariar m`i~lr rac`ikle`nl
ile nkmarae~ay g`ri`y na gqay~ra`nlray%
Ae iq`e~l ac amai~l na ke~ar`iikùe# ae ay~qnkly na kgz`i~l `gokae~`c qy`enl
glnacly na nkmqykùe `~glymërki`# c` ke~ar`iikùe na cly m`i~lray# o`hl ay~qnkl# zqana
nayiqorkr yk~q`ikleay pqa zqan`e `mai~`r ac ay~qnkl% @qepqa yk okae# cly glnacly na
nkmqykùe `~glymërki` `yqgae ilenkikleay ickg`~lcùbki`y qekmlrgay# el loy~`e~a#
agkykleay mqbk~k}`y l c`y nkmaraeik`y ae cly ~kzly na ~arraely ilgl `rae`# `rikcc`#
zkanr`y# `bq`# ~kzl na }aba~`ikùe# a~i%# zlr nlena z`y` c` zcqg` na c` ifkgaea`
zqanae li`ykle`r pqa cly m`i~lray o`hl ay~qnkl# ke~ar`i~øae%
Yk~q`ikleay ykgkc`ray zqanae liqrrkr ile nkyaðly m`i~lrk`cay `zcki`nly ` c`
`brkiqc~qr` iq`enl ya `zcki`e nly m`i~lray ilgl ~kzly na yagkcc` { ek}acay na
mar~kckx`e~ay% @pqç ya ca zqana `brab`r l~rl m`i~lr gày# nkb`gly# ac ek}ac na `bq`
z`r` f`iar qe nkyaðl m`i~lrk`c# ay naikr# ile ~ray m`i~lray% Yke ago`rbl# yk f`{
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 414/846
Nr% Fëi~lr Pqa}anl Qrç`y
2'>0
ke~ar`iikùe# ~`c }ax cly ~kzly na yqacly na c`y z`riac`y el ~kaeae c`y gkyg`y
i`r`i~arçy~ki`y# ay naikr# na fqgan`n# na ~kzly na yqacly# ~kzly na ~agzar`~qr`y#
~kzly na eq~rkae~ay# a~i%# ae iq{l i`yl f`{ pqa ragk~krely ` cly nkyaðly na oclpqay
ilgzca~`gae~a `ca`~lrkx`nly%
Ahagzcl )0:% Ae qe ay~qnkl fkzl~ë~kil na nkmqykùe `~glymërki`# ay naikr# qy`enl qe
glnacl na nkmqykùe `~glymërki`# ya fkikarle gankikleay ae iq`~rl nky~`eik`y
nkmarae~ay ` cl c`rbl na c` zcqg` &1::# 0:::# 0?:: { 01:: ga~rly+# ae nly `c~qr`y
nkmarae~ay# &1:: { 6:: ga~rly+# ile iq`~rl g`ri`y nkmarae~ay na yaeylray# { ile
~`g`ðly na gqay~r`y na 9 loyar}`ikleay z`r` i`n` qe` na c`y ilgoke`ikleay naek}acay na cly ~ray m`i~lray% Z`r` ay~l ya n` qe` `}`ex`n` na cly }`clray ae c`
ykbqkae~a mlrg`8 Yqg` na cly iq`nr`nly nac m`i~lr @ 7 YY` 7 0%1:# yqg` na cly
iq`nr`nly nac m`i~lr O 7 YY o 7 0;%91# yqg` na cly iq`nr`nly nac m`i~lr I 7 YYi 7
0>2%::# yqg` na cly iq`nr`nly na c` ke~ar`iikùe na m`i~lray @ { O 7 YY`o 7 :%::=#
yqg` na cly iq`nr`nly na c` ke~ar`iikùe na m`i~lray @ { I 7 YY`i 7 >%69# yqg` na
cly iq`nr`nly na c` ke~ar`iikle na O { I 7 YY oi 7 ?%=># yqg` na cly iq`nr`nly na c`
ke~ar`iikùe na cly m`i~lray @# O { I 7 YY`oi 7 :%21# yqg` ~l~`c na cly iq`nr`nly 7
YY~ 7 069%2:% @yqgkr qe ek}ac na ykbekmki`eik` na :%:1% Zrlo`r c`y fkzù~ayky na cly
amai~ly zrkeikz`cay# yùcl yk ~ln`y c`y ke~ar`iikleay el yle ykbekmki`~k}`y% F`iar cl
ykbqkae~a8
&`+ @ykbe`r cly ykgolckygly `zrlzk`nly z`r` i`n` qel na cly ilgzleae~ay na c`
mqae~a na }`rk`ikùe
&o+ F`iar qe` ~`oc` na `eàckyky na }`rk`ex` pqa keicq{` c` M irç~ki` { cly }`clray na
z
&i+ F`iar zrqao`y na ykbekmki`eik` ylora cly amai~ly zrkeikz`cay
&n+ F`iar qe` zrqao` na ykbekmki`eik` ylora ~ln`y c`y ke~ar`iikleay%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 415/846
Nr% Fëi~lr Pqa}anl Qrç`y
2'>?
Ylcqikùe8
&`+ C` nky~`eik` na cly gqay~ra`nlray yk~q`nly ` cl c`rbl na c` zcqg`# ay naikr#
}kae~l `o`hl# ay ac m`i~lr @ ile k 7 >% C`y `c~qr`y ` c`y pqa ay~àe yk~q`nl cly
gqay~ra`nlray ay ac m`i~lr O ile h 7 ?% Mke`cgae~a# cly gqay~ra`nlray yle ac m`i~lr
I ile d 7 >% Ac eøgarl na i`yly ay e 7 9% Zlr cl ~`e~l ac eøgarl na
ilgoke`ikleay ay >v?v> 7 9? { ac eøgarl ~l~`c na loyar}`ikleay ay 9?v9 7 ;=%
&o+ C` ~`oc` na `eàckyky na }`rk`ex` ya n` `o`hl%
^@OC@ 2%?>% ^`oc` gly~r`enl cly n`~ly { ac ccae`nl na cly m`c~`e~ay ae c` ~`oc`# na
`iqarnl ` cly n`~ly zrlzlrikle`nly zlr ac zrlocag`% &Ac`olr`ikùe zrlzk`+%Mqae~a na YY b%c% Iq`nr`nl nac Mi`ci% M~`o% ]`clr z
]`rk`ikùe zrlgankl
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
Amai~ly zrkeikz`cay
Naoknl ` @ 0%1: 9 %1: >%02 ?%2= %::;
Naoknl ` O 0;%>: 0 0;%>: 0=0%02 9%;> z 333 %::0
Naoknl ` I 0>2%:: 9 >;%:: >:6%99 ?%2= z 333 %::0
Ke~ar`iikùe na nly m`i~lray
Naoknl ` @O :%::= 9 :%::? :%:? ?%2= z 5 %0::
Naoknl ` @I >%69 ; :%1> >%1: 0%;2 z 3 %::0
Naoknl ` OI ?%=> 9 :%66 2%99 ?%2= z 3 %::0
Ke~ar`iikùe na ~ray m`i~lray
Naoknl ` @OI :%21 ; :%:6 :%=2 0%;2 z 5 %0::(
Arrlr 2%1; => :%0?
^l~`c 069%2? ;1
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 416/846
Nr% Fëi~lr Pqa}anl Qrç`y
2'>9
&i+ Ileicqykùe8 cly amai~ly zrkeikz`cay yle ykbekmki`e~ay yqy~ae~`nly ile }`clray na
z gq{ zapqaðly na %::; { z 333 %::0% @c hqxb`r zlr ay~ly }`clray na z# avky~ae
amai~ly zrkeikz`cay gq{ mqar~ay na nky~`eik`# `c~qr` { yaeylray% Zlr l~rl c`nl#
naoknl ` pqa M2 7 GY`oi,y?2 7 :%=2 3 M2T:%:14;#=>W 7 0%;2# c`y ke~ar`iikleay ae~ra cly
m`i~lray nky~`eik`# `c~qr` { yaeylray el yle na kgzlr~`eik`% Yke ago`rbl# c`y
ke~ar`iikleay @I { OI yle }`rk`ocay kgzlr~`e~ay nac avzarkgae~l%
Ae ac ~ùzkil na `eàckyky na }`rk`ex`# ~`gokëe f`{ cl pqa ya cc`g` nkyaðly
m`i~lrk`cay ile ~lnly cly m`i~lray ` nly ek}acay% @pqç ya keicq{ae ~ùzkily ilgl
ilgoke`ikleay lr~lble`cay ckea`cay# nkyaðly na razcki`ikleay mr`iikle`cay# nkyaðly`ekn`nly l haràrpqkily# iq`nr`nly c`~kely# a~i% Ay~`y mqeikleay# yke ago`rbl# el ya
nkyiq~kr`e `pqç%
Ac `eàckyky na }`rk`ex`# ~`gokëe ya zqana `zcki`r ` zrlocag`y na rabraykùe
ckea`c { gøc~kzca z`r` a}`cq`r c` ykbekmki`eik` ~l~`c na c` aiq`ikùe na rabraykùe# ay
naikr# zrlo`enl c` fkzù~ayky eqc` na pqa ~lnly cly ilamkikae~ay zloc`ikle`cay nac
glnacl na rabraykùe yle kbq`cay ` iarl% Ay~a ~ag`# yke ago`rbl# ya nkyiq~a ae ac
i`zç~qcl nanki`nl ` rabraykùe gøc~kzca%
Ahagzcl )00% Ay~a ay qe zrlocag` rac`ikle`nl ile qe avzarkgae~l m`i~lrk`c ile
nly m`i~lray na amai~ly mkhly &@ { O+ { ile ~`g`ðly na gqay~r`y kbq`cay% Zlr
ahagzcl# ac m`i~lr @ ~kaea ` ek}acay# gkae~r`y pqa ac m`i~lr O ~kaea o ek}acay% Ay~a
avzarkgae~l ay~à rac`ikle`nl ile qe ay~qnkl na nkmqykùe `~glymërki` z`r` gankr c`y
ileiae~r`ikleay nac ile~`gke`e~a nac `kra YL? zrl}aekae~ay na qe` mqae~a agkylr`
kenqy~rk`c% Z`r` ~`cay mkeay ya yk~q`rle nly yaeylray# `c `x`r ` iq`~rl nkmarae~ay
nky~`eik`y }kae~l `o`hl na c` ifkgaea` kenqy~rk`c# ay naikr# ` 1::# 0:::# 01:: {
?::: ga~rly { ` nly `c~qr`y nkmarae~ay# ay naikr# ` 0:: { ?:: ga~rly% Qy`r qe
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 417/846
Nr% Fëi~lr Pqa}anl Qrç`y
2'>>
z`pqa~a na ilgzq~`nlr`# z`r` ~`cay mkeay% C` ~`oc` na `o`hl zrlzlrikle` cly n`~ly
zar~keae~ay% Qy`r qe ek}ac na ykbekmki`eik` na ΰ 7 :%:1 { f`iar cl ykbqkae~a8
&`+ Iley~rqkr qe` ~`oc` na `eàckyky na }`rk`ex` mkhl ae nly ic`ykmki`ikleay
&o+ @e`ckx`r cly amai~ly zrkeikz`cay na c` nky~`eik` { c` `c~qr`
&i+ @e`ckx`r ac amai~l na ke~ar`iikùe { n`r avzcki`ikleay `c rayzai~l
&n+ F`iar qe `eàckyky rayknq`c z`r` a}`cq`r cl `zrlzk`nl nac glnacl na @EL]@
&a+ F`iar ay~qnkly loha~k}ky~`y na ay~`nçy~ki` z`r` a}`cq`r c` mknackn`n nac glnacl
na @EL]@
^@OC@ 2%?=% ^`oc` gly~r`enl c`y ileiae~r`ikleay na YL? &ae zzg+ ae mqeikùe na
iq`~rl nky~`eik`y }kae~l `o`hl na c` ifkgaea` { na c`y `c~qr`y na cly yaeylray%
&Ac`olr`ikùe zrlzk`+
Nky~`eik`y }kae~l `o`hl na c` mqae~a agkylr`
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
@c~qr`y na cly yaeylray 1:: g 0::: g 01:: g ?::: g
\\\\\\\\\ ↙ \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
0:: g 1:: 9:: 06: ;:
10: 9:1 061 ;0
>;1 9?: 02; 6;
>;; ?;; 0;: 66
?:: g >1: ?;: 02: 2:
>>; ?2: 0=: 2:
>96 ?=: 011 =;
>11 ?21 0=1 =6
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 418/846
Nr% Fëi~lr Pqa}anl Qrç`y
2'>1
Ylcqikùe8
Qy`enl ac zrlbr`g` Gkek~`o ya zrliana ` nkyað`r c` g`~rkx l c` ae~r`n` na cly
n`~ly gly~r`n` ae c` ~`oc` na `o`hl%
^@OC@ 2%?=% ^`oc` gly~r`enl c` g`~rkx l nkyzlykikùe lrnae`n` na cly n`~ly ae c`
zàbke` nac Gkek~`o z`r` c` kemlrg`ikùe na ay~a zrlocag`%
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ Ileiae~r`ikùe na YL? &zzg+ Nky~`eik`y &g+ @c~qr`y &g+
&Ilcqge` I0+ &Ilcqge` I?+ &Ilcqge` I9+
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ 1:: 1:: g 0:: g10: 1:: g 0:: g
>;1 1:: g 0:: g
>;; 1:: g 0:: g9:: 0::: g 0:: g
9:1 0::: g 0:: g
9?: 0::: g 0:: g
?;; 0::: g 0:: g06: 01:: g 0:: g
061 01:: g 0:: g
02; 01:: g 0:: g
0;: 01:: g 0:: g;: ?::: g 0:: g
;0 ?::: g 0:: g
6; ?::: g 0:: g66 ?::: g 0:: g
>1: 1:: g ?:: g
>>; 1:: g ?:: g
>96 1:: g ?:: g
>11 1:: g ?:: g?;: 0::: g ?:: g
?2: 0::: g ?:: g
?=: 0::: g ?:: g
?21 0::: g ?:: g02: 01:: g ?:: g
0=: 01:: g ?:: g
011 01:: g ?:: g0=1 01:: g ?:: g
2: ?::: g ?:: g
2: ?::: g ?:: g
=; ?::: g ?:: g=6 ?::: g ?:: g
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 419/846
Nr% Fëi~lr Pqa}anl Qrç`y
2'>=
Nayzqëy na kebray`r cly n`~ly na `rrko` ` c` zàbke` nac Gkek~`o zrliana ilgl8
Y~`~ ↟ @EL]@ ↟ ^l'R`{…
Ae c` }ae~`e` pqa `z`raia na –^l'R`{ @eàckyky lm ]`rk`eia‗ { nae~rl na c`
}ae~`ekcc` na –_ayzleya‗ zlear# ae c` ilcqge` I0# ~lnly cly }`clray na c` }`rk`oca
na rayzqay~`# ay naikr# ae ay~a i`yl# c`y ileiae~r`ikleay na YL?% Aeyabqkn`# ae c`
}ae~`ekcc` na –_l m`i~lr‗ nac m`i~lr @ &raebcleay+# zlear cly }`clray na c`y
nky~`eik`y { ga~ar ae c` }ae~`ekcc` na –Ilcqge m`i~lr‗ c` kemlrg`ikùe nac m`i~lr O
&ilcqge`y+# ay naikr# c`y `c~qr`y% Ay~` kemlrg`ikùe ya n` ae c` ^`oc` 2%?=%
Qe` }ax pqa ya ke~rlnqiae ~lnly cly ~ërgkely ykbqkaenl c`y key~rqiikleay`e~arklray# krya `8 Y~`~ ↟ @EL]@ ↟ ^l'R`{…# { ac zrlbr`g` baear`rà c` ~`oc`
na nao`hl na @EL]@ ilrrayzlenkae~a ` c` zrabqe~` nac keikyl &`+%
^@OC@ 2%?2% ^`oc` gly~r`enl cly rayqc~`nly na @EL]@ n`nly zlr ac Gkek~`o%
&Ac`olr`ikùe zrlzk`+
^l'`{ @EL]@8 Ilei% YL? &zzg+ }aryqy Nky~`eik`y &g+# @c~qr`y &g+
Ylqria NM YY GY M Z M~`o%
Nky~`eik`y &g+ 9 =;1=;= ?906;; >1:0%:2 :%::: 9%:0
@c~qr`y &g+ 0 6::0 6::0 011%9: :%::: >%?=
Ke~ar`iikùe 9 09;; >== ;%:1 :%::: 9%:0
Arrlr ?> 0?92 1?
^l~`c 90 2:=999
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
y 7 2%026 _'Yp 7 ;;%6?! _'Yp&`nh+ 7 ;;%22!
&o+ Na `iqarnl ` c` ~`oc` na @EL]@ na `rrko`# cly amai~ly zrkeikz`cay nac m`i~lr @
&nky~`eik`y+ { ac m`i~lr O &`c~qr`y+ yle gqifl gq{ ykbekmki`e~ay%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 420/846
Nr% Fëi~lr Pqa}anl Qrç`y
2'>2
&i+ Avky~a qe` ke~ar`iikùe ykbekmki`e~a ae~ra cly m`i~lray @ &nky~`eik`+ { O &`c~qr`y+%
C` ke~ar`iikùe ae ay~a i`yl# zqnl liqrrkr zlr gar` i`yq`ckn`n l ~`c }ax zqnl
naoarya ` `cbøe zrlocag` ae cly n`~ly# ay naikr# ae ~ërgkely na i`qy` { amai~l%
Mçyki`gae~a f`oc`enl# `cbøe m`i~lr pqa el ya zqnl ile~rlc`r zqnl li`ykle`r c`
ke~ar`iikùe ae~ra cly nly m`i~lray% Zlr ahagzcl# zqnl liqrrkr `cbøe g`c
mqeikle`gkae~l na cly yaeylray# pqa el gknkarle okae c`y ileiae~r`ikleay na YL?
ae qe glgae~l n`nl% L~r`y r`xleay zqnkarle rac`ikle`rya ile `cbøe i`gokl
ga~alrlcùbkil keqyk~`nl &`qepqa ac glnacl na nkmqykùe `yqga ilenkikleay
ga~alrlcùbki`y iley~`e~ay+# agkykleay mqbk~k}`y# ~arrael el qekmlrga zlr nlena z`y` c` zcqg`# a~i% Ay~`nçy~ki`gae~a f`oc`enl# c`y ke~ar`iikleay ~`gokëe zqanae
liqrrkr iq`enl cly amai~ly zrkeikz`cay yle gq{ br`enay &ilgl ac m`i~lr @ ae ay~a
i`yl# `qepqa yk okae# ay~l ya zqana ilrrabkr `gkelr`enl c`y nkmaraeik`y ae~ra cly
ek}acay na qe ~r`~`gkae~l# z`r` f`iar cly amai~ly zrkeikz`cay gaely `iae~q`nly+%
&n+ C`y bràmki`y na `o`hl gqay~r`e cly rayqc~`nly z`r` ay~a keikyl%
Loyar} ~kle Lrnar
_ a y k n q ` c
9?9:?6?=?>???:060=0>0?0:6=>?
?:
01
0:
1
:
'1
'0:
'01
_ayknq cy ]aryqy ~fa Lrnar lm ~fa N ~&rayzleya ky Ileiae~r ikle na YL? &g++
Mk~~an ] cqa
_ a y k n q ` c
1::>::9::?::0::
?:
01
0:
1
:
'1
'0:
'01
_ayknq cy ]aryqy ~fa Mk~~an ] cqay&rayzleya ky Ileiae~r ikle na YL? &g++
Mkbqr` 2%0:% Br`mki`y gly~r`enl cly }`clray rayknq`cay ae mqeikùe nac eøgarl na
loyar}`ikùe { na cly }`clray `hqy~`nly%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 421/846
Nr% Fëi~lr Pqa}anl Qrç`y
2'>6
_ayknq c
Z a r i a e
~
?:010:1:'1'0:'01
;;
;1
;:
6:
2:
=:
1:
>:
9:
?:
0:
1
0
Elrg c Zrlo okck~{ Zcl~ lm ~fa _ayknq cy&rayzleya ky Ileiae~r ikle na YL? &g++
I l e i %
Y L ? & z z g
+
Nky~ eik y &g+
@c~qr`y &g+
1:: g?::: g01:: g0::: g
?:: g0:: g?:: g0:: g?:: g0:: g?:: g0:: g
1::
>::
9::
?::
0::
Kenk}knq c ] cqa Zcl~ lm Ilei% YL? &zzg+ }y Nky~`eik y &g+# @c~qr y &g+
Mkbqr` 2%00% Mkbqr`y gly~r`enl c` zrqao` na elrg`ckn`n { c` bràmki` na c`y
ileiae~r`ikleay }aryqy `c~qr`y { nky~`eik`y
&a+ Cly `eàckyky loha~k}ky~`y ay~`nçy~kily kenki`e qe ilamkikae~a na na~argke`ikùe
gq{ `c~l# ay naikr# _ ?
7 ;;%6?! ile y 7 2%06# cl pqa yqbkara qe oqae `hqy~a na cly
n`~ly% @nagày# C` Mkbqr` 2%0: gqay~r`# `zrlvkg`n`gae~a# ac gkygl eøgarl na
i`yly zlyk~k}ly { eab`~k}ly# cl iq`c kenki` pqa ac glnacl ay `zrlzk`nl%
Ykgkc`rgae~a# c` Mkbqr` 2%00 gqay~r` qe oqae `hqy~a na cly n`~ly ile c` zrqao` na
elrg`ckn`n% Mke`cgae~a# c` Mkbqr` 2%00 ile c` bràmki` na c`y ileiae~r`ikleay }aryqy
c`y nky~`eik`y { c`y `c~qr`y gqay~r` c`y ke~ar`iikleay pqa liqrrae iq`enl Mi`ci% 5 M~`o%
Ahagzcl )0?% Ay~a ay qe aharikikl rac`ikle`nl ile qe avzarkgae~l na `eàckyky na
}`rk`ex` na ~ray yae~knly% Ay~a ahagzcl ay~à aei`gke`nl ` kcqy~r`r# iùgl ya
ay~rqi~qr` qe` g`~rkx ile cly n`~ly# pqa ya ke~rlnqiae ae ac zrlbr`g` Gkek~`o# z`r`
iley~rqkr qe` ~`oc` na @EL]@ na ~ray ic`ykmki`ikleay l ~ray yae~knly { yqy bràmki`yilrrayzlenkae~ay% @yqgkr qe ek}ac na ykbekmki`eik` na ΰ $ :%:1% Y`i`r
ileicqykleay% Cly n`~ly ya n`e ae c` ~`oc` na `o`hl%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 422/846
Nr% Fëi~lr Pqa}anl Qrç`y
2'>;
^@OC@ 2%?;% ^`oc` gly~r`enl c` kemlrg`ikùe z`r` ay~a aharikikl%
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
M`i~lr O0 M`i~lr O? \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
M`i~lr I0 M`i~lr I? M`i~lr I0 M`i~lr I?
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
M`i~lr @0 ?:%: 00%: 09%: 09%:
?:%: 0?%: 0?%: 0?%:
02%: 0:%: 0?%: 09%:
0;%: 0?%: 09%: 09%:
M`i~lr @? ?:%: 0=%: 0>%: 00%:?:%: 0;%: 02%: 0:%:
0;%: 02%: 0?%: 6%:
?:%: 06%: 09%: 6%:
M`i~lr @9 02%: ??%: ?:%: 0>%:
06%: ??%: ??%: 01%:
06%: ??%: ?0%: 0>%:
06%: ?0%: ?0%: 0=%:
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
Ylcqikùe8
@pqç# ac m`i~lr @ ~kaea ~ray ek}acay &k 7 0#…# ` 7 9+4 ac m`i~lr O ~kaea nly ek}acay &h
7 0#…# o 7 ?+ { ac m`i~lr I ~kaea nly ek}acay &d 7 0#…# i 7 ?+ l ya` 9v?v? 7 0?
ilgoke`ikleay na ~r`~`gkae~ly% @nagày f`{ c 7 0#…# e 7 > loyar}`ikleay ae i`n`
qel na c`y `oi ilgoke`ikleay na ~r`~`gkae~ly &iacn`y+ l ya` `oie 7 9v?v?v> 7 >6
loyar}`ikleay%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 423/846
Nr% Fëi~lr Pqa}anl Qrç`y
2'1:
Zrliankgkae~l z`r` iley~rqkr qe` ~`oc` na @EL]@ na ~ray m`i~lray l ae ~ray
yae~knly qy`enl ac zrlbr`g` Gkek~`o
0% Zrkgar`gae~a# ya namkeae ic`r`gae~a cly ~r`~`gkae~ly &ae ilcqge`y {
raebcleay+# ay naikr ac m`i~lr @# { cly m`i~lray O { I ile yqy ilrrayzlenkae~ay
ek}acay na i`n` qel na ay~ly m`i~lray# ilgl ya nayirkokù `rrko`% @nagày# f`{ pqa
na~argke`r e l ya` ac eøgarl na n`~ly ae i`n` iacn`%
?% Qe` }ax faifl cl `e~arklr# f`{ pqa baear`r c` g`~rkx na n`~ly pqa ya ke~rlnqikrà
ae c` flh` nac Gkek~`o% Z`r` f`iar ay~l# krya `8
I`ci ↟ G`da Z`~~arean N`~` ↟ Ykgzca Ya~ lm EqgoaryF`ikaenl ay~l `z`raia qe` }ae~`e` pqa ya ccae` `yç8
Ae c` }ae~`ekcc` na –Y~lra Z`~~arean N`~` ke‗ zlear @ &l ya` ac m`i~lr @+
Ae c` }ae~`ekcc` –Mrlg Mkry~ ]`cqa‗ zlear 0 &ac zqe~l na z`r~kn` na c` yaiqaeik`+
Ae c` }ae~`ekcc` –^l C`y~ ]`cqa‗ zlear 9 &` 7 9 ek}acay nac m`i~lr @+
Ae c` }ae~`ekcc` –Ke Y~azy lm‗ zlear 0
Ae c` }ae~`ekcc` –Cky~ A`if ]`cqa‗ zlear 0
Ae c` }ae~`ekcc` na –Cky~ ~fa Rflca Yapqaeia‗ zlear 0= &l ya` ac zrlnqi~l na oie
7 ?v?v> 7 0=+
Aeyabqkn`# zlear LD { zraykle`r c` ~aic` m9 z`r` olrr`r ~lnl cl `e~arklr { zrlyabqkr
ile ac ykbqkae~a z`yl%
?% @flr`# krya `8
I`ci ↟ G`da Z`~~arean N`~` ↟ Ykgzca Ya~ lm Eqgoary%
F`ikaenl ay~l `z`raia qe` }ae~`e` { ya ccae` `yç8
Ae c` }ae~`ekcc` na –Y~lra Z`~~arean N`~` Ke‗ zlear O &ac m`i~lr O+
Ae c` }ae~`ekcc` na –Mrlg Mkry~ ]`cqa‗ zlear 0 &zqe~l na z`r~kn`+
Ae c` }ae~`ekcc` –^l C`y~ ]`cqa‗ zlear ? &o 7 ? ek}acay na O+
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 424/846
Nr% Fëi~lr Pqa}anl Qrç`y
2'10
Ae c` }ae~`ekcc` –Ke Y~azy lm‗ zlear 0
Ae c` }ae~`ekcc` –Cky~ A`if ]`cqa‗ zlear 9 &` 7 9+
Ae c` }ae~`ekcc` na –Cky~ ~fa Rflca Yapqaeia‗ zlear 6 &l ya` ac zrlnqi~l na ie 7
?v> 7 6+
Zlear LD { cqabl zraykle`r c` ~aic` m9 z`r` olrr`r ~lnl cl `e~arklr { zrlyabqkr ile
ac ykbqkae~a z`yl%
9% @flr`# krya `8
I`ci ↟ G`da Z`~~arean N`~` ↟ Ykgzca Ya~ lm Eqgoary
F`ikaenl ay~l `z`raia qe` }ae~`e` { ya ccae` `yç8Ae c` }ae~`ekcc` na –Y~lra Z`~~arean N`~` Ke‗ zlear8 I &ac m`i~lr I+
Ae c` }ae~`ekcc` na –Mrlg Mkry~ ]`cqa‗ zlear8 0 &zqe~l na z`r~kn`+
Ae c` }ae~`ekcc` –^l C`y~ ]`cqa‗ zlear8 ? &i 7 ? ek}acay na I+
Ae c` }ae~`ekcc` –Ke Y~azy lm‗ zlear8 0
Ae c` }ae~`ekcc` –Cky~ A`if ]`cqa‗ zlear8 = &ac zrlnqi~l na `o 7 9v? 7 =+
Ae c` }ae~`ekcc` na –Cky~ ~fa Rflca Yapqaeia‗ zlear > &e 7 >+
Zlear LD { zraykle`r c` ~aic` m9 z`r` olrr`r ~lnl cl `e~arklr { zrlyabqkr ile ac
ykbqkae~a z`yl%
>% Nayzqëy na ~lnl cl `e~arklr# qe` }ax pqa {` ay~àe ccae`y c`y ilcqge`y I0# I? {
I9 &M`i~lray @# O { I# rayzai~k}`gae~a+# ya ~r`~` na ga~ar cly n`~ly na c` }`rk`oca
na rayzqay~`# [ &Ilcqge` I>+% Ay~l ya zqana f`iar g`eq`cgae~a zlekaenl i`n`
}`clr na [# &qy`enl cly n`~ly na c` ^@OC@ 2%?;+ ae yq ilrrayzlenkae~a zlykikùe
na @# O { I# ilgl ya gqay~r` ae c` ~`oc` na `o`hl%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 425/846
Nr% Fëi~lr Pqa}anl Qrç`y
2'1?
C` ^@OC@ 2%9:% ^`oc` gly~r`enl cly n`~ly%
\\\\\\\\\\\\\\\\\\\\\\\\\\\\ @ O I [
\\\\\\\\\\\\\\\\\\\\\\\\\\\\
0 0 0 ?:%:? 0 0 ?:%:9 0 0 02%:
0 ? 0 09%:? ? 0 0>%:9 ? 0 ?:%:0 0 ? 00%:
? 0 ? 0=%:9 0 ? ??%:0 ? ? 09%:? ? ? 00%:
9 ? ? 0>%:0 0 0 ?:%:? 0 0 ?:%:9 0 0 06%:
0 ? 0 0?%:? ? 0 02%:9 ? 0 ??%:0 0 ? 0?%:
? 0 ? 0;%:9 0 ? ??%:0 ? ? 0?%:
? ? ? 0:%:9 ? ? 01%:0 0 0 02%:
? 0 0 0;%:9 0 0 06%:0 ? 0 0?%:? ? 0 0?%:
9 ? 0 ?0%:0 0 ? 0:%:? 0 ? 02%:9 0 ? ??%:0 ? ? 09%:
? ? ? 6%:9 ? ? 0>%:0 0 0 0;%:? 0 0 ?:%:
9 0 0 06%:0 ? 0 09%:? ? 0 09%:9 ? 0 ?0%:
0 0 ? 0?%:? 0 ? 06%:
9 0 ? ?0%:0 ? ? 09%:? ? ? 6%:9 ? ? 0=%:
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
1% Qe` }ax ke~rlnqiknly cly n`~ly na [ gly~r`nly ae c` ^@OC@ 2%06# krya `8
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 426/846
Nr% Fëi~lr Pqa}anl Qrç`y
2'19
Y~`~ ↟ @EL]@ ↟ Baear`c Ckea`r Glnac
Ay~` lrnae baear` c` }ae~`e` –Baear`c Ckea`r Glnac‗%
Ae c` }ae~`ekcc` na –_ayzleyay‗ zlear8 [
Ae c` }ae~`ekcc` na –Glnac‗ zlear8 @OI @(O @(I O(I @(O(I
Ae c` }ae~`e` na –Baear`c Ckea`r Glnac Ilgz`rkyley‗ zqe~a`r8 –Z`krkya
Ilgz`rkyley l ^qda{‗# a~i%
Ae c` }ae~`e` na –Baear`c Ckea`r Glnac _ayqc~y‗ zqe~a`r8 –Ke `nnk~kle
Ilammkikae~y mlr `cc ^argy‗# a~i%
Ae c` }ae~`e` na –Baear`c Ckea`r Glnac'M`i~lr‗ ae~r`r ae c` }ae~`ekcc` na –G`keZcl~ Ammai~y‗ { zlear8 @ O I
Ae c` }ae~`ekcc` na –Ke~ar`i~kle Zcl~y‗ zlear8 @ O I# a~i%
^lnly ay~ly gl}kgkae~ly baear`r`e c` ^`oc` 2%90 na @EL]@ { c`y bràmki`y%
^@OC@ 2%90% ^`oc` gly~r`enl cly rayqc~`nly na @EL]@ na ~ray yae~knly%
Baear`c Ckea`r Glnac8 [ }aryqy @# O# I
M`i~lr ^{za Ca}acy ]`cqay@ mkvan 9 0# ?# 9
O mkvan ? 0# ?I mkvan ? 0# ?
@e`c{yky lm ]`rk`eia mlr [ &_ayzqay~`+# qykeb @nhqy~an YY mlr ^ay~y%
Ylqria NM Yap YY @nh YY @nh GY M Z Mirç~ki`
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ @ ? ?0:%621 ?0:%621 0:1%>96 62%2= :%::: 9%?9
O 0 02?%1?0 02?%1?0 02?%1?0 0>9%=: :%::: >%:6
I 0 ;9%1?0 ;9%1?0 ;9%1?0 22%6> :%::: >%:6
@(O ? =?%1>? =?%1>? 90%?20 ?=%:9 :%::: 9%?9
@(I ? 0=%2;? 0=%2;? 6%9;= =%;; :%::9 9%?9O(I 0 2%1?0 2%1?0 2%1?0 =%?= :%:02 >%:6
@(O(I ? 0=2%2;? 0=2%2;? 69%6;= =;%69 :%::: 9%?9Arrlr 9= >9%?1: >9%?1: 0%?:0
^l~`c >2 22>%609
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 427/846
Nr% Fëi~lr Pqa}anl Qrç`y
2'1>
y 7 0%:;=:6 _'Yp 7 ;>%>?! _'Yp&`nh+ 7 ;?%20!
^arg Ilam YA Ilam ^ Z
Iley~`e~ 01%;921 :%016? 0::%2> :%:::
@
0 '?%:=?1 :%??92 ';%?? :%:::
? ':%60?1 :%??92 '9%=9 :%::0
O
0 0%6;16 :%016? 00%;6 :%:::
I
0 0%9;16 :%016? 6%6? :%:::
@(O
0 0 ':%=>16 :%??92 '?%6; :%::2
? 0 0%=:>? :%??92 2%02 :%:::
@(I
0 0 :%>2;? :%??92 ?%0> :%:9;
? 0 :%91>? :%??92 0%16 :%0??O(I
0 0 ':%9;16 :%016? '?%1: :%:02
@(O(I
0 0 0 ?%9;16 :%??92 0:%20 :%:::
? 0 0 ':%??;? :%??92 '0%:? :%909
G a ` e l
m [ & r a y z q a y
~ `
+
9?0
0;%1
06%:
0=%1
01%:
?0
?0
0;%1
06%:
0=%1
01%:
@ O
I
Br`mki`y na cly amai~ly zrkeikz cay @# O { I
Mkbqr` 2%0?% Bràmki` gly~r`enl cly amai~ly zrkeikz`cay na @# O { I%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 428/846
Nr% Fëi~lr Pqa}anl Qrç`y
2'11
@
?:
0=
0?
O
I
?0
?0
?:
0=
0?
9?0
?:
0=
0?
@
9
0
?
O
0
?
I
0
?
Br`mki` na c`y ke~ar`iikleay na cly m`i~lray @#O#I
Mkbqr` 2%09% Bràmki`y gly~r`enl cly amai~ly na ke~ar`iikùe ae~ra cly m`i~lray @# O {
I% Ya ca zkna `c cai~lr ke~arzra~`r ay~`y ke~ar`iikleay%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 429/846
Nr% Fëi~lr Pqa}anl Qrç`y
2'1=
Y~`en`rnkxan _ayknq`c
Z a r i a e ~
9%:0%1:%:'0%1'9%:
;;
;:
1:
0:
0
Mk~~an ]`cqa
Y ~ ` e n ` r n k x a n _ a y k n q ` c
?:010:
?
:
'?
Y~`en`rnkxan _ayknq`c
M r a p q a e i {
9?0:'0'?
0=
0?
6
>
:
Loyar}`~kle Lrnar
Y ~ ` e n ` r n k x a n _ a y k n q `
c
>1>:919:?1?:010:10
?
:
'?
Elrg`c Zrlo`okck~{ Zcl~ lm ~fa _ayknq`cy _ayknq`cy ]aryqy ~fa Mk~~an ]`cqay
Fky~lbr`g lm ~fa _ayknq`cy _ayknq`cy ]aryqy ~fa Lrnar lm ~fa N`~`
_ayknq`c Zcl~y mlr [
Mkbqr` 2%0>% Bràmki`y gly~r`enl c` zrqao` na elrg`ckn`n# cly rayknq`cay
ay~`en`r`kx`nly# ae mqeikùe na cly }`clray `hqy~`nly# fky~lbr`g` na cly rayknq`cay {
cly rayknq`cay ae mqeikle na cly ùrnaeay loyar}`nly% @pqç ay na el~`rya pqa# na
`iqarnl ` ay~ly bràmkily yqoha~k}ly# ac glnacl `hqy~` okae cly n`~ly%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 430/846
Nr% Fëi~lr Pqa}anl Qrç`y
2'12
Aharikikly I`zç~qcl 2
2%0% Cly ykbqkae~ay n`~ly ya lo~q}karle na qe gqay~ral `~glymërkil na ùvknly na
`xqmra &YL?+ zrl}aekae~a na > cqb`ray nkmarae~ay% F`iar qe `eàckyky na }`rk`ex` ile
qe ek}ac na ykbekmki`eik` na ΰ 7 :%:1% ]ar yk f`{ nkmaraeik`y ae~ra cly > yk~kly% Qy`r
c` zrqao` na ilgz`r`ikleay gøc~kzcay z`r` }ar iq`cay yle kbq`cay { iq`cay yle
naykbq`cay yk ay pqa `yç ay%
^`oc` gly~r`enl cly n`~ly nac YL?% &Ac`olr`ikùe zrlzk`+
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
Yk~kl )0 Yk~kl )? Yk~kl )9 Yk~kl ) > \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
?: ?1 ?6 90
02 ?1 90 01
06 ?= 9> 0?
0: 0> 02 ?>
^`oc` gly~r`enl cly rayqc~`nly qy`enl qe z`pqa~a na ilgzq~`nlr` ilgl AVIAC%
Ccae`r cly m`c~`e~ay na c` ~`oc`% &Ac`olr`ikùe zrlzk`+
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ Mqae~a na YY b% c% Iq`nr`nl nac Mi`ci% M~`o% ]`clr na z
}`rk`ikùe zrlgankl
Naoknl `c ?=0%=; 9 0%;9
~r`~`gkae~l
_ayknq`c 1>9%21 >1%90
&Arrlr avzarkgae~`c+^l~`c 01
2%?% Qe ke}ay~kb`nlr naya` ay~qnk`r ac amai~l na iq`~rl mar~kckx`e~ay nkmarae~ay z`r`
}ar yqy amai~ly ae c` zrlnqiikùe na g`çx% Z`r` ay~l# ya nk}knkù qe` xle` `brçilc` ae
?> z`riac`y nac gkygl ~`g`ðl { mlrg`% Qy`r qe ek}ac na ykbekmki`eik` na :%:1%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 431/846
Nr% Fëi~lr Pqa}anl Qrç`y
2'16
Zrlo`r pqa el f`{ nkmaraeik` ae~ra cly iq`~rl ~r`~`gkae~ly% Qy`r ac zrlbr`g`
Gkek~`o%
^`oc` gly~r`enl c` zrlnqiikùe na g`çx o`hl iq`~rl nkmarae~ay ~r`~`gkae~ly na
mar~kckx`e~ay% &Ac`olr`ikùe zrlzk`+
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
^r`~`gkae~ly _aenkgkae~ly
Yke `zcki`ikùe na mar~kckx`e~a &0+ ;; >: =0 2? 2= 6>
Ile `zcki`ikùe na mar~kckx`e~a &?+ ;= 6> 6? 0:> ;; 0:1
Ile `zcki`ikùe na mar~kckx`e~a &9+ =9 12 60 1; => 2?
Ile `zcki`ikùe na mar~kckx`e~a &>+ 2; ;? ;0 62 26 20
C`y yqzlykikleay yle pqa c`y > zloc`ikleay nac raenkgkae~l na g`çx ay~àe
elrg`cgae~a nky~rkoqkn`y# ile c`y }`rk`ex`y na c`y zloc`ikleay kbq`cay { ile c`y
loyar}`ikleay kenazaenkae~ay%
2%9% Z`r` ilgz`r`r c` amai~k}kn`n na 9 gqay~ra`nlray na b`yay# ay naikr# qy`enl
gë~lnly @# O { I ya yacaiikle`rle gqay~r`y na ~`g`ðl iq`~rl { ya rabky~r`rle cly
ykbqkae~ay rayqc~`nly ae zzg%
^`oc` gly~r`enl c` kemlrg`ikùe rapqarkn`% &Ac`olr`ikùe zrlzk`+
Gë~lnl @ Gë~lnl O Gë~lnl I
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
20 ;: 2?
21 6: 22
=1 6= 2=
=; 6> 2;
Zrlo`r c` fkzù~ayky na pqa el f`{ nkmaraeik`y ae~ra cly ~ray zrlgankly
zloc`ikle`cay ile ΰ 7 :%:1% Qy`r cly zrlbr`g`y na AVIAC# EIYY l Y@Y {
ilgzca~`r cly m`c~`e~ay na c` ~`oc`%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 432/846
Nr% Fëi~lr Pqa}anl Qrç`y
2'1;
^`oc` na `eàckyky na }`rk`ex`% &Ac`olr`ikùe zrlzk`+
Mqae~a na b%c% Yqg` na cly Iq`nr`nl nac Mi`ci% M~`o% ]`clr z
}`rk`ikùe iq`nr`nly &YY+ zrlgankl
Naoknl `c ~r`~`gkae~l ? ??6%: 01%26
&}`rk`ikùe ae~ra
cly brqzly+
_ayknq`c &arrlr+ 09:%: 0>%>
&}`rk`ikùe nae~rl na
cly brqzly+
^l~`c 00 16=%:
: M~`o% 7 >%9 Mi`ci% 7 01%26
Bràmki` gly~r`enl c`y àra`y na `iaz~`ikùe { raif`xl z`r` ac zrlocag` na `rrko`%
&Ac`olr`ikùe zrlzk`+
2%>% Yqzùeb`ya pqa iq`~rl c`olr`~lrkly `gokae~`cay ay~àe `e`ckx`enl qe` gqay~r`
na qe mkc~rl ile z`r~çiqc`y na zclgl `~glymërkil zrl}aekae~ay na qe ilgzcahl
kenqy~rk`c% Z`r` ay~l# ya pqkara y`oar c` amai~k}kn`n ae~ra cly gë~lnly na `eàckyky
qy`nly zlr ay~ly iq`~rl c`olr`~lrkly nkmarae~ay% F`iar cly ykbqkae~ay iàciqcly8
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 433/846
Nr% Fëi~lr Pqa}anl Qrç`y
2'=:
&`+ Zrlo`r c` fkzù~ayky eqc` Fl8±0 7 ±? 7 ±9 7 ±> ay naikr pqa el f`{ nkmaraeik`y ae
cly zrlgankly zloc`ikle`cay na cly rayqc~`nly na cly `eàckyky na cly > gë~lnly
nkmarae~ay qy`nly zlr cly c`olr`~lrkly% Ay~`ocaiar c` fkzù~ayky `c~are`~k}` na ay~a
zrlocag`%
&o+ I`ciqc`r ac }`clr na c` zrlo`okckn`n z%
C` ~`oc` na `o`hl gqay~r` cly }`clray lo~aeknly zlr cly > c`olr`~lrkly zlr cly ~ray
gë~lnly qy`nly zlr ay~ly iq`~rl c`olr`~lrkly% Ay~a ay qe ahagzcl na `eàckyky na
}`rk`ex` ile nly m`i~lray%
^`oc` gly~r`enl c`y ay~kg`ikleay na cly iàciqcly na cly 0? rayqc~`nly zlr cly 9gë~lnly nkmarae~ay qy`nly zlr cly iq`~rl c`olr`~lrkly nky~ke~ly% &Ac`olr`ikùe
zrlzk`+
Gë~lnl na `eàckyky Yqg` na cly raebcleay
„„„„„„„„„„„„„„„„„„„„
C`olr`~lrkl 0 ? 9 k
0 0= 0; ?> 1;
? ?0 ?: ?0 =?
9 06 ?0 ?? =0
> 09 ?: ?1 16
Yqg` na c`y =6 6: ;? ?>:
ilcqge`y & h+
Y`i`r c`y ileicqykleay naokn`y na c` fkzù~ayky z`r` cly ~ray gë~lnly na `eàckyky {
naikr yk ya raif`x` l ya ra~kaea c` fkzù~ayky%
2%1% C` ~`oc` na `o`hl gqay~r` qe` kemlrg`ikùe pqa ya rai`où na qe gqay~ral na qe
ile~`gke`e~a `~glymërkil &lxlel+ zrl}aekae~a na 1 gqay~ra`nlray cli`ckx`nly ae
ikeil cqb`ray nkmarae~ay% F`iar cly ykbqkae~ay iàciqcly%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 434/846
Nr% Fëi~lr Pqa}anl Qrç`y
2'=0
&`+ Zrlo`r c` fkzù~ayky eqc` na pqa el f`{ nkmaraeik`y ae~ra c`y 1 zloc`ikleay
gqay~ra`n`y# Fl8 ±0 7 ±? 7 ±9 7 ±> 7 ±1# ile qe ek}ac ykbekmki`e~a na ΰ 7 :%:1%
&o+ F`iar qe` ~`oc` na @EL]@%
&i+ I`ciqc`r ac }`clr na c` zrlo`okckn`n z%
^`oc` ile cly n`~ly na lxlel ile cly eøgarly na cly gqay~ra`nlray% &Ac`olr`ikùe
zrlzk`+
0 ? 9 > 1
110 1;1 =9; >02 1=9
>12 16: =01 >>; =90
>1: 1:6 100 102 1??290 169 129 >96 =09
>;; =99 =>6 >01 =1=
=9? 102 =22 111 =2;
2%= Yqzùeb`ya pqa 01 zaryle`y f`e yknl yacaiikle`nly `ca`~lrk`gae~a na qe`
zloc`ikùe na loayly { f`e yknl yaz`r`nly `c `x`r nae~rl ~ray brqzly% I`n` brqzl na
loayly mqa `ckgae~`nl ile ~ray ~kzly na ilgkn`y nkmarae~ay z`r` zarnar zayl# ay
naikr# `ckgae~ly &0+# &?+ { &9+% Nayzqëy na `cbøe ~kagzl# cly zayly pqa zarnkarle
cly z`r~kikz`e~ay na cly ~ray brqzly ya rabky~r`rle% Cly zayly ya n`e ae c` ~`oc` na
`o`hl8
^`oc` gly~r`enl cly zayly zarnknly &br`gly+ na cly z`r~kikz`e~ay% &Ac`olr`ikùe
zrlzk`+
^kzly na ilgkn`y
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
^kzl &0+ ^kzl &?+ ^kzl &9+
>? 00? 2:;= ;= 02
60 66 >;
;1 091 ?>
2= 00; >:
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 435/846
Nr% Fëi~lr Pqa}anl Qrç`y
2'=?
Ay~ly n`~ly ay~àe ae ilemlrgkn`n ile qe m`i~lr na qe nkyaðl ilgzca~`gae~a
`ca`~lrkl% Qe m`i~lr ay ac `ckgae~l n`nl ` cly loayly% Ay~l ay qe nkyaðl
ilgzca~`gae~a `ca`~lrkl# zlrpqa c`y qekn`nay avzarkgae~`cay# na cly 01 yqha~ly#
f`e yknl `ykbe`n`y `ca`~lrk`gae~a ` cly ~ray ~kzly na ilgkn`y% Zrlo`r c` fkzù~ayky
eqc` na Fl8 ±0 7 ±? 7 ±9 { c` fkzù~ayky `c~are`~k}` na pqa# iq`enl gaely qel na cly
zrlgankly ay nkmarae~a na cly l~rly% F08±0 ≪ ±? ≪ ±9% Yk c` fkzù~ayky eqc` ay ikar~`#
ae~leiay c`y ~ray zloc`ikleay na cly zayly zarnknly zlr cly loayly yle kbq`cay%
Yqbaraeik`8 z`r` ay~kg`r ac zrlgankl nac iq`nr`nl nae~rl na cly ~r`~`gkae~ly l nac
arrlr avzarkgae~`c qy`r c`y rac`ikleay8d e
y?Z 7 Ω Ω & V kh ' V k+
?, d &e ' 0+
k70 h70
F`iar c` ~`oc` nac `eàckyky na }`rk`ex` z`r` cly loayly { y`i`r ileicqykleay `c
rayzai~l% @nagày# ra}ar~kr ay~a zrlocag` ` qe nkyaðl na oclpqay `ca`~lrkly { }ar yk
f`{ `cbqe` gahlrç` ae ac arrlr avzarkgae~`c%
2%2% Cly n`~ly na `o`hl razrayae~`e ac eøgarl na flr`y na `ck}kl z`ck`~k}l n`nl zlr
1 ~`oca~`y nkmarae~ay @# O# I# N# A# z`r` ac nlclr na i`oax`# pqa ya cay
`ngkeky~r`rle ` ?1 yqha~ly pqkaeay avzarkgae~`o`e nlclray na i`oax` &gkbr`ð`y+%
F`iar qe `eàckyky na }`rk`ex` z`r` zrlo`r c` fkzù~ayky `c ek}ac na ykbekmki`eik` na
:%:1 na pqa ac eøgarl zrlgankl na flr`y na `ck}kl z`ck`~k}l n`nl zlr c`y ~`oca~`y
ay ac gkygl z`r` c`y ikeil ~`oca~`y qy`n`y% I`ciqc`r ac }`clr na c` zrlo`okckn`n z%
Cly n`~ly ya n`e ae c` ~`oc` na `o`hl% I`ciqc`r c` ~`oc` na `eàckyky na }`rk`ex`%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 436/846
Nr% Fëi~lr Pqa}anl Qrç`y
2'=9
^`oc` gly~r`enl c`y flr`y na `ck}kl ile c`y 1 ~`oca~`y na `yzkrke`y% &Ac`olr`ikùe
zrlzk`+
^kzly na ~`oca~`y
„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„ @ O I N A
„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„
1 ; 9 ? 2
> 2 1 9 =
6 6 ? > ;
= = 9 0 >
9 ; 2 > 2
2%6% Ae qe ay~qnkl na ile~`gke`ikùe na ilrrkae~ay# ile ac loha~l na ra}ky`r pqa el
fqokar` nayi`rb`y kenqy~rk`cay# zra}kl ` qe zrl{ai~l na nkcqikùe# ya `e`ckxù c`
nag`en` oklpqçgki` na lvçbaely na 1 nç`y &NOL+1 ae gb,C { ya lo~q}karle cly
ykbqkae~ay n`~ly &gly~r`nly ae c` ~`oc` na `o`hl+ nac gqay~ral pqa ya fkxl ` cl
c`rbl na c` ilrrkae~a# ay naikr# ae > cqb`ray nkmarae~ay% F`iar qe `eàckyky na }`rk`ex`
qy`enl qe ek}ac na ykbekmki`eik` na :%:1% ]ar yk f`{ nkmaraeik`y ae~ra c`y
ileiae~r`ikleay na NOL na cly iq`~rl cqb`ray gqay~ra`nly &¼Na `iqarnl `
ay~qnkly na ile~`gke`ikùe na ilrrkae~ay# ile pqë loha~l ya ~aenrç` pqa f`iar ay~l<+
@nagày# qy`r c` zrqao` na ilgz`r`ikleay gøc~kzcay z`r` }ar iq`cay yk~kly na
gqay~ral yle kbq`cay { iq`cay yle naykbq`cay# yk ay pqa ay~l ay `yç% Yqbaraeik`8
Qy`r ac z`pqa~a na AVIAC l ac zrlbr`g` GKEK^@O z`r` raylc}ar ay~a zrlocag`%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 437/846
Nr% Fëi~lr Pqa}anl Qrç`y
2'=>
^`oc` gly~r`enl cly rayqc~`nly nac gqay~ral nac NOL ae gb,C% &Ac`olr`ikùe
zrlzk`+%
Yk~kl )0 Yk~kl )? Yk~kl )9 Yk~kl )>
?: ?1 ?6 90
02 ?1 90 01
06 ?= 9> 0?
0: 0> 02 ?>
2%;% Ya n`e cly ykbqkae~ay n`~ly gly~r`nly ae c` ~`oc` na `o`hl%
F`iar qe` ~`oc` na @EL]@ { y`i`r ~ln`y c`y ileicqykleay naokn`y%&Mi`ci% 7 >%9;#
M~`o% 7 9%6;+
^`oc` gly~r`enl c`y ~agzar`~qr`y nac iqarzl &lM+ ic`ykmki`n`y zlr an`nay%
&Ac`olr`ikùe zrlzk`+
06 ` ?: `ðly ?0 ` ?; `ðly G`{lray pqa 9: `ðly
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
;6%: ;;%= ;6%=
;6%> ;;%1 ;6%=
;2%2 ;;%: ;6%:
;6%1 ;6%6 ;2%2
;2%0 ;2%; ;2%1
2%0:% Ya f`ia qe ay~qnkl ae~ra ac ek}ac na `iknax ae ~ërgkely na zF &M`i~lr @+ { c`
ileiae~r`ikùe na iclrl &m`i~lr O+ ae ac `bq`% @pqç ya `yqga qe `eàckyky na }`rk`ex`
na nly }ç`y ile qe nkyaðl `ca`~lrkl ilgzca~`gae~a `ca`~lrkx`nl% Cly n`~ly ya n`e
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 438/846
Nr% Fëi~lr Pqa}anl Qrç`y
2'=1
ae c` ~`oc` na `o`hl% F`iar qe` ~`oc` na `eàckyky na }`rk`ex` { y`i`r c`y
ileicqykleay naokn`y%
^`oc` gly~r`enl cly n`~ly na ay~a zrlocag` nac zF { c` ileiae~r`ikùe na iclrl%
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
Ek}ac nac zF
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
zF 7 2%: zF 7 2%? zF 7 2%> zF 7 2%=
Ek}a na ileiae~r`ikùe na iclrl
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
O`h` ?? 02 6 =Gank`e` ; 00 2 >
@c~` 6 6 = 1
2%00% Ae qe ay~qnkl na kebaekarç` nac `kra# ae qe aymqarxl zlr zrl~abar ac gankl
`gokae~a &ae~ra gaely ilgoqy~koca ileyqg` qe `q~l# gaely ya na~arklr` ac
`gokae~a+# ya ay~qnk`rle iq`~rl glnacly na `q~ly &@# O# I# N+# z`r` zrlo`r acileyqgl na b`ylcke`% Z`r` i`n` `q~l# av`i~`gae~a# qe b`cùe na b`ylcke` ya zqyl ae
ac ~`epqa { ac `q~l ya g`eahù f`y~` pqa ya ileyqgkù ~ln` c` b`ylcke`% C`y
nky~`eik`y ae gkcc`y n`n`y zlr i`n` ilifa ya n`e ae c` ~`oc` na `o`hl% Ile qe ek}ac
na ykbekmki`eik` na :%:0 zrlo`r pqa ~lnly cly zrlgankly zloc`ikle`cay yle kbq`cay%
I`ciqc`r ac }`clr na c` zrlo`okckn`n z% Yk Qy~an pqkykar` zrl~abar eqay~rly raiqryly
e`~qr`cay l yar `i~k}ky~` ac gankl `gokae~a ¼Yarç`# z`r` Qy~an kbq`c pqa ya
yacaiikle`r` iq`cpqkar` na cly > glnacly< &Mi`ci% 7 ?9%1# Mirç~ki` 7 ?%62# z 7 0%9v0:'6
+
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 439/846
Nr% Fëi~lr Pqa}anl Qrç`y
2'==
^`oc` gly~r`enl cly n`~ly nac zrlocag`% &Ac`olr`ikùe zrlzk`+
Glnacly @ O I N „„„„„„„„„„„„„„„„„„„„„„„„„
Gkcc`y 0> 9 02 0=0= 1 ?: 0606 0? ?? ?:
0> 6 ?> 02
?? 2 ?= ?0; = 06 0=
= ; ?? 02
> 00 ?0 ??2 00 ?: 0;
0= ; 06 0=
„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„
2%0?% Ya ay~qnk` ac na~arklrl i`qy`nl ` > ~kzly na ~ac`y &0# ?# 9# >+ qy`n`y ae ac
apqkzl na ile~rlc na mkc~rly% Cly mkc~rly l o`bflqyay# z`r` ac ile~rlc na z`r~çiqc`y ya
~kaeae pqa y`iqnkr zarkùnki`gae~a# iq`enl f`{ i`çn` na zraykùe naoknl ` c`
loy~rqiikùe na cly lrkmkikly na c`y ~ac`y% Yk el ya f`ia ay~l# iq`enl f`{ gqif`
i`çn` na zraykùe# c` ~ac` ya na~arklr` zrag`~qr`gae~a# ay~l ay# nazaenkaenl nac ~kzl
na z`r~çiqc`y { nagày }`rk`ocay g`eah`n`y%
^`oc` gly~r`enl cly n`~ly na ay~a zrlocag`% &Ac`olr`ikùe zrlzk`+%
^kzl na ~ac` I`çn` na zraykùe &ckor`y zlr zqcb`n` iq`nr`n`+
&0+ 90?; 9::: ?6=1 ?6;:
&?+ 9?:: 99:: ?;21 901:
&9+ ?6:: ?;:: ?;61 9:1:
&>+ ?=:: ?2:: ?=:: ?2=1
F`iar cly ykbqkae~ay iàciqcly8
&`+ Zrlo`r c` fkzù~ayky eqc` na Fl8±0 7 ±? 7 ±9 7 ±># ile~r` c` fkzù~ayky `c~are`~k}`
na F08±0 ≪ ±? ≪ ±9 ≪ ±># ay naikr# na pqa el f`{ nkmaraeik`y ae~ra cly zrlgankly
zloc`ikle`cay na c`y > ~ac`y qy`n`y% Qy`r qe ek}ac ykbekmki`e~a naΰ 7 :%1
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 440/846
Nr% Fëi~lr Pqa}anl Qrç`y
2'=2
&o+ F`iar qe` ~`oc` na `eàckyky na }`rk`ex` pqa keicq{` cly }`clray na c` M
i`ciqc`n`# c` M ~`oqc`n` { ac }`clr na z%
2%09% Ac ckorl na Gle~blgar{ a~ `c% Zrlo`okckn`n { Ay~`nçy~ki` @zcki`n`y ` c`
Kebaekarç` nkyiq~a qe` ke}ay~kb`ikùe z`r` na~argke`r ac ileyqgl na b`ylcke` &ae
gkcc`y zlr ck~rl+ na > ilifay% Z`r` ay~l# ya `brqz`e cly > ~kzly na `q~ly ~r`~`enl na
flglbaekx`r l na ile~rlc c`y }`rk`ocay pqa zqnkar`e `mai~`r ac ileyqgl na
b`ylcke` &oclpqal z`r` ackgke`r c`y }`rk`ocay el naya`ocay+% C`y }`rk`ocay
ile~rlc`n`y yle i`o`cc`hay nac gl~lr# gkygl rln`ha na cc`e~`y# gkygl ~kzl na
i`roqr`nlr# gkygl ~kzl na `iak~a# gkygl ~kzl na g`e~aekgkae~l# gkygl zayl#gkyg`y ~agzar`~qr`y `gokae~`cay# gkygl gkcc`ha# an`n nac gl~lr# ~`g`ðl nac
gl~lr# a~i% Zrlo`r c` fkzù~ayky na pqa el f`{ nkmaraeik`y ae ac gkcc`ha na cly ilifay
zrlo`nly qy`enl ΰ 7 :%:0% I`ciqc`r ac }`clr na z% &Gle~blgar{# 0;;=+%
^`oc` na n`~ly na cly gkcc`hay zlr ck~rl na cly > ilifay zrlo`nly%
Gkcc`ha ^l~`cay zlr Zrlgankly zlr
Ilifa el% ~r`~`gkae~l ~r`~`gkae~l
0 ? 9 > 1 [ k% [ k%
&0+ 0%9 0%= :%1 0%? 0%0 1%2 0%0>
&?+ ?%? ?%> :%> ?%: 0%6 6%6 0%2=
&9+ 0%6 0%2 :%= 0%1 0%9 =%; 0%96
&>+ 9%; >%> ?%: >%0 9%> 02%6 9%1=
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
&Mqae~a8 Gle~blgar{ a~ `c% 0;;=+
2%0>% Ac ckorl na Gle~blgar{ a~ `c% Zrlo`okckn`n { Ay~`nçy~ki` @zcki`n`y ` c`
Kebaekarç` &0;;=+ na c` zàbke` =2?# ik~` qe `r~çiqcl zqocki`nl ae ac @garki`e
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 441/846
Nr% Fëi~lr Pqa}anl Qrç`y
2'=6
Kenqy~rk`c fkbkaea @yylik`~kle Hlqre`c &}lc% 92# 0;2=# z`by% >06'>??+# c` iq`c
nayirkoa qe` zrqao` na i`gzl z`r` na~ai~`r c` zrayaeik` na `ryëekil ae gqay~r`y
na lrke`% C` zrqao` f` yknl zrlzqay~` z`r` yq qyl ae~ra ~r`o`h`nlray mlray~`cay
naoknl `c qyl i`n` }ax g`{lr na `ryëekily lrbàekily ae nkif` kenqy~rk`% Ac
avzarkgae~l ilgz`r` cly rayqc~`nly lo~aeknly ile c` zrqao` `c yar amai~q`n` zlr
qe keavzar~l { qe ae~rae`nlr avzarkgae~`nl ile ac `eàckyky amai~q`nl ae qe
c`olr`~lrkl ragl~l% Z`r` c` zrqao` ya ayilbae iq`~rl yqha~ly# cly iq`cay yle
ileyknar`nly ilgl oclpqay% C` }`rk`oca na rayzqay~` ay ac ile~aeknl na `ryëekil
&ae zzg+ ae c` lrke` nac yqha~l% Cly n`~ly yle cly ykbqkae~ay8^`oc` gly~r`enl cly n`~ly nac zrlocag`%
Yqha~l
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
Zrqao` 0 ? 9 >
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
Keavzar~l :%:1 :%:1 :%:> :%01
Avzar~l :%:1 :%:1 :%:> :%02
C`olr`~lrkl :%:> :%:> :%:9 :%0: \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
Mqae~a8 Gle~blgar{ a~ `c% Zrlo`okckn`n { Ay~`nçy~ki` @zcki`n`y ` c` Kebaekarç`
&0;;=+
&`+ ¼Avky~a nkmaraeik` `cbqe` ae ac zrliankgkae~l na zrqao` na `ryëekil<
&o+ @e`ckx`r cly rayknqly na ay~a avzarkgae~l
2%01% Iq`~rl ek}acay na mar~kckx`e~ay mqarle qy`nly ae qe avzarkgae~l `brçilc` ile
nly ek}acay na `bq`# ay naikr mrqb`c { `oqen`e~a% Cly lifl ~r`~`gkae~ly mqarle
`ykbe`nly `ca`~lrk`gae~a ` lifl z`riac`y% C` rayzqay~` ay ae ~leac`n`y zlr
fai~àra`% C` ~`oc` na `o`hl n` c` kemlrg`ikùe rapqarkn`%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 442/846
Nr% Fëi~lr Pqa}anl Qrç`y
2'=;
^`oc` gly~r`enl cly n`~ly na ay~a zrlocag`%
Ek}ac na mar~kckx`e~a
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
Ek}ac na `bq` E`n` O`hl Gank`el @c~l
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
Zli` `bq` 9%: 9%9 9%2 9%0
Gqif` `bq` ?%9 >%: >%9 1%:
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
&`+ Qy`r ac glnacl gày `zrlzk`nl na @EL]@%&o+ F`iar qe` ~`oc` na `eàckyky na }`rk`ex`
&i+ Naikr yk f`{ amai~ly ykbekmki`~k}ly ae cly mar~kckx`e~ay { cly ek}acay na `bq`
&n+ Naikr yk f`{ qe` ke~ar`iikùe ykbekmki`e~a
2%0=% Ya f`ia qe ay~qnkl fkzl~ë~kil na nkmqykùe `~glymërki` yk~q`enl cly yaeylray
z`r` gankr c` i`ckn`n nac `kra ile rayzai~l ` YL? ` ~ray nkmarae~ay nky~`eik`y { ` ~ray
nkmarae~ay `c~qr`y% Cly n`~ly ya n`e `o`hl% @yqgkaenl ΰ 7 :%:1 f`iar cl ykbqkae~a8
&`+ Qy`r ac glnacl na @EL]@ gày `zrlzk`nl z`r` ay~a zrlocag`
&o+ @e`ckx`r c` bràmki` na cly n`~ly z`r` ay~qnk`r c` ke~ar`iikùe zlykoca pqa zqnkar`
liqrrkr ae~ra c`y nky~`eik`y { c`y `c~qr`y%
^`oc` gly~r`enl c`y ileiae~r`ikleay &zzg+ na YL? z`r` ay~a zrlocag`%
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
Nky~`eik`y ae ga~rly
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
@c~qr`y 0::: 01:: ?::: \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
@ ek}ac nac g`r 91: ?1: 0::
9:: ga~rly ?6: ?0: ;:
1:: ga~rly ?1: 0;: 2:
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 443/846
Nr% Fëi~lr Pqa}anl Qrç`y
2'2:
2%02% Ae qe ay~qnkl na `eàckyky na }`rk`ex` na ~ray }ç`y ya n`e cly ykbqkae~ay n`~ly8
YY` 7 ??%=9# YY o 7 %::9# YYi 7 %>:# YY`o 7 %>:# YY`i 7 %:2# YY oi 7 %:%:=9# YYa 7%::0 { YY~ 7 %;:% Z`r` ac m`i~lr @ ya qy`rle iq`~rl ek}acay# z`r` ac m`i~lr O ya
qy`rle nly ek}acay { z`r` ac m`i~lr I ya qy`rle ? ek}acay% @yqgkr nly gqay~r`y ile
i`n` ek}ac% Qy`enl ΰ 7 :%:1# f`iar cl ykbqkae~a8
&`+ Iley~rqkr qe` ~`oc` na @EL]@
&o+ Knae~kmki`r c`y ke~ar`iikleay ykbekmki`~k}`y a ke~arzra~`rc`y `ilrnagae~a
2%06% Ay~a ay qe ay~qnkl nac ~av~l na @zzckan Y~`~ky~kiy8 @eàckyky lm ]`rk`eia `en
_abrayykle na Nqee { Ic`rd% Ay~` ke}ay~kb`ikùe ay~à rac`ikle`n` ile qe ay~qnkl na
c` ke~ackbaeik` na cly ekðly ile yçe~lg`y i`rnk`ily na ~kzly `ik`eù~kily {
ik`eù~kily% Z`r` ay~l# cly i`gokly ae ac ilamkikae~a na ke~ackbaeik` ya gknkarle# ay
naikr# nayzqëy na lzar`rya { `e~ay na lzar`rya% Cly rayqc~`nly ya n`e ilgl ykbqa8
^`oc` gly~r`enl cly rayqc~`nly na cly i`gokly ae ac ilamkikae~a na ke~ackbaeik`%
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
Lzar`ikùe @ik`eù~kily Ik`eù~kily \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
El ; ?
'0 0
'0: '>
9 '1
'? :
Yk '2 1
'2 0:
'0? ;'09 ?
'0? 01
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\&
`+ Qy`r ac glnacl na @EL]@ gày `zrlzk`nl z`r` ay~a ay~qnkl
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 444/846
Nr% Fëi~lr Pqa}anl Qrç`y
2'20
&o+ Iley~rqkr qe` ~`oc` na @EL]@ { `e`ckx`r cly rayqc~`nly
&i+ Y`i`r ~ln`y c`y ileicqykleay `c rayzai~l
2%0;% Ae qe avzarkgae~l `brçilc` ya ay~qnkù ac raenkgkae~l na ~rkbl qy`enl ~ray
ek}acay nkmarae~ay na mar~kckx`e~ay mlym`~`nly# ay naikr# o`hl# gank`el { `c~l% Ilgl
yabqenl m`i~lr ya qy`rle ~ray }`rkan`nay nkmarae~ay na yagkcc`y na ~rkbl &0# ?# 9+
f`ikaenl# ile ay~l# qe ~l~`c na ; ilgoke`ikleay na ~r`~`gkae~ly% Na ay~` g`ear`
i`n` ilgoke`ikùe na ~r`~`gkae~l ya `ykbeù `ca`~lrk`gae~a ` qe` na c`y ?2 z`riac`y
&na av~aeykleay na nly fai~àra`y+# na ~`c g`ear` pqa ~ray z`riac`y raikokarle i`n`
~r`~`gkae~l% Cly raenkgkae~ly na ~rkbl# ae ~leac`n`y gë~rki`y ya n`e `o`hl%^`oc` gly~r`enl ac raenkgkae~l na c` ilyaif` na ~rkbl ae ~leac`n`y gë~rki`y%
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
Ek}ac nac mar~kckx`e~a
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
]`rkan`n na c` yagkcc` O`hl Gank`el @c~l
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
0 2 0: 0?
0: 0: 0>
; 0? 0?
? 6 0? 02
0: 0> 0=
6 09 02
9 ; 0> 0=
0: 0> 06
0? 0= ?0
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
@zcki`r c` mqeikùe na @EL]@ gày `zrlzk`n` z`r` ay~a avzarkgae~l { y`i`r c`y
ileicqykleay naokn`y%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 445/846
Nr% Fëi~lr Pqa}anl Qrç`y
2'2?
2%?:% Ay~a ay~qnkl ay~à aei`gke`nl z`r` pqa ac cai~lr `npqkar` nay~rax` ae ac
iqgzckgkae~l nac ccae`nl na ~`oc`y na `eàckyky na }`rk`ex`% Z`r` ay~l ilgzca~`r c`
ykbqkae~a ~`oc` na @EL]@ { naikr pqa nkyaðl ya qyù%
^`oc` gly~r`enl cly n`~ly nac zrlocag`%
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
Mqae~a na YY b%c% GY Mi`ci% M~`o% ]`clr z
]`rk`ikùe
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
Naoknl ` cly ?::: 0:
~r`~`gkae~ly
Naoknl ` c`y 0?::
ilcqge`yNaoknl ` cly 2>:: 1
raebcleay
_ayknq`c ?1
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
^l~`c 0?::: >:
2%?0% Ilgzca~`r c` ykbqkae~a ~`oc` na @EL]@ { naikr pqë nkyaðl ya qyù%
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
Mqae~a na YY b%c% GY Mi`ci% M~`o% ]`clr z]`rk`ikùe
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
Naoknl ` @ 0?%: ?
Naoknl ` O 0;%1
Ke~ar`iikùe @O 6%1 2
^r`~`gkae~ly 9;%2 0?
_ayknq`c 00
^l~`c 6>%2 91
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
2%??% Ya ra`ckx` qe nkyaðl na nly m`i~lray ae qe nkyaðl ilgzca~`gae~a
`ca`~lrkx`nl# ae ac iq`c ya `zcki`e iq`~rl ek}acay nac m`i~lr @ { ~ray ek}acay nac
m`i~lr O% Cly n`~ly n`nly yle YY 7 01%::# YY o 7 >0%::# YY`o 7 ?9%:1# YY~ 7 ;?%6 {
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 446/846
Nr% Fëi~lr Pqa}anl Qrç`y
2'29
e 7 ?% Ile ay~` kemlrg`ikùe f`iar qe` ~`oc` na `eàckyky na }`rk`ex` { y`i`r c`y
ileicqykleay naokn`y%
2%?9% Ya f`ia qe ay~qnkl fkzl~ë~kil rac`ikle`nl ile c` gankikùe na ileiae~r`ikleay
na z`r~çiqc`y `~glymërki`y agk~kn`y zlr qe` mqae~a kenqy~rk`c# ay~l ay# qy`enl qe
glnacl na nkmqykùe `~glymërki`% Z`r` ~`cay mkeay ya yacaiikle`rle nly ~kzly
nkmarae~ay na gqay~ra`nlray# iq`~rl nkmarae~ay `c~qr`y { iq`~rl nkmarae~ay nky~`eik`y
}kae~l `o`hl na c` mqae~a agkylr`% Zlr ahagzcl# cc`gagly c`y iq`~rl nky~`eik`y ac
m`i~lr @ &ay naikr K 7 > nky~`eik`y na 1::# 0:::# 01:: { ?::: ga~rly+% C`y
ileiae~r`ikleay ya gknkarle ile nly ~kzly g`ri`y nkmarae~ay na yaeylray# iq{lm`i~lr cl cc`g`ragly O &ay naikr# H 7 ?+% @nagày# ya yacaiikle`rle iq`~rl `c~qr`y
nkmarae~ay iq{l m`i~lr cl naelgke`ragly I &D 7 > `c~qr`y na 0::# ?::# 9:: { 1::
ga~rly+% Z`r` ~lnl ay~l# ya fkikarle C7 9 loyar}`ikleay z`r` i`n` qe` na c`y 9?
ilgoke`ikleay na ek}acay na cly ~ray m`i~lray &> v ? v >+ { z`r` qe ~l~`c na ;=
loyar}`ikleay% C` ~`oc` na `o`hl gqay~r` cly rayqc~`nly na c`y gankikleay% @yqgkr
ΰ 7 :%:1% Z`r` ay~l# f`iar cly ykbqkae~ay iàciqcly8
&`+ Ay~`ocaiar ac glnacl `zrlzk`nl ile c`y yqzlykikleay
&o+ F`iar zrqao`y na ykbekmki`eik` ylora cly m`i~lray zrkeikz`cay# }%b%# nky~`eik`#
g`ri`y na yaeylray { zlykikùe na cly yaeylray
&i+ F`iar zrqao`y na ykbekmki`eik` ylora ~ln`y c`y ke~ar`iikleay
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 447/846
Nr% Fëi~lr Pqa}anl Qrç`y
2'2>
^`oc` gly~r`enl c`y ileiae~r`ikleay na z`r~çiqc`y `~glymërki`y &ae zzg+ agk~kn`y
zlr c` ifkgaea` kenqy~rk`c# ae mqeikùe na c` nky~`eik`# `c~qr` { g`ri`y na yaeylray%
&Ac`olr`ikùe zrlzk`+
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
Gqay~ra`nlr g`ri` O0 Gqay~ra`nlr g`ri` O?
@c~qr`y 0:: g ?:: g 9:: g 1:: g 0:: g ?:: g 9::g 1:: g
\\\\\\\\\\\\\\\\\\\\\\\\\\ \\\\\\\\\\\\\\\\\\\\\\\\\\
Nky~`eik`y
1:: g >1: 9:: ?;1 ?;: >=1 9:0 ?;2 ?66
>1; 9:2 ?;: ?2; >2: 90: ?;0 ?6:
>=: 90: ?61 ?=: >2: 9:: ?61 ?2:
0::: g 91: ?6: ?26 ?:: 9>1 ?21 ?11 ?1:
9>= ?1= ?2: 06= 99> ?=1 ?1: ?0:
99; ?1= ?=6 01; 9:: ?1; ?12 ?0:
01:: g 9:: ?2: ?=? 0;6 90: ?1: ?9: ?::
?6; ?=9 ?1= 0=: 9:: ?>9 ??1 0;1
?;; ?=: ?=1 02; 9:1 ?=: ?>1 06:
?::: g 0=: 0=2 01: 0>0 011 0>1 096 09;
0=: 0>1 0>: 09> 01: 092 09> 0?;
0>6 09; 01? 0?> 0>2 01? 09: 0?1
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
2%?>% Ac ~av~l @zzckan Ckea r Y~`~ky~ki`c Glnacy na cly `q~lray Dq~ear# E`if~yfakg#
Ga~ar { Ck avzcki` qe zrlocag` rac`ikle`nl ile qe m`orki`e~a na `q~lgù}kcay#
pqkae naya` ay~qnk`r cly amai~ly ae~ra nkmarae~ay ilenqi~lray na `q~ly &m`i~lr @# k 7
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 448/846
Nr% Fëi~lr Pqa}anl Qrç`y
2'21
>+ { c`y nkmaraeik`y ae~ra `q~ly &m`i~lr O# h 7 1+ rac`ikle`nl ile ac ileyqgl na
b`ylcke`% Z`r` ay~l# ya yacaiikle`rle iq`~rl ilenqi~lray `ca`~lrk`gae~a% Na c`
gkyg` g`ear` ya yacaiikle`rle `ca`~lrk`gae~a ikeil `q~ly nac gkygl glnacl ile
~r`eygkykùe g`eq`c% I`n` ilenqi~lr g`eahù i`n` `q~l nly }aiay ae qe` zrqao` na
>: gkcc`y { c`y gkcc`y zlr b`cùe n`n`y ya rabky~r`rle% @yqgkaenl qe glnacl na
@EL]@ `ca`~lrkl { ~`g`ðly na gqay~r`y kbq`cay# zrliay`r cly n`~ly na c` ~`oc` na
`o`hl%
^`oc` gly~r`enl c` nayirkzikùe na cly nly m`i~lray qy`nly ae ac ay~qnkl%
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
M`i~lr O &`q~ly+ \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
M`i~lr @ &iflmaray+ h 7 0 h 7 ? h 7 9 h 7 > h 7 1
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
k 7 0 ?1%9 ?6%; ?>%6 ?6%> ?2%0
?1%? 9:%: ?1%0 ?2%; ?=%=
k 7 ? 99%= 9=%2 90%2 91%= 99%2
9?%; 9=%1 90%; 91%: 99%;
k 7 9 ?2%2 9:%2 ?=%; ?;%2 ?;%?
?6%1 9:%> ?=%9 9:%? ?6%;
k 7 > ?;%? 9?%> ?2%2 90%6 9:%9
?;%9 9?%> ?6%; 9:%2 ?;%;
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
Mqae~a8 Dq~ear a~ `c% ?::?
&`+ Iley~rqkr qe` ~`oc` na @EL]@
&o+ _a}ky`r cly amai~ly na cly m`i~lray @ { O%
&i+ @e`ckx`r c` ke~ar`iikùe na cly m`i~lray @ { O
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 449/846
Nr% Fëi~lr Pqa}anl Qrç`y
2'2=
2%?1% Ay~a ay qe zrlocag` `n`z~`nl nac ckorl Zrlo`okckn`n { Ay~`nçy~ki` z`r`
Kebaekarly na cly `q~lray R`czlca a~ `c% &0;;6+ ac iq`c n` qe ahagzcl pqa keicq{a
~ray m`i~lray naelgke`nly m`i~lr @# m`i~lr O { m`i~lr I# ile ~lnly cly amai~ly mkhly%
@yqgkr ΰ 7 :%:1% C` ~`oc` na `o`hl gqay~r` c` kemlrg`ikùe rapqarkn` z`r` ay~a
zrlocag`%
^`oc` gly~r`enl cly n`~ly zar~keae~ay z`r` ay~a zrlocag`%
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
I0 I? I9
\\\\\\\\\\\\\\
O0 O? O9 O0 O? O9 O0 O? O9
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ @0 01%: 0>%6 01%; 0=%6 0>%? 09%? 01%6 01%1 0;%?
06%1 09%= 0>%6 01%> 0?%; 00%= 0>%9 09%2 09%1
??%0 0?%? 09%= 0>%9 09%: 0:%0 09%: 0?%= 00%0
@? 00%9 02%? 0=%0 06%; 01%> 0?%> 0?%2 02%9 2%6
0>%= 01%1 0>%2 02%9 02%: 09%= 0>%? 01%6 00%1
06%? 0>%? 09%> 0=%0 06%= 01%? 01%; 0>%= 0?%?
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
Mqae~a8 R`czlca a~ `c% &0;;6+&`+ Baear`r c` g`~rkx na n`~ly# ke~rlnqikrcly `c zrlbr`g` Gkek~`o { iley~rqkr qe`
~`oc` na `eàckyky na }`rk`ex` { f`iar zrqao`y na ykbekmki`eik` ylora cly amai~ly
zrkeikz`cay &M`i~lray @# O { I+
&o+ F`iar zrqao`y na ykbekmki`eik` ylora ~ln`y c`y ke~ar`iikleay
&i+ Avzcki`r zlrpqa qe` ke~ar`iikùe ykbekmki`~k}` aeiqora ac amai~l nac m`i~lr I
_ayzqay~`y8
&`+ @8 M 7 :%1>4 el ykbekmki`~k}`% O8 M 7 =%614 ykbekmki`~k}`% I8 M 7 ?%014 el
ykbekmki`~k}`%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 450/846
Nr% Fëi~lr Pqa}anl Qrç`y
2'22
&o+ @O8 M 7 9%694 ykbekmki`~k}`% @I8 M 7 9%2;4 ykbekmki`~k}`% OI8 M 0%904 el
ykbekmki`~k}`% @OI8 M 7 0%=94 el ykbekmki`e~a%
2%?=% Ae cly `iiknae~ay kenqy~rk`cay# i`n` `ðl ya zkarnae gqifly gkcay na gkccleay
na nùc`ray naoknl ` `iiknae~ay na ~r`o`hl% Zlr cl ~`e~l# c` kenqy~rk` ke}kar~a gqifl
nkearl ae zrlbr`g`y rac`ikle`nly ile fkbkaea kenqy~rk`c { yabqrkn`n% Qe`
ilgz`ðç` nay`rrlccù nly eqa}ly zrlbr`g`y na fkbkaea kenqy~rk`c { yabqrkn`n# z`r`
ae~rae`r ` yqy ~r`o`h`nlray ae yky~ag`y na yabqrkn`n% Z`r` na~argke`r c` amkikaeik`
na ay~ly zrlbr`g`y# ya aey`goc`rle ~ray brqzly na lorarly% I`n` brqzl na lorarly
~lgù qe zrlbr`g` na ae~rae`gkae~l nkmarae~a% Nayzqëy# cly ~r`o`h`nlray mqarle`ykbe`nly ` ~`ra`y knëe~ki`y% Ac eøgarl flr`y'flgora gaeyq`cay zarnkn`y# ilgl
rayqc~`nl na `iiknae~ay c`olr`cay# ya rabky~rù z`r` cly ykbqkae~ay 0? gayay% Ay~`
n`~` ya gqay~r` ae c` ~`oc` na `o`hl%
^`oc` gly~r`enl c`y Flr`y'flgora zarnkn`y gaeyq`cgae~a% &Ac`olr`ikùe zrlzk`+
Zrlbr`g`y na ae~rae`gkae~l na fkbkaea kenqy~rk`c { yabqrkn`n
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
Gayay 0 ? 9Aearl ?6 ?6 9:Maorarl 91 ?= ?6G`rxl 9? ?2 ?=@orkc 0? 0: =G`{l 0= 0: 0?Hqekl ; 0? 0:Hqckl 01 ; 00@bly~l 06 01 0=Yaz~kagora ?? 09 00
Li~qora ?2 ?: 06 El}kagora ?2 02 0;Nkikagora 90 ?9 ?0
Y`i`r ileicqykleay `iari` na c` amai~k}kn`n na ay~ly ~ray zrlbr`g`y
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 451/846
Nr% Fëi~lr Pqa}anl Qrç`y
2'26
2%?2% Ay~a ay qe zrlocag` pqa ke}lcqir` qe avzarkgae~l na @EL]@ na nly
yae~knly# ae ac iq`c ya qy`rle yaky ek}acay z`r` ac m`i~lr @ { ikeil ek}acay z`r` ac
m`i~lr O% Z`r` ay~l ya n` qe` `}`ex`n` na cly n`~ly8 YY` 7 2;# YY o 7 == { YY~ 7 06>%
Iley~rqkr qe` ~`oc` na @EL]@ { y`i`r ~ln`y c`y ileicqykleay zar~keae~ay%
2%?6% Ya n` c` ~`oc` na @EL]@ na `o`hl ccae`r ~lnly cly m`c~`e~ay { naikr yk f`{
ke~ar`iikleay {# yk ëy~`y yle kgzlr~`e~ay% ¼@ pqë ek}acay ay~àe i`n` qel na cly
m`i~lray<
^`oc` na `eàckyky na }`rk`ex` keilgzca~`% Ccae`r cly m`c~`e~ay% &Ac`olr`ikùe zrlzk`+
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
Mqae~a na Yqg` na cly Br`nly na Zrlgankl na cly Mi`ci% M~`o% ]`clr na z}`rk`ikùe iq`nr`nly ckoar~`n iq`nr`nly
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ M`i~lr @ >=%: 0
M`i~lr O 00%: 1%1
M`i~lr I 9%:@O 6%: 0
@I :%0 0
OI 0%= 0@OI 1%: :%69
_ayknq`c =
^l~`c ;>%: 01 \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
2%?;% Ay~a ay qe zrlocag` `n`z~`nl nac ~av~l Y~`~ky~kiy mlr Ae}krlegae~`c
Aebkeaary na Oar~flqv { Orle &0;;>+% Ay~a aharikikl ay qe `eàckyky na }`rk`ex` na
> ic`ykmki`ikleay# { ay~à rac`ikle`nl ile c` lzar`ikùe na qe keikear`nlr gqekikz`c#
ae ac iq`c ya q~kckx`rle nly ~kzly nkmarae~ay na gqay~ra`nlray pqa ~lg`rle gqay~r`y
ykgqc~àea`y nqr`e~a iq`~rl zarklnly na 9%1 flr`y ylora qe zarklnl na ~ray nç`y%
I`n` gqay~ra`nlr mqa `e`ckx`nl z`r` ikeil brqzly na yqy~`eik`y ~ùvki`y zackbrly`y#
ay naikr# zay~kikn`y &fknrli`roqrly fa~ariçickily cly iq`cay liqrrae ilgl kgzqrax`y
~ùvki`y zaryky~ae~ay ae farokikn`y+ { na cçpqknly kemc`g`ocay &I>F>L+ qy`nly ae
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 452/846
Nr% Fëi~lr Pqa}anl Qrç`y
2'2;
yçe~ayky lrbàeki`y% C`y ayzaikay na i`n` brqzl mqarle iclrke`n`y ` br`nly
nkmarae~ay# ay naikr# ile ># 1# =# 2 { 6 à~lgly na iclrl zlr glcëiqc`% @yqgkr ΰ 7
:%:1% Ae ay~a `eàckyky na }`rk`ex` ya qy`rle iq`~rl m`i~lray# ilgl ykbqa8
0% Nly ~kzly na gqay~ra`nlray &Y+
?% Iq`~rl zarklnly na gqay~ral &Z+
9% Nly brqzly na yqy~`eik`y ~ùvki`y &NM+# ilgl c`y nayirk~`y `rrko`
>% Ikeil ek}acay na iclrke`ikùe nae~rl na i`n` qel na cly nly brqzly &Ic+%
Ay~l n` qe ~l~`c na e 7 ?v>v?v1 7 6: gankikleay%
C` ~`oc` na `o`hl gqay~r` c` kemlrg`ikùe rapqarkn`%^`oc` gly~r`enl c`y agkykleay na c`y 0: yqy~`eik`y ~ùvki`y# aeqgar`n`y `o`hl# ae
qekn`nay na eb,g9
na b`y yail elrg`c ae zlriae~`hay ra`cay na IL?%
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ Zarklnly na gqay~ral 0 ? 9 >
\\\\\\\\\\\ \\\\\\\\\\\\ \\\\\\\\\\\ \\\\\\\\\\\\
Gqay~ra`nlr @ O @ O @ O @ O \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
Zay~kikn`y
Yqg ^INN :%> 0%; :%1 0%2 :%9 :%2 0%: ?%:Yqg ZaINN 0%6 ?6 9%: 2%9 ?%2 1%1 2%: 00
Yqg FvINN ?%1 ?> ?%= 2%9 9%6 1%0 >%2 =%:
Yqg FzINN 02 011 0= =? ?; >1 9: >:LINN 2%> 11 2%9 ?6 0> ?0 0? 02
CçpqknlyKemc`g`ocay
Yqg ^INM >%; ?= 2%6 06 1%6 ; 09 09
Yqg ZaINM >%? 90 00 ?? 2%: 0? 02 ?>
Yqg FvINM 9%1 90 00 ?6 6%: 0> 06 0;
Yqg FzINM ;%0 0:9 9? 6: 9? >0 >2 =?LINM 9%6 0; =%> 06 =%= 2%: =%2 =%2
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
Mqae~a8 Oar~flqav# Z% G`i { C% I% Orle% Y~`~ky~kiy mlr Ae}krlegae~`c AebkeaaryCaky Zqockyfary% I_I Zrayy# Kei% &0;;>+%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 453/846
Nr% Fëi~lr Pqa}anl Qrç`y
2'6:
&`+ Baear`r qe` g`~rkx ile cly n`~ly na c` ~`oc` na `rrko` a ke~rlnqikrcly ae ac
zrlbr`g` Gkek~`o%
&o+ F`iar qe` ~`oc` na `eàckyky na }`rk`ex`
&i+ F`iar zrqao`y na ykbekmki`eik` ylora cly iq`~rl m`i~lray zrkeikz`cay# }%b%#
zarklnly na ~kagzl# gqay~ra`nlray# brqzly na yqy~`eik`y ~ùvki`y { ek}acay na
iclrke`ikle
&n+ F`iar zrqao`y na ykbekmki`eik` ylora ~ln`y c`y ke~ar`iikleay
&a+ Y`i`r ~ln`y c`y ileicqykleay zar~keae~ay
2%9: Ya f`ia qe ay~qnkl nac ile~rlc na c` ile~`gke`ikùe nac `kra# ay naikr# qy`enl
yky~ag`y na ile~rlc na z`r~çiqc`y z`r` flrely na iagae~l% Z`r` ay~l ya qy`e
nkmarae~ay ~kzly na zraikzk~`nlray acai~rly~à~kily &m`i~lr @+# ay naikr# zraikzk~`nlray
na zc`i` na `c`gora# zraikzk~`nlray na zc`i` zc`e` { zraikzk~`nlray ~qoqc`ray%
@nagày ya qy`rle aemrk`nlray na `yzarykùe { ilcai~lray gaiàekily &m`i~lr O+% C`
mke`ckn`n na ay~a avzarkgae~l m`i~lrk`c mqa z`r` }ar c` amkikaeik` na ilcaiikùe na c`y
z`r~çiqc`y qy`enl cly `e~arklray m`i~lray% Iley~rqkr qe` ~`oc` na `eàckyky na
}`rk`ex` ~lg`enl ae ileyknar`ikùe c` ykbqkae~a kemlrg`ikùe8 Iq`nr`nl gankl nac
zrkgar m`i~lr mqa kbq`c ` ?%9:4 ac iq`nr`nl gankl nac yabqenl m`i~lr gankl mqa
kbq`c ` 1%::4 iq`nr`nl gankl na c` ke~ar`iikùe mqa na :%0?4 iq`nr`nl gankl nac
arrlr mqa na :%:21% @yqgkr ΰ 7 :%:1% Ilgzca~`r c` ~`oc` na @EL]@ na `o`hl
i`ciqc`enl cly ykbqkae~ay }`clray%
&`+ Cly }`clray na Mi`ci% z`r` cly amai~ly zrkeikz`cay { z`r` ac amai~l na ke~ar`iikùe&o+ Cly }`clray na M~`o% z`r` cly nly m`i~lray zrkeikz`cay { z`r` c` ke~ar`iikùe
&i+ Cly }`clray na z z`r` i`n` qel na cly m`i~lray zrkeikz`cay { z`r` c` ke~ar`iikùe
&n+ Naikr yk cly amai~ly zrkeikz`cay `mai~`e c` amkikaeik` na cly m`i~lray @ { O
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 454/846
Nr% Fëi~lr Pqa}anl Qrç`y
2'60
&a+ Naikr yk f`{ ke~ar`iikùe ae~ra cly m`i~lray o`hl ileyknar`ikùe {# yk c` f`{#
avzcki`r zlrpqa liqrrkù `yç%
^`oc` na `eàckyky na }`rk`ex` z`r` ac avzarkgae~l na cly zraikzk~`nlray
acai~rly~à~kily%
Mqae~a na b% c% Yqg` na Iq`nr`nl Mi`ci% M~`o% ]`clr
}`rk`ikùe iq`nr`nly gankl na z
Zrkgar m`i~lr ?%9:
Yabqenl m`i~lr 1%::
Ke~ar`iikùe :%0?
Arrlr 0? :%:21
^l~`c
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 455/846
Nr% Fëi~lr Pqa}anl Qrç`y
6'0
I@ZK^QCL 6
_abraykùe ckea`c ykgzca { gøc~kzca
Yqzlykikleay nac glnacl na rabraykùe ckea`c%' Aiq`ikleay elrg`cay z`r`
i`ciqc`r ac ke~ariaz~l ae c` lrnae`n` ` { c` zaenkae~a o na c` iqr}` l cçea` na
rabraykùe%' Ilamkikae~a na na~argke`ikùe gøc~kzca _ ?
na c` gqay~r` pqa ay~kg`
` π?
ac ilamkikae~a na na~argke`ikùe zloc`ikle`c%' Ilamkikae~a na ilrrac`ikùe _
na c` gqay~r` pqa ay~kg` ` π# ac ilamkikae~a na ilrrac`ikùe zloc`ikle`c%'
Ke~ar}`cl na ilemk`ex` z`r` ac ilamkikae~a zloc`ikle`c ο ilgzleae~a na c`
cçea` na rabraykùe ±[sV 7 ΰ $ ο V # ay~kg`nl zlr o# c` zaenkae~a na c` cçea`%'
Ke~ar}`cl na ilemk`ex` z`r` ac z`ràga~rl zloc`ikle`c ΰ# ac ke~ariaz~l na c`
lrnae`n` na c` cçea` na rabraykùe ±[sV 7 ΰ $ ο V # iq{l ay~kg`nlr ay `%' Fkzù~ayky
eqc` na Fl8ο 7 οl ile~r` c`y fkzù~ayky `c~are`~k}`y na F08ο 3 0 { F?8ο 5 0%'
Fkzù~ayky eqc` na Fl8ΰ 7 ΰl ile~r` c`y fkzù~ayky `c~are`~k}`y na F08ΰ ≪ ΰl# F?8ΰ
5 ΰl# { na F98 ΰ 3 ΰl%' Ke~ar}`cl na ilemk`ex` z`r` ±[sV na c` cçea` na rabraykùe
zloc`ikle`c ay~kg`n` zlr [%' _abraykùe { ilrrac`ikùe gøc~kzca%' Gë~lnly z`r`
}`ckn`r ac glnacl na rabraykùe ckea`c ykgzca { gøc~kzca8 ` ~r`}ëy na ay~`nçy~ki` na
kemaraeik`y { ` ~r`}ëy nac `eàckyky bràmkil na cly rayknq`cay ay~`en`rkx`nly%
Zrliankgkae~l na rabraykùe gøc~kzca qy`enl ac zrlbr`g` Gkek~`o%'
Ac loha~k}l na ay~qnk`r rabraykùe ckea`c ykgzca ay z`r` lo~aear ac glnacl na
rabraykùe gày `zrlzk`nl# ay naikr# qe` aiq`ikùe na rabraykùe ckea`c ykgzca l
gøc~kzca z`r` mkeay na zrankiikùe { ay~kg`ikùe% Cly ilgzleae~ay na ay~` aiq`ikùe
na rabraykùe ckea`c# ile ylcl qe` }`rk`oca kenazaenkae~a# ~`gokëe cc`g`nl glnacl
ckea`c na zrkgar lrnae# yle c` }`rk`oca nazaenkae~a [¾ l mqeikùe na rayzqay~` {# c`
}`rk`oca kenazaenkae~a V % Ac glnacl na ay~` aiq`ikùe# pqa nayirkoa c` rac`ikùe na
c` }`rk`oca V ile c` }`rk`oca [ # ya cc`g` c` aiq`ikùe na rabraykùe na [ ylora V {# c`
bràmki` na ay~` mqeikùe# ya cc`g` c` iqr}` na rabraykùe%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 456/846
Nr% Fëi~lr Pqa}anl Qrç`y
6'?
Ac glnacl na rabraykùe ckea`c zloc`ikle`c pqa nayirkoa c` rac`ikùe ae~ra c`
rayzqay~` l }`rk`oca nazaenkae~a [ {# c` }`rk`oca kenazaenkae~a l rabraylr` V ay8
[ 7 οl $ ο0v0 $ α k 7 0# ?# …%# e &6'0+
Nlena8
[ 7 }`rk`oca nazaenkae~a zloc`ikle`c &~`gokëe ya qy` c` `el~`ikùe {+
οl 7 ke~ariaz~l ae c` lrnae`n`
ο0 7 zaenkae~a na c` cçea`
v0 7 }`rk`oca kenazaenkae~a
α 7 arrlr `ca`~lrkl ile zrlgankl na : { }`rk`ex` ω?
iley~`e~a% Ay~a }`clr na α ay c`nkmaraeik` ae~ra ac }`clr ~aùrkil na [ k { ac }`clr na [ i`ciqc`nl q loyar}`nl% C`y
ilenkikleay na α yle na pqa ay~a z`ràga~rl naoa ay~`r elrg`cgae~a nky~rkoqknl4 yqy
}`clray naoae na yar kenazaenkae~ay qel nac l~rl { c` }`rk`ex` na α ay ]`r&α+ 7 ω?α
e 7 eøgarl na & v# {+ z`ray na loyar}`ikleay
C` aiq`ikùe na c` cçea` na rabraykùe gqay~r`c pqa ay~kg` ` glnacl na rabraykùe
zloc`ikle`c &6'0+ na `rrko` ya n` ilgl8
[ 7 ` $ ov $ a &6'?+
Nlena8
[ 7 }`clr na c` }`rk`oca nazaenkae~a na c` gqay~r`
` 7 ke~ariaz~l ae c` lrnae`n`
o 7 zaenkae~a na c` cçea`
a 7 arrlr l rayknq`c na c` gqay~r` nael~`nl zlr ak 7 {k ' [ k% Ay~` ay~`nçy~ki` ay c`
ay~kg`nlr` nac z`ràga~rl α
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 457/846
Nr% Fëi~lr Pqa}anl Qrç`y
6'9
Yqzlykikleay nac glnacl na rabraykùe ckea`c
0% Cly }`clray na [ yle kenazaenkae~ay qel nac l~rl# ay naikr# el naoae na ay~`r
ilrrac`ikle`nly%?% C`y nky~rkoqikleay ilenkikle`cay na zrlo`okckn`n na [ n`nl V yle elrg`cay%
9% C` }`rk`ex` nac arrlr ay ω? { ay iley~`e~a%
>% Cly ilamkikae~ay οl { ο0 yle nayileliknly { naoae na ay~kg`rya%
Z`r` ay~kg`r c` aiq`ikùe na rabraykùe ckea`c ykgzca { gøc~kzca ya qy` cl pqa
ya cc`g` ac gë~lnl na cly iq`nr`nly gçekgly pqa `hqy~` cly n`~ly na c` gqay~r` `
c` cçea` na rabraykùe% Ay~` ay qe` na c`y ~ëieki`y gày qy`n`y ae ke}ay~kb`ikleay
ikae~çmki`y# z`r` aeile~r`r c` rac`ikùe ae~ra nly l gày }`rk`ocay pqa ay~àe
i`yq`cgae~a rac`ikle`n`y%
Ae ay~` yaiikùe }aragly ac zrlocag` na rabraykùe ckea`c na qe` }`rk`oca
nazaenkae~a &[ + l~r` kenazaenkae~a & V +# ile mkeay na zrankiikùe { ay~kg`ikùe% Yke
ago`rbl# qe` }ax pqa ya lo~kaea c` aiq`ikùe na rabraykùe ckea`c# ëy~` ya ~kaea pqa
a}`cq`r l }`ckn`r z`r` }ar pqë ~`e~` ilemk`okckn`n ya ca zqana zlear `c glnacl z`r`
qyly na zrankiikùe% Ay~l ya f`ia qy`enl aemlpqay loha~k}ly { yqoha~k}ly% Zlr
ahagzcl# ac aemlpqa loha~k}l ya f`ia f`ikaenl zrqao`y ay~`nçy~ki`y na kemaraeik`%
Ay~a aemlpqa ya ilgzcagae~` qy`enl aemlpqay yqoha~k}ly# ay naikr# `e`ckx`enl c`y
bràmki`y na cly rayknq`cay ay~`en`rkx`nly l el ay~`en`rkx`nly# ` ~r`}ëy na
keyzaiikleay }kyq`cay%
Zlr ahagzcl# c`y ilenkikleay l yqzlykikleay rapqarkn`y z`r` }`ckn`r ac
glnacl# yqoha~k}`gae~a# ya f`ia ` ~r`}ëy na cly `eàckyky na cly rayknqly irqnly lay~`en`rkx`nly &z`r` nkmaraeik`rcly na cly rayknqly ay~`en`rkx`nly+% Cly cc`g`nly
rayknqly ya namkeae ilgl c`y nkmaraeik`y ae~ra ac }`clr `i~q`c na [ { ac }`clr
zrlely~ki`nl na [ zlr ac glnacl na rabraykùe ay~kg`nl% Cly rayknqly ya nael~`e
zlr ak# ay~l ay# ak 7 [ k ’ [¾ k% Ae }arn`n# c`y bràmki`y na cly rayknqly n`e kemlrg`ikùe
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 458/846
Nr% Fëi~lr Pqa}anl Qrç`y
6'>
gq{ kgzlr~`e~a# `iari` na c` e`~qr`cax` { mqarx` na c` rac`ikùe ae~ra c`y }`rk`ocay%
C` mkbqr` na `o`hl gqay~r` cly rayknqly pqa yle c`y nkmaraeik`y ae~ra cly }`clray na
[ 0 # [ ? # [ 9 #…#[ d { cly }`clray loyar}`nly na [¾ 0 # [¾ ? # [¾ 9 #…#[¾ d na c` cçea` narabraykùe na c` gqay~r`% Zlr l~r` z`r~a# cly rayknqly ay~`en`rkx`nly ya lo~kaeae
nk}knkëenlcly zlr yqy rayzai~k}`y nay}k`ikleay ay~àen`ray%
Mkbqr`% 6%:% Bràmki` gly~r`enl cly rayknqly na qe ahagzcl% &Ac`olr`ikùe zrlzk`+
C`y yqzlykikleay na cly }`clray rayknq`cay yle8
&`+ Cly rayknq`cay ak ay~àe elrg`cgae~a nky~rkoqknly &αk ay~àe elrg`cgae~a
nky~rkoqknly+%
&o+ Cly rayknqly ~kaeae c` gkyg` }`rk`ex` &αk yle iley~`e~ay+%
&i+ Cly rayknq`cay ak el ay~àe ilrrac`ikle`nly# ay naikr# yle kenazaenkae~ay%
L~rl gë~lnl gaely zlzqc`r pqa ac `eàckyky na cly rayknqly# z`r` a}`cq`r c`
aiq`ikùe na rabraykùe ay ilgz`r`enl ac nk`br`g` ayz`riknl na cly zqe~ly# ilerayzai~l ` c` cçea` na rabraykùe# ile c` bràmki` na cly zqe~ly ile rayzai~l `c
zrlgankl na { % Ay~l ya naoa ` pqë# yke kgzlr~`r ac }`clr na V # ac zrlgankl {
ykagzra zarg`eaia iley~`e~a &cçea` flrkxle~`c ~r`x`n` ae ac nk`br`g` ayz`riknl na
c` bràmki`+% Na ay~` g`ear`# yk c` nkyzarykùe na cly zqe~ly ile rac`ikùe ` c` cçea` na
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 459/846
Nr% Fëi~lr Pqa}anl Qrç`y
6'1
rabraykùe ay gqifl gaelr# pqa c` nkyzarykùe na cly zqe~ly ile rayzai~l ` c` cçea`
flrkxle~`c na { # ae~leiay# ya zqana ileicqkr pqa c` aiq`ikùe na c` cçea` na
rabraykùe n` qe oqae `hqy~a z`r` cly n`~ly na c` gqay~r` &N`ekac a~ `c% 0;6;+%Ilgl ya nkhl `e~ay# ac aemlpqa loha~k}ky~` ay c` l~r` g`ear` pqa ya qy` z`r`
a}`cq`r ac glnacl na rabraykùe ckea`c# ay~l ay# ` ~r`}ëy na `eàckyky ay~`nçy~kily% Z`r`
ay~l# ya zqanae qy`r c`y ykbqkae~ay mqeikleay ay~`nçy~ki`y8
&`+ Ilamkikae~a na na~argke`ikùe ckea`c _ ? &l r ?+# ac ilamkikae~a na ilrrac`ikùe ckea`c
_# y { Z_AYY%
&o+ @eàckyky na }`rk`ex` ykgzca &@EL]@+# z`r` zrlo`r cly ilamkikae~ay nac glnaclna rabraykùe &ο+# z`r` π# a~i%
&i+ Ke~ar}`cly na ilemk`ex` z`r` π?# z`r` οl# οk# ± {sv# a~i%
^kzly na ilrrac`ikùe ckea`c
0% Ilrrac`ikùe ykgzca pqa ileyky~a na nly }`rk`ocay# qe` nazaenkae~a &[ + { c` l~r`
kenazaenkae~a & V +% Nae~rl na ay~` i`~ablrç` ~aeagly8
&`+ Ilrrac`ikùe nkrai~`% Ay~` ilrrac`ikùe ileyky~a ae ac keiragae~l ae qe` }`rk`oca c`
iq`c ay `ilgz`ð`n` zlr ac keiragae~l na l~r` }`rk`oca &ilrrac`ikùe zlyk~k}`+%
&o+ Ilrrac`ikùe ke}ary`% Ay~` ilrrac`ikùe ileyky~a ae ac keiragae~l na qe` }`rk`oca c`
iq`c ay `ilgz`ð`n` zlr ac keiragae~l na l~r` &ilrrac`ikùe eab`~k}`+%
&i+ Ilrrac`ikùe el ckea`c% Ae ay~` ilrrac`ikùe el f`{ ekebqe` `ylik`ikùe ae~ra c`y nly
}`rk`ocay%
?% Ilrrac`ikùe gøc~kzca% @pqç# f`{ gày na nly }`rk`ocay% Qe` }`rk`oca ay
nazaenkae~a &[ +# gkae~r`y pqa c`y l~r`y yle kenazaenkae~ay V 0 # V ? #…# V d # a~i%C`y mkbqr`y na `o`hl razrayae~`e }`rkly ~kzly na ilrrac`ikleay%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 460/846
Nr% Fëi~lr Pqa}anl Qrç`y
6'=
Mkb% 6%0% Nk`br`g`y ayz`riknly ile cçea`y na iq`nr`nly gçekgly% C` Mkbqr` &`+razrayae~` qe` cçea` rai~` ile V mkh`4 c` Mkbqr` &o+ razrayae~` cçea` el rai~` ile V mkh`4
c` Mkbqr` &i+ razrayae~` qe` nky~rkoqikùe `nhqe~` ile cçea` rai~`4 c` Mkbqr` &n+
razrayae~` qe` nky~rkoqikùe `nhqe~` ile cçea` el rai~`4 c` Mkbqr` &a+ razrayae~` qe
nk`br`g` nlena el f`{ `ylik`ikùe ae~ra c`y nly }`rk`oca {4 c` Mkbqr` &m+ razrayae~`
qe` rac`ikùe i`qy`c% C`y l~r`y nly bràmki`y razrayae~`e ilrrac`ikleay zarmai~`y%
&Ac`olr`ikùe zrlzk`+
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 461/846
Nr% Fëi~lr Pqa}anl Qrç`y
6'2
^kzly na iqr}`y gày ilgqeay
Mkbqr` 6%?% C` mkbqr` &`+ razrayae~` c` mqeikùe avzleaeik`c4 c` mkbqr` &o+ razrayae~` c`
mqeikùe na zl~aeik`# c` mkbqr` &i+ razrayae~` qe` mqeikùe raiçzrli` {# c` mkbqr` &n+
razrayae~` qe` mqeikùe fkzaroùcki`% &Ac`olr`ikùe zrlzk`+
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 462/846
Nr% Fëi~lr Pqa}anl Qrç`y
6'6
Aiq`ikleay elrg`cay z`r` i`ciqc`r ac ke~ariaz~l ae c` lrnae`n` ` { c` zaenkae~a
o na c` iqr}` l cçea` na rabraykùe
C`y }`rk`ocay ` { o ya lo~kaeae na c`y aiq`ikleay elrg`cay na `o`hl# ay naikr#raylc}këenlc`y ykgqc~àea`gae~a8
Ω[ 7 ` e $ o Ω V &6'9+
Ω V[ 7 ` Ω V $ o ΩV &6'>+
@c raylc}arya ykgqc~àea`gae~a n`e ac ke~ariaz~l# ` ae c` lrnae`n` {# c` zaenkae~a na
c` cçea`# o8
Ke~ariaz~l 7 ` 7 [ ’ o V &6'1+
Zaenkae~a 7 o 7 Te Ω V[ ’ &Ω V +&Ω[ +W , Te Ω V ? ’ &Ω V +? W &6'=+
7 Ω v{ , Ω v? &6'2+
Nlena8
Ω v{ { Ω v? ya n`e zlr c`y aiq`ikleay &6'6+ { &6';+ na `o`hl%
El~` 0% C`y ykbqkae~ay aiq`ikleay yle gq{ kgzlr~`e~ay%
Ω v? 7 Y vv 7 Ω V ? ’ &Ω V +? , e &6'6+
Ω v{ 7 Y v{ 7 Ω V[ ’ Ω V Ω[ , e &6';+
Ω {? 7 Y {{ 7 Ω[ ? ’ &Ω[ +? , e &6'0:+
El~` ?% Ay gq{ kgzlr~`e~a el~ r c`y nkmaraeik`y ae~ra ac qyl na c`y }`rk ocay
gkeøyiqc`y { c`y g`{øyiqc`y ae c`y aiq`ikleay na `rrko`%
Ilamkikae~a na na~argke`ikùe _ ?
na c` gqay~r` pqa ay~kg` ` π?
ac ilamkikae~a na
na~argke`ikùe zloc`ikle`c
Ac iàciqcl nac ilamkikae~a na na~argke`ikùe gøc~kzca _ ?
ay qe` zrqao` loha~k}ky~`na ay~`nçy~ki`% Ay~` ay qe` mqeikùe ay~`nçy~ki` gq{ kgzlr~`e~a# z`r` }`ckn`r ac
glnacl na rabraykùe ckea`c% Ay~a ilamkikae~a _ ? gkna c` zrlzlrikùe na }`rk`ikùe ae
c` }`rk`oca nazaenkae~a [ avzcki`n` zlr c` }`rk`oca kenazaenkae~a V % Cly }`clray na
_ ? }`rç`e na : ` 0% Zlr ahagzcl# qe }`clr iari`el ` : kenki` pqa el f`{ qe`
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 463/846
Nr% Fëi~lr Pqa}anl Qrç`y
6';
rac`ikùe ckea`c ae~ra [ { V # gkae~r`y pqa qe }`clr iari`el ` qel kenki` qe `hqy~a
ckea`c zarmai~l% @pqç# yke ago`rbl# ay eaiay`rkl `ic`r`r pqa# qe }`clr `c~l na _ ?# el
eaiay`rk`gae~a kenki` qe oqae `hqy~a nac glnacl na rabraykùe# ykel f`y~` pqa yaf`iae ~ln`y c`y zrqao`y loha~k}ky~`y { yqoha~k}`y% C` mqeikùe pqa i`ciqc` _ ? ay8
_ ? 7 &Ω v{+? , Ω v?Ω {? &6'00+
7 0 ’ YYa , YY~ &6'0?+
Nlena Ω v{# Ω v? { Ω {? ya n`e zlr c`y aiq`ikleay &6'6+# &6'6+ { &6'0:+ nayirk~`y z`r`
c` aiq`ikùe &6'00+% @nagày# z`r` c` aiq`ikùe &6'0?+ YYa ay c` yqg` na cly
iq`nr`nly nac arrlr l rayknq`c { YY~ ay c` yqg` na cly iq`nr`nly nac ~l~`c# gkygly
pqa ya nayirkoae ae ac mlrg`~l na c` ~`oc` na @EL]@%
^`gokëe f`{ ac cc`g`nl ilamkikae~a _ ? na na~argke`ikùe `hqy~`nl% Ay~` ay
qe` }arykùe `hqy~`n` na _ ?# ac iq`c oqyi` ragl}ar c` nky~lrykùe naokn` ` qe ~`g`ðl
na gqay~r` zapqaðl% Ya namkea ilgl8
_ ?`hqy~`n` 7 0 ’ T&0 ’ _ ?+ &e ’ 0+,&e ’ ?+W &6'09+
Nlena _ ? {` ya namkekù { e ay ac ~`g`ðl na c` gqay~r`
Ilamkikae~a na ilrrac`ikùe _ na c` gqay~r` pqa ay~kg` ` π# ac ilamkikae~a na
ilrrac`ikùe zloc`ikle`c
Ac ilamkikae~a na ilrrac`ikùe _# pqa ay~kg` ` π# ~`gokëe ya cc`g` ilamkikae~a na
ilrrac`ikùe na Za`ryle% Ay~a ilamkikae~a ay qe çenkia na c` mqarx` na c` `ylik`ikùe
ckea`c ae~ra c`y }`rk`ocay V a [ % Ac ilamkikae~a na ilrrac`ikùe _ ay8
_ 7
∐∐
∐
{ v
v{
??
&6'0>+
Nlena8 Ω v{# Ω v? { Ω {? ya n`e zlr c`y aiq`ikleay &6'6+# &6';+ { &6'0:+
El~`8 Ac ilamkikae~a na ilrrac`ikùe _ avzcki` ac br`nl na `ylik`ikùe ae~ra c`y
}`rk`ocay V a [ % Ay~a ilamkikae~a _ }`rç` na ’0 ` :# yk c` ilrrac`ikùe ay eab`~k}`# ay
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 464/846
Nr% Fëi~lr Pqa}anl Qrç`y
6'0:
naikr# ile zaenkae~a eab`~k}`% Zarl# yk c` ilrrac`ikùe ay zlyk~k}`# ae~leiay# _ }`rç`
na : ` 0% @yç# ` gankn` pqa _ ya `zrlvkg` ` °0# gahlr `ylik`ikùe f`orà ae~ra c`y
}`rk`ocay V a [ % Eù~aya pqa# ae i`yl na c` rabraykùe ckea`c gøc~kzca# ~aeagly cl pqaya cc`g`e ilamkikae~ay z`rik`cay na rabraykùe qy`nly z`r` gankr c` rac`ikùe ckea`c
ae~ra c` }`rk`oca nazaenkae~a { c` }`rk`oca kenazaenkae~a ayzaikmki`n`%
Ke~ar}`cl na ilemk`ex` z`r` ac ilamkikae~a zloc`ikle`c ο ilgzleae~a na c`
cçea` na rabraykùe ±[sV 7 ΰ $ ο V # ay~kg`nl zlr o# c` zaenkae~a na c` cçea`%
o ’ ~T0'ΰ,?4e'?W y , ∐ v? 3 ο 3 o $ ~T0'ΰ,?4e'?W y , ∐ v
? &6'01+
Nlena8o 7 Ω v{ , Ω v?
~T0'ΰ,?4e'?W 7 }`clr na c` nky~rkoqikùe na ~ na Ay~qnk`e~a
Ω v? 7 Ω V ? ’ &Ω V +? , e
y 7 ?
+&?
∟
∟∐ ∐e
v{o {&6'0=+
7 YYA,&e ’ ?+ 7 e
{ {∐ ∐∟ +&??
' &oΩ V[ ' Ω V Ω[ ,e+W , e'?
C` aiq`ikùe na c` }`rk`ex` ay8 y? 7 &Ω {? ’ oΩ v{+ , &e ’ ?+ &6'02+
ο 7 ilamkikae~a zloc`ikle`c na c` zaenkae~a na c` cçea`# ac iq`c ay ay~kg`nl zlr o 7
Ω v{ , Ω v? l ya` ac ilamkikae~a na c` cçea` na rabraykùe gqay~r`c%
Ke~ar}`cl na ilemk`ex` z`r` ac z`ràga~rl zloc`ikle`c ΰ# ac ke~ariaz~l na c`
lrnae`n` na c` cçea` na rabraykùe ±[sV 7 ΰ $ ο V # iq{l ay~kg`nlr ay `
&6'06+
Nlena8
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 465/846
Nr% Fëi~lr Pqa}anl Qrç`y
6'00
` {` ya namkekù `e~arklrgae~a
~T0'ΰ,?4e'?W 7 ` qe }`clr qy`enl c` nky~rkoqikùe na ~ na ay~qnk`e~a ile η 7 e ’ ? br`nly
na ckoar~`ny 7 na c` aiq`ikùe &6'0=+
Yvv 7 Ωv{ &na c` aiq`ikùe &6';++
Fkzù~ayky eqc` Fl8ο 7 οl ile~r` c`y fkzù~ayky `c~are`~k}`y F08ο 3 0 { F?8ο 5 0%
Z`r` ay~` zrqao` ~`gokëe ya qy` c` nky~rkoqikùe na ~ na Ay~qnk`e~a ile η 7 e ’ ?
br`nly na ckoar~`n# ay naikr8
~ 7 &o ’ οl+ , y,Ω v? &6'0;+
Nlena8
~ 7 c` ay~`nçy~ki` na c` nky~rkoqikùe na ~ na Ay~qnk`e~a
οl 7 qe }`clr n`nl
o 7 zaenkae~a na c` cçea`
Fkzù~ayky eqc` Fl8ΰ 7 ΰl ile~r` c`y fkzù~ayky `c~are`~k}`y F08ΰ ≪ ΰl# F?8ΰ 5 ΰl#
{ F98 ΰ 3 ΰl @pqç# eqa}`gae~a# ya qy` c` nky~rkoqikùe na ~ na Ay~qnk`e~a ile br`nly na ckoar~`n#
η 7 e ’ ?% Z`r` ay~l ya qy` c` mùrgqc` na `o`hl8
&6'?:+
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 466/846
Nr% Fëi~lr Pqa}anl Qrç`y
6'0?
Nlena8
ΰl 7 qe }`clr n`nl
y 7 {` namkekn` `e~arklrgae~a
` {` ya namkekù `e~arklrgae~a
Ke~ar}`cl na ilemk`ex` z`r` ±[sV na c` cçea` zloc`ikle`c ay~kg`n` zlr [
Ac ke~ar}`cl na ilemk`ex` z`r` ac }`clr na ±[sV ya f`ia ay qy`enl c` mùrgqc` &6'?0+ na
`o`hl8
[l‛’ ~Tΰ,?4 ηW ye0 $ &Vl ' V +?,Ωv? 3 ±[sV 3 [l‛$ ~Tΰ,?4 ηW y
e0 $ &Vl ' V +?,Ωv? &6'?0+
Nlena8
[ l‛ 7 ` $ o V l 7 }`clr na c` cçea` na rabraykùe ile qe }`clr na V l n`nl &6'??+
~Tΰ,?4e'?W 7 }`clr na c` nky~rkoqikùe na ~ ile qe ek}ac na ykbekmki`eik` na ΰ 7 %:1 l okae
:%:0 ile η 7 e ’ ? br`nly na ckoar~`n
` 7 {` namkekn` `e~arklrgae~a
y 7 {` namkekn` e~arklrgae~a
V l 7 qe }`clr n`nl
V 7 zrlgankl na c` gqay~r`
Fkzù~ayky eqc` Fl8ο 7 : ile~r` c`y fkzù~ayky `c~are`~k}`y F08ο 5 : { F?8ο 3 :
Z`r` f`iar ay~` zrqao` qy`gly c` nky~rkoqikùe na ~ na Ay~qnk`e~a ile η 7 e ’ ? br`nly
na ckoar~`n% C` mqeikùe ay~`nçy~ki` qy`n` z`r` ~`cay mkeay ay8~ 7 &o ’ ol+ , y , ∐ v
? &6'?9+
Nlena8
y 7 {` namkekn` `e~arklrgae~a
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 467/846
Nr% Fëi~lr Pqa}anl Qrç`y
6'09
o 7 ke~ariaz~l ae c` lrnae`n` [
ol 7 qe }`clr n`nl
Ω {?
7 Ω[ ?
’ &Ω[ +?
,eΩ v{ 7 Ω V[ ’ Ω V Ω[ ,e
οl 7 :
@pqç# ~`gokëe ya ~kaeae pqa i`ciqc`r c`y rabkleay irç~ki`y qy`enl c` nky~rkoqikùe
na ~# ay naikr# ~T0'ΰ,?4 ηW# nlena ΰ ay ac ek}ac na ykbekmki`eik` naya`nl {# η ay ac eøgarl na
br`nly na ckoar~`n# ay naikr# e ' 0% Nayzqëy na ay~l# ya ilgz`r` ac }`clr na ~ i`ci%# ile ac
}`clr irç~kil na ~~`o% { ya ykbqa ac gkygl zrliankgkae~l z`r` iq`cpqkar zrqao` na
fkzù~ayky%
Fkzù~ayky eqc` na Fl8ΰ 7 ΰl ile~r` c`y fkzù~ayky `c~are`~k}`y F08ΰ 5 : { F?8ΰ 3 :
Z`r` f`iar ay~` zrqao` na fkzù~ayky ya qy` c` ay~`nçy~ki` na ~ na Ay~qnk`e~a gly~r`n`
`o`hl8
&6'?>+
Nlena8
y 7 {` namkekn` `e~arklrgae~a
Nlena8
Ω {? 7 Ω[ ? ’ &Ω[ +?,e
Ω v{ 7 Ω V[ ’ Ω V Ω[ ,e
o 7 {` namkekn` `e~arklrgae~a@pqç# ~`gokëe ya ~kaea pqa ay~`ocaiar c`y rabkleay irç~ki`y qy`enl c` nky~rkoqikùe na ~
na Ay~qnk`e~a% Ay~`y rabkleay irç~ki`y yle8 ~T0'ΰ,?4 ηW# nlena ΰ ay ac ek}ac na ykbekmki`eik`
qy`nl%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 468/846
Nr% Fëi~lr Pqa}anl Qrç`y
6'0>
Zrqao`y na fkzù~ayky Fl8π 7 :# ile~r` c` fkzù~ayky `c~are`~k}`y F08π ≪ :# z`r` ac
ilamkikae~a na ilrrac`ikùe zloc`ikle`cπ ay~kg`nl zlr _% &Nqee a~ `c% 0;2>+
Z`r` ay~ly mkeay ya qy` c` ay~`nçy~ki` na ~ na Ay~qnk`e~a8
~ 7 ∘ η _ , _?
0∟ &6'?1+
Nlena8
_ 7 Ω v{ , ∐∐ { v?? &6'?=+
η 7 e ’ ? br`nly na ckoar~`n
@pqç# eqa}`gae~a# z`r` i`ciqc`r c`y rabkleay irç~ki`y ya qy` c` ~ na Ay~qnk`e~a# aynaikr# ~Tΰ,?4e'?W%
Ahagzcly na zrlocag`y qy`enl rabraykùe { ilrrac`ikùe ckea`c ykgzca
Ahagzcl )0% Ay~a zrlocag` ay~à rac`ikle`nl ile qe ay~qnkl `iari` na c` i`e~kn`n na
zraikzk~ ikùe zcq}k c { c` i`e~kn`n na ile~ gke`ikùe `~glymërki %
^@OC@ 6%:% ^`oc` gly~r`enl cly n`~ly% &Ac`olr`ikùe zrlzk`+ \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
Ccq}k` &:%:?=‗+ s 06 2 0> 90 ?0 1 00 0= ?= ?;_aglikùe na ile~`gke`ikùe s 11 02 9= 61 =? 06 99 >0 =9 62
F`iar c`y ykbqkae~ay ay~kg`ikleay8
&`+ Knae~kmki`r c` }`rk`oca nazaenkae~a { c` }`rk`oca kenazaenkae~a% F`iar qe` bràmki`
pqa }`{` ae mqeikùe na c` }`rk`oca nazaenkae~a [ # { c` }`rk`oca kenazaenkae~a V %
&o+ I`ciqc`r cly }`clray na c` ay~`nçy~ki` nayirkz~k}` na cly n`~ly%
&i+ Lo~aear c` aiq`ikùe na rabraykùe ckea`c ykgzca { ~r`x`rc` ae c` bràmki`%
&n+ ]`ckn`r c` ilemk`okckn`n nac glnacl na rabraykùe# ay naikr# ` ~r`}ëy na c` agkykùe
na qe hqkikl yqoha~k}l `e`ckx`enl cly }`clray na cly rayknqly ay~`en`rkx`nly# na c`
ykbqkae~a g`ear`8
0% F`iar qe` bràmki` pqa gqay~ra c` zrqao` na elrg`ckn`n%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 469/846
Nr% Fëi~lr Pqa}anl Qrç`y
6'01
?% F`iar qe` bràmki` ile cly rayknq`cay ay~`en`rkx`nly }aryqy }`clray `hqy~`nly na [ %
&Ac }`clr zranaiknl l `hqy~`nl na [ k ay ac }`clr na [ pqa ya ayzar`rç` iq`enl ya qy` c`
cçea` na rabraykùe% Ae l~r`y z`c`or`y# cly }`clray `hqy~`nly na [ 0# [ ?#%%# [ e ya lo~kaeaeyqy~k~q{aenl# yqiayk}`gae~a# v0# v?# %%# ve ae c` aiq`ikùe na c` cçea` na rabraykùe
ay~kg`n`8 [ k 7 οl $ ο0vk# %%# οl $ ο0ve%
9% F`iar qe fky~lbr`g` na rayknq`cay%
>% F`iar qe` br`mki` pqa gqay~ra cly rayknq`cay ay~`en`rkx`nly }aryqy raebcleay%
&a+ Ilgzcagae~`r c` a}`cq`ikùe nac glnacl ile kemaraeik`y ay~`nçy~ki`y# ilgl8
0% Iàciqcl nac ilamkikae~a na na~argke`ikùe _ ? { ac ilamkikae~a na ilrrac`ikùe _%
?% F`iar qe` ~`oc` na `eàckyky na }`rk`ex` &@EL]@+%
9% F`iar qe` ~`oc` ile cly ilamkikae~ay# cly arrlray ay~àen`ray# c`y zrqao`y na ~# cly
}`clray na z# { cly ke~ar}`cly na ilemk`ex` z`r` ac ke~ariaz~l { c` zaenkae~a%
Ylcqikùe8
&`+ C` }`rk`oca nazaenkae~a ay c` raglikùe na ile~`gke`e~ay &[+ { c` }`rk`oca
kenazaenkae~a ay c` i`e~kn`n na ccq}k` &V+% C` mkbqr` na `o`hl gqay~r` ay~` ylcqikùe8
Mkbqr` 6%9% Bràmki` gly~r`enl [ }aryqy V# ile qe` cçea` rai~` flrkxle~`c
ilrrayzlenkae~a `c }`clr nac zrlgankl na [ 7 >;%2:::% &Ac`olr`ikùe zrlzk`+
&o+ Cly }`clray na c` ay~`nçy~ki` nayirkz~k}` yle8
V 7 02%6:::# [ 7 >;%2:::% Cly }`clray gàvkgly { gçekgly na cly }`clray na [ yle
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 470/846
Nr% Fëi~lr Pqa}anl Qrç`y
6'0=
62%::: { 02%:::# rayzai~k}`gae~a% Cly }`clray gàvkgly { gçekgly na cly }`clray na
V yle 90%::: { 1%::::# rayzai~k}`gae~a% Iq`nr`nl gankl nac arrlr 7 y? {sv 7 ?=%==24
arrlr iq`nrà~kil gankl ay y {sv 7 1%0=>&i+ Qy`enl qe zrlbr`g` na ilgzq~`nlr` ya ay~kg`e cly }`clray nac ke~ariaz~l ae c`
lrnae`n` { c` zaenkae~a% Ay~ly yle8 ke~ariaz~l 7 ` 7 0%:?09# zaenkae~a na c` cçea` 7 o
7 ?%29>6% Yqy~k~q{aenl ay~ly }`clray n`e c` cçea` na rabraykùe gqay~r`c &gkyg` pqa ya
}a ae c` Mkbqr` 6%9+# n`%
[ 7 ` $ oV
[ 7 0%:?09 $ ?%29>6& V +
&n+ Z`r` ay~a keikyl c` Mkbqr` 6%> gqay~r` c` kemlrg`ikùe rapqarkn`%
_ayknq`c
Z a r i a e ~
0:1:'1'0:
;;
;:
1:
0:
0
Mk~~an ]`cqa
_ a y k n q ` c
6:=:>:?:
1
:
'1
'0:
_ayknq`c
M r a p q a e i {
>:'>'6
9
?
0
:
Loyar}`~kle Lrnar
_ a y k n q ` c
0:;62=1>9?0
1
:
'1
'0:
Elrg`c Zrlo`okck~{ Zcl~ lm ~fa _ayknq`cy _ayknq`cy ]aryqy ~fa Mk~~an ]`cqay
Fky~lbr`g lm ~fa _ayknq`cy _ayknq`cy ]aryqy ~fa Lrnar lm ~fa N`~`
_ayknq`c Zcl~y mlr _aglikle na ile~`gke`~eay &[+
Mkbqr` 6%>% Bràmki`y gly~r`enl c`y rayzqay~`y z`r` ac keikyl &n+%
Ilgl ya }a ae c` Mkbqr` 6%> c` mkbqr` yqzarklr kxpqkarn` gqay~r` c` zrqao` na
elrg`ckn`n ile ~lnly cly zqe~ly mlrg`enl qe` ckea` rai~`% Ay~l kenki` pqa c`
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 471/846
Nr% Fëi~lr Pqa}anl Qrç`y
6'02
nky~rkoqikùe na cly n`~ly ay elrg`c% Kbq`cgae~a# c` mkbqr` yqzarklr naraif` gqay~r`
cly rayknq`cay ae mqeikùe na cly }`clray `hqy~`nly na [% @pqç# f`{ `ca`~lrkan`n ae
c` nky~rkoqikùe na cly zqe~ly ile c` gkyg` i`e~kn`n na zqe~ly eab`~k}ly { zlyk~k}ly# cl pqa kenki` pqa el f`{ ilrrac`ikle na cly n`~ly% C` mkbqr` kemarklr
kxpqkarn` gqay~r` c` mraiqaeik` }aryqy cly rayknq`cay% Mke`cgae~a# c` mkbqr` kemarklr
naraif` gqay~r` cly rayknq`cay ae mqeikùe na cly ùrnaeay na c`y loyar}`ikleay%
@pqç# ae ay~` mkbqr` f`{ `ca`~lrkan`n { ac gkygl eqgarl na zqe~ly zlyk~k}ly {
eab`~k}ly# cl pqa yqbkara pqa el f`{ ilckea`ckn`n l ilrrac`ikle ae yarka na c`
kemlrg`ikùe yqgkeky~r`n`%
&a+ Z`r` ilgzcagae~`r ac ay~qnkl loha~k}ky~`# ay~l ya f`ia f`ikaenl zrqao`y
ay~`nçy~ki`y na kemaraeik`%
&0+ Ilgl ya nkhl `e~ay# ac ilamkikae~a na na~argke`ikùe _ ? ay qe aemlpqa loha~k}ky~`#
pqa ykr}a z`r` }`ckn`r ac glnacl na rabraykùe% Ay~a ilamkikae~a na na~argke`ikùe _ ?#
gkna c` mqarx` rac`~k}` na c` rac`ikùe ckea`c ae~ra V a [ &gkna c` zrlzlrikùe na
}`rk`ikùe ae [ pqa zqana yar avzcki`n` zlr c` }`rk`ikùe ae V + ay n`nl zlr c` aiq`ikùe
&6'00+ { zlr c`y aiq`ikleay &6'=+# &6'2+ { &6'6+# rayzai~k}`gae~a8
_ ? 7 :%;=?:
Ac iàciqcl nac ilamkikae~a na ilrrac`ikùe _ ay8
_ 7 ? _ 7 :%;6:6
&?+ Z`r` ac `eàckyky na }`rk`ex` &@EL]@+# pqa ~`gokëe ykr}a z`r` }`ckn`r ac glnacl
na rabraykùe# ay qe` mqeikùe ay~`nçy~ki` loha~k}ky~` pqa zrqao` c` fkzù~ayky eqc` na pqa
c` zaenkae~a ay kbq`c ` :% @pqç ya }arà pqa# qe }`clr br`ena na M kenki` pqa ac glnaclna rabraykùe yacaiikle`nl ay q~kc% Yke ago`rbl# ay eaiay`rkl `e`ckx`r ~lnly cly nagày
irk~arkly `e~ay na agk~kr qe hqkikl mke`c% C` ~`oc` na @EL]@ na `o`hl n` cly
rayqc~`nly%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 472/846
Nr% Fëi~lr Pqa}anl Qrç`y
6'06
^@OC@ 6%0% `oc` na eàckyky na }`rk`ex` &@EL]@+ z`r` ac ahagzcl%&Ac`olr`ikùe zrlzk`+
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
Mqae~a na Yqg` na cly b%c% Iq`nr`nl nac Mi`ci% M~`o% ]`clr na z}`rk`ikùe iq`nr`nly zrlgankl \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ Naoknl `c 1#9;=%22 0 1#9;=%22 ?:?%96 1%9? :%::::0~r`~`gkae~l_ayknq`c &arrlr+ ?09%99 6 ?=%=2
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ ^l~`c 1#=0:%0 ;
Ac }`clr na M~`o% ya y`i` ileyqc~`enl c` ~`oc` na c` nky~rkoqikùe na M# ay~l ay Mΰ4 η0# η?# aciq`c n` M%;140#6 7 1%9?% @pqç# naoknl ` pqa ac }`clr na Mi`ci% 7 ?:?%96 555 1%9?# ya
raif`x` c` fkzù~ayky yqy~ae~`n` na pqa Fl8ο0 7 : { ya keicke` zlr Fl8ο0 ≪ :% C`
ileicqykùe ay na pqa c` zaenkae~a na c` cçea` el ay kbq`c ` : q flrkxle~`c%
&9+ C` ~`oc` na `o`hl gqay~r` cly }`clray nac ke~ariaz~l ae c` lrnae`n`# ac br`nkae~a
na c` cçea` na rabraykùe# cly arrlray ay~àen`r# c` zrqao`y na fkzù~ayky qy`enl c` ~ na
ay~qnk`e~a# cly }`clray na c` zrlo`okckn`n z { cly ke~ar}`cly na ilemk`ex` &;1!+ z`r`
οl &ke~ariaz~l+ { ο0 &zaenkae~a+%
^@OC@ 6%?% ^`oc` gly~r`enl cly }`clray nac ke~ariaz~l# zaenkae~a# zrqao`y na ~ naAy~qnk`e~a# }`clr nac ek}ac na z { yqy ke~ar}`cly% &Ac`olr`ikùe zrlzk`+
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ Ilamkikae~a Arrlr Zrqao` ~ ]`clr z Cçgk~a Cçgk~a
ay~àen`r kemarklr yqzarklr \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ Ke~ariaz~l 0%:? 9%2; :%?2 :%2; '2%22? ;%2=
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\Zaenkae~a ?%29 :%0; 0>%?9 1%6v0:'2 ?%?; 9%06 \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
@pqç# eù~aya pqa ac ke~ar}`cl na ilemk`ex` z`r` ac ke~ariaz~l ay gq{ `gzckl { c`
fkzù~ayky el ya zqana raif`x`r# zqay~l pqa ac }`clr na ~ ay gq{ zapqaðl { ac }`clr na
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 473/846
Nr% Fëi~lr Pqa}anl Qrç`y
6'0;
z 7 :%2; ay br`ena% Ay~l ay `zl{`nl zlr ac }`clr na :%2; na z { zlr qe arrlr ay~àen`r
na 9%2;# rac`~k}`gae~a br`ena4 cl ile~r`rkl liqrra ile c`y zrqao`y ay~`nçy~ki`y na c`
zaenkae~a# iq{l }`clr na ~ ay br`ena { iq{l }`clr na z ay gq{ zapqaðl%Ahagzcl )?% Ae qe ay~qnkl na gkirloklclbç` `gokae~`c# ae gqay~r`y na `bq`# ya
nkarle cly ykbqkae~ay n`~ly na c` ~`oc` na `o`hl% Ay~ly n`~ly ya ramkarae `c iraikgkae~l
na qe` ilclek` na o`i~ark`y ae qe gankl na iqc~k}l%
^@OC@ 6%9% ^`oc` gly~r`enl cly n`~ly% &Ac`olr`ikùe zrlzk`+
^kagzl ae nç`y na s 9 = ; 0? 01 06keliqc`ikùe & V +
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ El% o`i~ark`y &[ + s 001#::: 0>2#::: 06;#::: ?91#=:: ?12#;:: ?6=#>::
F`iar cly ykbqkae~ay iàciqcly8
&`+ I`ciqc`r c` cçea` na rabraykùe%
&o+ I`ciqc`r ac ilamkikae~a na na~argke`ikùe _ ? { ac ilamkikae~a na ilrrac`ikùe _%
&i+ Ile c` aiq`ikùe na rabraykùe# ay~kg`r ac eøgarl na o`i~ark`y nayzqëy na ?: nç`y
&n+ Aeile~r`r cly ke~ar}`cly na ilemk`ex` z`r` ΰ { ο qy`enl ac z`pqa~a na AVIAC%&a+ Qy`r ac zrlbr`g` Gkek~`o { ay~kg`r cly }`clray rayknq`cay { `e`ckx`rcly
yqoha~k}`gae~a# z`r` ra}ky`r zlr c` i`ckn`n nac glnacl na rabraykùe%
Ylcqikùe8
&`+ C` aiq`ikùe na c` cçea` na rabraykùe ay8
[ 7 60#1?:%:: $ 00#22>%?; V
&o+ Ac ilamkikae~a na na~argke`ikùe ckea`c gøc~kzca _ ? ay kbq`c ` :%;66:% Ac ilamkikae~a
na ilrrac`ikùe _ ay kbq`c ` :%;;>:%
&i+ Iq`enl V 7 ?: nç`y# ac eøgarl na o`i~ark`y ay na8
[ 7 60#1?: $ 00#22>%?; &?:+ ≆ 902#::= o`i~ark`y
&n+ Ae iq`e~l ` cly ke~ar}`cly na ilemk`ex` z`r` ΰ { ο# ac zrlbr`g` na ilgzq~`nlr` na
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 474/846
Nr% Fëi~lr Pqa}anl Qrç`y
6'?:
AVIAC rrlh` cly ykbqkae~ay rayqc~`nly8
Ke~ar}`cl na ilemk`ex` na ;1! z`r` ΰ8 =0#?1;%>1 3 ΰ 3 0:0#26:%=4 }`clr na c`
zrlo`okckn`n z 7 :%:::>4 Ke~ar}`cl na ilemk`ex` na ;1! z`r` ο ay8 0::>:%0> 3 ο 3091:6%>9# ile qe }`clr na c` zrlo`okckn`n z 7 :%::::>=
&a+% C`y mkbqr`y na `o`hl gqay~r`e c`y bràmki`y pqa ~r`~`e na }`ckn`r ac glnacl na
rabraykùe ckea`c# ile nac eøgarl na o`i~ark`y ae mqeikùe nac ~kagzl na keiqo`ikùe%
Mkbqr` 6%1% Mkbqr`y gly~r`enl cly rayqc~`nly nac eøgarl na o`i~ark`y }aryqy ac ~kagzlna keiqo`ikùe% C` bràmki` &`+ gqay~r` c` rac`ikùe ae~ra [ { V # ile c` cçea` rai~` na [ 4
c` bràmki` &o+ gqay~r` cly rayknqly irqnly }aryqy V 4 c` bràmki` &i+ gqay~r` cly rayknqly
irqnly }aryqy cly raebcleay {# c` bràmki` &n+ gqay~r` cly rayknqly irqnly }aryqy
rayknqly rax`b`nly &Ac`olr`ikùe zrlzk`+%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 475/846
Nr% Fëi~lr Pqa}anl Qrç`y
6'?0
^ln`y ay~`y bràmki`y yqbkarae# yqoha~k}`gae~a# pqa ac glnacl na rabraykùe ckea`c ay
ilemk`oca% ¼Zlr pqë<
Ahagzcl )9% Ae qe ay~qnkl na `brkiqc~qr`# rac`ikle`nl ile c` ykagor` na `cblnùe# aeikar~l ay~`nl na c` Qekùe @garki`e`# c` zraikzk~`ikùe `eq`c { ac raenkgkae~l na c`
ilyaif` na `cblnùe yle ilgl ykbqa%
^@OC@ 6%>% ^`oc` gly~r`enl cly n`~ly% &Ac`olr`ikùe zrlzk`+
Zraikzk~`ikùe s 2%0? =9%1> >2%96 >1%;? 6%=6 1:%6= >>%>=ae zqcb`n`y
& V +
_aenkgkae~l na s 0:92 96: >0= >?2 =0; 966 9?0c` ilyaif` aeckor`y,`ira
&[ +
F`iar cly ykbqkae~ay iàciqcly8
&`+ I`ciqc`r cly }`clray nac ke~ariaz~l ` { c` zaenkae~a o%
&o+ Ayirkokr c` aiq`ikùe na c` cçea` na rabraykùe%
&i+ I`ciqc`r ac ilamkikae~a na na~argke`ikùe _
?
{ ac ilamkikae~a na ilrrac`ikùe _%&n+ Zranaikr ac raenkgkae~l na c` ilyaif` na `cblnùe# yk c` zraikzk~`ikùe ay na 9:
zqcb`n`y%
&a+ F`iar qe` ~`oc` na `eàckyky na }`rk`ex`%
Ylcqikùe8
&`+ Qy`enl qe z`pqa~a na ilgzq~`nlr` ilgl ac Aviac n`8
Ke~ariaz~l ae c` lrnae`n` 7 ` 7 66:%>:
Zaenkae~a na c` cçea` 7 o 7 ';%=0
&o+ Zlr cl ~`e~l# c` aiq`ikùe na c` cçea` na rabraykùe ay8
[ 7 66:%>: ’ ;%=0 & V +
&i+ Ac ilamkikae~a na na~argke`ikùe 7 _ ? 7 :%=;;0
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 476/846
Nr% Fëi~lr Pqa}anl Qrç`y
6'??
Ac ilamkikae~a na ilrrac`ikùe 7 _ 7 :%69=0
&n+ Iq`enl c` zraikzk~`ikùe na ccq}k` ay na 9: zqcb`n`y# ac raenkgkae~l na c`
ilyaif` ya i`ciqc` qy`enl ac glnacl na rabraykùe lo~aeknl# ay naikr yqy~k~q{aenl ac}`clr na V 7 9:% Na ay~` g`ear`# qy`enl c` aiq`ikùe na rabraykùe n`n` `rrko` {
yqy~k~q{aenl ac }`clr na V 7 9: ely n`8
[ 7 66:%> ’ ;%=0 &9:+ 7 1;?%0
&a+ C` ~`oc` na `eàckyky na }`rk`ex` n`n` zlr ac z`pqa~a Aviac ya n` `o`hl%
^@OC@ 6%1% ^`oc` na `eàckyky na }`rk`ex` &@EL]@+% &Ac`olr`ikùe zrlzk`+
Mqae~a na }`rk`ikùe b%c% YY GY Mi`ci% M~`o% ]`clr na zNaoknl ` c` _abraykùe 0 ?=:#=?6%? ?=:#=?6%? 00%=? 1%9? :%:0;
_ayknql 1 00?#0=1%1 ??#>99%00
^l~`c = 92?2;9%2
Ae ileicqykùe# `c ilgz`r`r ac }`clr na c` ay~`nçy~ki` i`ciqc`n` M ile ac }`clr irç~kil na
M ya raif`x` c` fkzù~ayky yqy~ae~`n` ile qe }`clr na z kbq`c ` :%:0;%
Ahagzcl )>% Ac ckorl @zzckan Y~`~ky~kiy8 @eàckyky lm ]`rk`eia `en _abrayykle na Nqee
{ Ic`rd &0;2>+ nayirkoa qe ay~qnkl na mçyki`# ay naikr# na ùz~ki`# nlena ya lo~q}karle
cly n`~ly na `o`hl pqa gqay~r`e cly nkàga~rly na c`y mkor`y ùz~ki`y &ae gkir`y+ ae
mqeikùe na c` mqarx` na rlgzkgkae~l na ëy~`y% Z`r` ay~a zrlocag` f`iar cly ykbqkae~ay
iàciqcly
&`+ F`iar ~lnly cly i`ciqcly zrackgke`ray { i`ciqc`r c` aiq`ikùe na c` cçea` na
rabraykùe gqay~r`c pqa ay~kg` ` c` aiq`ikùe na rabraykùe zloc`ikle`c ±[sV 7 ΰ $ ο V %&o+ Qy`enl qe z`pqa~a na ilgzq~`nlr`# aeile~r`r ac ke~ar}`cl na ilemk`ex` z`r` ac
ilamkikae~a na rabraykùe zloc`ikle`c ΰ &ke~ariaz~l ae [ +# pqa ay~kg` ` `%
&i+ Ae mlrg` `eàclb` pqa ile ae ac keikyl &o+# aeile~r`r ac ke~ar}`cl na ilemk`ex` z`r`
ac ilamkikae~a na rabraykùe ο &c` zaenkae~a na c` cçea`+ iq{l ay~kg`nlr ay o%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 477/846
Nr% Fëi~lr Pqa}anl Qrç`y
6'?9
&n+ Zrlo`r c` fkzù~ayky eqc` na Fl8ο 7 οl# ay naikr# ο 7 : ile~r` c` fkzù~ayky `c~are`~k}`
na F08ο 5 : { F?8ο 3 :% I`ciqc`r ac }`clr na c` zrlo`okckn`n z%
&a+ F`iar qe ke~ar}`cl na ilemk`ex` z`r` ±[sVl% &m+ I`ciqc`r cly irk~arkly a}`cq`nlray nac glnacl na rabraykle# }% b%# _ ?# Z_AYY { y%
&b+ F`iar qe` zrqao` na fkzù~ayky z`r` ac ilamkikae~a na ilrrac`ikùe zloc`ikle`c π%
&f+ Br`mki`r cly n`~ly { ~r`x`r c` aiq`ikùe na c` cçea` na rabraykùe ylora c` bràmki` {
~r`x`r c` cçea` flrkxle~`c ilrrayzlenkae~a `c }`clr nac zrlgankl [ %
&k+ Agk~kr qe hqkikl yqoha~k}l pqa `{qna ` }`ckn`r ac qyl nac glnacl na rabraykùe%
C` ~`oc` na `o`hl gqay~r` cly n`~ly%
^@OC@ 6%=% ^`oc` gly~r`enl ac nkàga~rl na mkor`y }y% mqarx` na rlgzkgkae~l%
Nkàga~rl na c` mkor` & V + Clb na c` mqarx` na rlgzkgkae~l &[ +
??%1 %0;?6%: %=??2%1 %10?1%1 %19??%: %?>
9:%1 %62?9%: %?1?1%: %?1?9%1 %92?2%: %9??0%1 %09??%: %91?;%: %19?:%1 %???2%: %=1
&Mqae~a8 Nqee a~ `c% 0;2>% @zzckan Y~`~ky~kiy8 @e`c{yky lm ]`rk`eia `en _abrayykle+
Ylcqikùe8
&`+ Cly iàciqcly zrackgke`ray yle8
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 478/846
Nr% Fëi~lr Pqa}anl Qrç`y
6'?>
e 7 01# ΩV 7 92>%1# &ΩV+?,e 7 ;#91:%:# Ω[ 7 =%:9# &Ω[+?,e 7 ?%>?# ΩV[ 7 016%?1# ΩV? 7 ;#>6?%21# Ω[ ? 7 9%:9# &ΩVΩ[+,e 7 ?#?16%?>,01 7 01:%11# V 7 ?>%;2# [ 7 :%>:?#
Ωv? 7 ΩV ? ’ &ΩV+?,e 7 ;#>6?%21 ’ ;#91:%: 7 09?%21# Ωv{ 7 ΩV[ ’ ΩVΩ[,e 7 016%?1 ’ 01:%11 7 2%2:# Ω{? 7 Ω[ ? ’ &Ω[+?,e 7 9%:9 ’ &=%:9+?,01 7 %=:2>
Z`r` i`ciqc`r c` cçea` na rabraykùe na c` gqay~r`# zrkgarl i`ciqc`gly g`eq`cgae~a#
cly ilamkikae~ay ` { o na c` cçea` na rabraykùe gqay~r`c pqa ay~kg`e ` ΰ { ο%
o 7 Ω v{,Ω v? 7 2%2:,09?%21
7 %:16
` 7 [ ’ o V 7 :%>:? ’ &:%:16+&?>%;2+ 7 '0%:>=Zlr cl ~`e~l# c` cçea` na rabraykùe gqay~r`c ay8
{ 7 ` $ o& V +
{ 7 '0%:>= $ :%:16& V +
&o+ Ac ke~ar}`cl na ilemk`ex` z`r` ΰ ay qy`enl c` mqeikùe &6'06+ l qy`enl qe z`pqa~a
na ilgzq~`nlr` ilgl Aviac zrliankaenl ilgl8 ^llcy ↟ N`~` eàckyky ↟ _abrayykle
{ LD% Aeyabqkn`# nayzqëy na pqa cly n`~ly ya ke~rlnqharle ae c`y ilcqge`y @ { O na
c` flh` na Aviac krya ` c` }ae~`ekcc` na –Kezq~ [ _`eba‗ { –Kezq~ V _`eba‗# cl pqa
baear` c` ^@OC@ 6%2 na `o`hl%
^@OC@ 6%2% ^`oc` gly~r`enl ac }`clr nac ke~ariaz~l# c` zaenkae~a# cly }`clray na ~ { z
{ cly ke~ar}`cly na ilemk`ex` z`r` ΰ { ο%
Zlr cl ~`e~l# ac ke~ar}`cl na ilemk`ex` z`r` ac ke~ariaz~l &ΰ+ ya caa na c` ~`oc` ilgl8
'0%12:= 3 ΰ 3 ':%1??>
&i+ Ae mlrg` `eàclb` ac ke~ar}`cl na ilemk`ex` z`r` ο ya caa na c` ^@OC@ 6%2 ilgl8
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 479/846
Nr% Fëi~lr Pqa}anl Qrç`y
6'?1
:%:266 5 ο 5 :%:920
&n+ Z`r` zrlo`r c` fkzù~ayky eqc` Fl8ο 7 οl ay naikr# ο 7 :# ile~r` F08ο 5 : { F?8ο 3 :
qy`gly c` nky~rkoqikùe na ~ na ay~qnk`e~a ile η 7 e ’ ? 7 01 ’ ? 7 09 b%c% C` mùrgqc`ay8 ~ 7 &o ’ οl+ , y,∘ Ω v?% Yqy~k~q{aenl ~lnly cly }`clray na οl 7 : { nagày }`clray ae
c` mùrgqc` na `rrko` n`8
~ 7 &:%:16 ’ :+ , :%0?, 09?%29
7 1%6
C`y rabkleay irç~ki`y yle8 ~ 7 ° ?%0=%
Ae ileicqykùe8 naoknl ` pqa ~i`ci%
7 1%6 5 ~~`o%
7 ?%0=# ya raif`x` c` fkzù~ayky eqc` na
Fl8ο 7 : { ya keicke` zlr F08ο 5 :% Ac }`clr na c` zrlo`okckn`n ya i`ciqc` qy`enl c`
mùrgqc` na ke~arzlc`ikùe &='0:+8 &δ? ’ δ0+,&~? ’ ~0+ 7 &δ? ’ V+,&~? ’ ~i`ci%+
Yqy~k~q{aenl cly }`clray `zrlzk`nly na c` ~`oc` na ~ ely n`8
&%::::0 ' %::::?+,&=%?62 ’ 1%=:2+ 7 &%::::0 ’ V+,&=%?62 ’ 1%6+
Cl pqa n` V 7 z 7 %::::?% Zarl ilgl c` zrqao` ay okc`~ar`c# cl gqc~kzcki`gly zlr ? {
n` z 7 %::::>% Ay~a }`clr `zl{`# gq{ ile~qenae~a# c` fkzù~ayky `c~are`~k}` na F08ο 5
:%
&a+ Ac ke~ar}`cl na ilemk`ex` z`r` c` }`rk`oca nazaenkae~a na c` cçea` na rabraykùe
zloc ikle`c# ±[sV ay~kg`n` zlr [ # ile ek}ac na ykbekmki`eik` na ΰ 7 :%:1# n`r }`rkly
}`clray V l% Z`r` f`iar ay~l# ya qy` c` mqeikùe na `o`hl8
[¾ l ' ~Tΰ,?4e'?W ye
0 $& V l ’ V +?,Ω v? 3 ±[sV 3 [¾ ù $ ~Tΰ,?4e'?W ye
0 $& V l ’ V +?,Ω v? &6'?6+
Nlena8 V 7 zrlgankl
~Tΰ,?4e'?W 7 }`clr na ~ ile η 7 e ’ ? b%c%
~T%:?1409W 7 ° ?%0=
V l 7 cly nkmarae~ay }`clray pqa ya ca nae ` V l z`r` iley~rqkr cly cçgk~ay l o`en`y na
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 480/846
Nr% Fëi~lr Pqa}anl Qrç`y
6'?=
ilemk`ex` z`r` ±[sV
@flr` okae# ile cly }`clray na8 ` 7 '0%:>2# V 7 ?>%;2# Ωv? 7 09?%29# y 7 :%0?# ~%:%?1409
7 ° ?%0= { `ykbeàenlca }`clray ` V l# nkb`gly na 0;# ?6# 9:%:# a~i%# ya zrliana na c`ykbqkae~a g`ear`8
Z`r` V l 7 0;%:4 [‛ l 7 '0%:>2 $ :%:16&0;%:+ 7 :%:11# a~i%
Aeyabqkn`# qy`enl c` mùrgqc` &6'?6+ { yqy~k~q{aenl cly }`clray# ay naikr# z`r` V l 7 0;
n`8
%:11’?%0=&:%0?+01
0 $&0;%:'?>%;2+?,09?%29 3 ±[s0; 3 %:11$?%0=&:%0?+01
0 $&0;%:'?>%;2+?,09?%29
Ac iq`c ya ykgzckmki` `8 :%991 5 ±[s0; 5 :%?;;
@yç ya zqana ile~keq`r n`enl nkmarae~ay }`clray na V l { yqy~k~q{ëenlcly# ilgl ya
fkxl `rrko`# z`r`# mke`cgae~a# f`iar c`y o`en`y na ilemk`ex` z`r` ±[sV %
&m+ Z`r` i`ciqc`r cly }`clray na _# _ ?# y { Z_AYY ya zqanae f`iar ile qe z`pqa~a na
ilgzq~`nlr`% Zlr ahagzcl# yk ya f`ia g`eq`cgae~a# ac ilamkikae~a _ ya i`ciqc` qy`enl
c` aiq`ikle &6'0>+# a~i% Na l~r` g`ear`# yk ya qy` ac Gke~`o zrlianar ilgl8
Y~`~ ↟ _abrayykle↟ _abrayykle
Ae c` }ae~`e` na –_ayzleya‗ zlear c` }`rk`oca nazaenkae~a# { ae c` }ae~`e` na
–Zranki~lry‗ zlear c` }`rk`oca kenazaenkae~a% ^`gokëe ya zqanae qy`r c`y }ae~`e`y na
–Br`zfy‗# –Lz~kley‗ { –_ayqc~y‗ z`r` lo~aear kemlrg`ikùe `nkikle`c% Zlr ahagzcl cly
}`clray na c`y ay~`nçy~ki`y loha~k}ky~`y na kemaraeik` n`n`y zlr ac zrlbr`g` yle8
_ ? 7 29%=!# _ 7 :%616# y 7 :%000?# Z_AYY 7 :%??:>% Zlr ahagzcl# ac }`clr na _ 7
:%612= kenki` kenki` qe` ilrrac`ikùe zlyk~k}` pqa }` na `iqarnl ile c` zaenkae~a
zlyk~k}` na c` iqr}` na %:16% Cly }`clray ~`e zapqaņly na y { na Z_AYY kenki`e qe
oqae hqy~a na cly n ~ly c glnacl na rabraykùe%
&b+ Z`r` c` zrqao` na fkzù~ayky Fl8π 7 :# ay naikr# z`r` ac ilamkikae~a na ilrrac`ikùe
zloc ikle`c# π ile ΰ 7 :%:1# ile~r` c` fkzù~ayky `c~are`~k}` na F08≪ :# ay~l ay# F?8π 5 :
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 481/846
Nr% Fëi~lr Pqa}anl Qrç`y
6'?2
{ F98π 3 : ya qy`e c`y ykbqkae~ay ay~`nçy~ki`y8
&0+ Qy`enl c` ay~`nçy~ki` na ~ na Ay~qnk`e~a &6'?1+8
~ 7 ?∟e _ , _?0∟
Nlena8
_ 7 {` namkekn`Z`r` i`ciqc`r c`y rabkleay irç~ki`y ya qy` c` nky~rkoqikùe na ~# ay naikr# ~Tΰ,?4e'?W 7 ~%:?1409 7° ?%0=Ae~leiay# qy`enl c` mùrgqc` na `o`hl { yqy~k~q{aenl cly }`clray n`8
_ 7Ω v{
,∐∐ { v
?? 7 2%2:0 , %=:2>+&09?%29+&: 7 :%6=
{ _ ? 7 :%29;=
@flr`# qy`enl c` ay~`nçy~ki` na `o`hl { yqy~k~q{aenl n`
~ 7 ?∟e _ , _?
0∟
~ 7 09 &:%6=+ , ?=:>% 7 =%:2
Yk ya naya` y`i`r ac }`clr na z ya oqyi` =%:2 ae c` ~`oc` na c` nky~rkoqikùe na ~ ile η 7
09 { ile ΰ 7 %:1# cl pqa n` %:?1 3 z 3 %:1%
&f+ Z`r` br`mki`r cly n`~ly `qe`nly ` c` aiq`ikùe na c` cçea` na rabraykùe ile qe` cçea`
flrkxle~`c ilrrayzlenkae~a `c }`clr nac zrlgankl [ ya f`ia qy`enl qe z`pqa~a na
ilgzq~`nlr`%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 482/846
Nr% Fëi~lr Pqa}anl Qrç`y
6'?6
Mkbqr` 6%=% Bràmki` gly~r`enl c` mqarx` na rlgzkgkae~l &clb0:+ ae mqeikùe nac
nkàga~rl na c` mkor`# ile c` aiq`ikùe na c` ckea` na rabraykùe [ 7 '0%:>= $ :%:16&V+ {
ile ac zrlgankl [ 7 :%>:?% &Ac`olr`ikùe zrlzk`+%&k+ Agk~kr qe hqkikl yqoha~k}l pqa `{qna ` }`ckn`r ac qyl nac glnacl na rabraykùe% Z`r`
rayzlenar ay~` zrabqe~` ya f`iae cly ykbqkae~ay bràmkily8
? > = 6 0: 0? 0>
'?
'0
:
0
?
Loyar}`~kle Lrnar
Y ~ ` e n ` r n k x a n _ a y k n q ` c
_ayknq`cy ]aryqy ~fa Lrnar lm ~fa N`~`
&rayzleya ky Clb mqar+
Mkbqr` 6%2`% Bràmki` gly~r`enl cly rayknqly ay~`en`rkx`nly }aryqy ac lrnae na c`
loyar}`ikùe% Ay~` ay qe` bràmki` pqa gqay~r` ~lnly cly rayknq`cay ae ac lrnae ae aciq`c cly n`~ly mqarle ilcaiikle`nly% @pqç f`{ ac gkygl eøgarl na n`~ly zlyk~k}ly
{ eab`~k}ly% Ay~` bràmki` ~`gokëe ykr}a z`r` aeile~r`r arrlray el `ca`~lrkly#
ayzaik`cgae~a# ae amai~ly rac`ikle`nly ile ac ~kagzl%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 483/846
Nr% Fëi~lr Pqa}anl Qrç`y
6'?;
:%2:%=:%1:%>:%9:%?:%0
?
0
:
'0
'?
Mk~~an ]`cqa
Y ~ ` e n ` r n k x a n _ a y k n q ` c
_ayknq`cy ]aryqy ~fa Mk~~an ]`cqay
&rayzleya ky Clb mqar+
Mkbqr` 6%2o% Ay~à bràmki` gqay~r` cly rayknq`cay }aryqy }`clray `hqy~`nly% Z`r` pqaac glnacl na rabraykùe ya` `iaz~`oca# ya rapqkara pqa8 cly zqe~ly ae c` bràmki` ya`e
`ca`~lrkly ae `goly c`nly na :4 el naoa f`oar yarkay na zqe~ly pqa `qgae~ae l
nkygkeq{`e4 el naoa f`oar zranlgke`eik` na rayknq`cay zlyk~k}ly l eab`~k}ly# ek
~`gzlil naoa f`oar z`~rleay na rayknq`cay pqa `qgae~ae ile }`clray `hqy~`nly
pqa `qgae~ae% Ilgl ya }a# ~ln`y ay~`y ilenkikleay ay~àe okae yqy~ae~`n`y%
'? '0 : 0 ?
'?
'0
:
0
?
E l r g ` c Y i l r a
Y~ en`rnkxan _ayknq c
Elrg`c Zrlo`okck~{ Zc l~ lm ~fa _ayknq`cy
&rayzleya ky Clb mqar+
Mkbqr` 6%2i% Bràmki` gly~r`enl c` zrqao` na elrg`ckn`n% Cly n`~ly naoae mlrg`r
qe` cçea` rai~` yk cly rayknq`cay ay~àe elrg`cgae~a nky~rkoqknly &yk~q`ikùe pqa
liqrra `pqç+% Na l~r` g`ear`# c` yqzlykikùe na elrg`ckn`n ya ke}àckn`%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 484/846
Nr% Fëi~lr Pqa}anl Qrç`y
6'9:
Ilgl ya loyar}` ae ay~`y bràmki`y# c` agkykle na qe hqkikl yqoha~k}l ay `iaz~`oca#
zlrpqa ac glnacl na rabraykùe yacaiikle`nl `hqy~ okae cly n`~ly% Ay~l ya naoa ` pqa#
ae c` Mkbqr` 6%2` f`{ `ca`~lrkn`n ae cly n`~ly# ay naikr# ile ac gkygl eøgarl na}`clray zlyk~k}ly { eab`~k}ly% @nagày# ae c` Mkbqr` 6%2o c` nayirkzikùe na ëy~`#
yqbkara qe glnacl na rabraykùe razrayae~`~k}l na c` kemlrg`ikùe n`n`% Yk~q`ikùe
ykgkc`r liqrra ile c` nayirkzikùe na c` Mkbqr` 6%2i%
Ahagzcl )1% Ae qe ay~qnkl na kebaekarç` nac `bq` rac`ikle`nl ile c`y ranqiikleay na
cly yùcknly yqyzaenknly# ae mqeikùe na c` nag`en` pqçgki` na lvçbael &NPL+# ya
y`iù qe` gqay~r` `ca`~lrk`# iq{ly n`~ly ya n`e ae c` ~`oc` na `o`hl% Z`r` cl ykbqkae~a8
&`+ Knae~kmki`r c` }`rk`oca nazaenkae~a { c` kenazaenkae~a { f`iar qe` bràmki` na NPL
}aryqy ranqiikùe na yùcknly%
&o+ I`ciqc`r c` aiq`ikùe na c` cçea` na rabraykùe%
&i+ F`iar qe` ~`oc` na `eàckyky na }`rk`ex` pqa keicq{` c` M irç~ki` { ac }`clr na z%
&n+ ]`ckn`r ac glnacl i`enkn`~l# ` ~r`}ëy na ay~`nçy~ki`y ilgl _ ?# Z_AYY# y { na c`
ay~`nçy~ki` na Nqroke'R`~yle &z`r` c` zrqao` na q~lilrrac`ikùe na rayknq`cay+%
&a+ A}`cq`r c` q~kckn`n nac glnacl ` ~r`}ëy na bràmkily yqoha~k}ly8
^@OC@ 6%6% ^`oc` gly~r`enl c`y gankikleay na yùcknly { c` nag`en` pqçgki` nalvçbael% &Ac`olr`ikùe zrlzk`+ \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
Yùcknly yqzaenknly NPL \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
9: ?; 99 92 ?1 9? ?; ?2 90 9= ?1 909: 9: 99 9: 91 90 ?; ?6 9? ?; 9: 9:
?; 9: 9> 9: 9= 9: ?6 ?; 9> ?; 9> ?;
9> 90 9= ?; 90 9: 99 9: 91 ?6 9: ?6?6 90 9= ?6 99 9? ?= 9: 9> ?6 9: 90?2 9? 9= ?2 90 9? ?2 9? 9> ?= ?; 90
Ylcqikùe8
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 485/846
Nr% Fëi~lr Pqa}anl Qrç`y
6'90
&`+ C` }`rk`oca nazaenkae~a ay NPL { c` }`rk`oca kenazaenkae~a ay ranqiikùe na
yùcknly yqyzaenknly% C` mkbqr` 6%6 na `o`hl gqay~r` c`y ileiae~r`ikleay na NPL
}aryqy ranqiikùe na yùcknly yqyzaenknly%
?2 9? 92
?1
9:
91
Ylc knly yqyzaenk nly &V+
N P L & [
+
Mkbqr` gly~r`enl c` br`mki` na NPL { ylcknly yqyzaenkly%
Mkbqr` 6%6% Bràmki` gly~r`enl ac NPL }aryqy ranqiikùe na yùcknly%
&Ac`olr`ikùe zrlzk`+
&o+ C` aiq`ikùe na c` cçea` na rabraykùe ay8
NPL &[+ 7 0%19 $ :%;:; V&yùcknly yqyzaenknly+
C` zaenkae~a ay kbq`c ` :%;:; { ac ke~ariaz~l ay 0%19
&i+ C` ~`oc` na `o`hl gqay~r` c` kemlrg`ikùe na @EL]@%
^@OC@ 6%;% ^`oc` na @EL]@ na yùcknly yqyzaenknly { NPL%Mqae~a na YY b%c% GY Mi`ci% Mirç~ki` ]`clr na z]`rk`ikùe
Ae~ra cly brqzly 9?%:: 0 9?%:: >%91 9%;6 :%:>_ayknq`c &arrlr+ 101%>> 2: 2%91^l~`c 1>=%>> 20
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
&n+ y 7 :%;:9; _ ? 7 66%6! _ ?&`hqy~`n`+ 7 66%1!Z_AYY 7 90%6;?6 _ ?&zranaikn`+ 7 62%09! Nqroke'R`~yle y~`~ky~ki 7 0%=2
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 486/846
Nr% Fëi~lr Pqa}anl Qrç`y
6'9?
@pqç# ac ilamkikae~a na na~argke`ikle _ ?# gkna# pqë ~`e okae ac glnacl na
rabraykùe `hqy~` cly n`~ly% @eàclb`gae~a# ac ay~`nçy~kil Z_AYY &yqg` na
iq`nr`nly na arrlr na zrankiikùe+ gkna c` i`ckn`n nac glnacl na rabraykùe% Aeiq`e~l ` c` ay~`nçy~ki` Nqroke'R`~yle# yk ay~à iari`e` ` ? el f`{ `q~lilrrac`ikleay
ae yarkay zlyk~k}`y l eab`~k}`y% C` }`rk`ikùe na cly n`~ly c` n` c` ay~`nçy~ki` y%
&a+ C` Mkbqr` 6%; n` c` kemlrg`ikùe yqoha~k}` z`r` c` a}`cq`ikùe nac glnacl%
&`+
?1 9: 91
'>
'9
'?
'0
:
0
?
Mk~~an ]`cqa
Y ~ ` e n ` r n k x a n _ a y k n q ` c
_ayknq`cy ]aryqy ~fa Mk~~an ]`cqay
&rayzleya ky NPL &[++
&o+
'> '9 '? '0 : 0 ?
'?
'0
:
0
?
E l r g ` c Y i l r a
Y~ en`rnkxan _ayknq c
Elrg`c Zrlo`okck~{ Zc l~ lm ~fa _ayknq`cy
&rayzleya ky NPL &[++
Mkbqr` 6%;% C` mkbqr` &`+ zrqao` zlr c` `q~lilrrac`ikùe l m`c~` na kenazaenaeik` na cly
n`~ly% @nagày# c` mkbqr` &o+ zrqao` zlr c` elrg`ckn`n na cly n`~ly%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 487/846
Nr% Fëi~lr Pqa}anl Qrç`y
6'99
_abraykùe { ilrrac`ikùe ckea`c gøc~kzca
Gqif`y `zcki`ikleay nac `eàckyky na rabraykùe ke}lcqir`e yk~q`ikleay nlena ya ~kaea
gày na qe` }`rk`oca kenazaenkae~a% Ae c` g`{lr z`r~a na cly zrlocag`y nake}ay~kb`ikùe ya eaiayk~`e }`rk`y }`rk`ocay kenazaenkae~ay z`r` }ar ac amai~l ae c`
}`rk`oca nazaenkae~a% C` }`rk`oca nazaenkae~a l na rayzqay~` &[ + zqana ay~`r
rac`ikle`n` ile gqif`y }`rk`ocay kenazaenkae~ay l rabraylr`y V 0# V ?# a~i%
Ae ac ay~qnkl na rabraykùe ckea`c gøc~kzca ya zqanae qy`r ac aemlpqa g`~rkik`c%
^`gokëe ya zqanae f`iar zrqao`y na fkzù~ayky# ke~ar}`cly na ilemk`ex`# `eàckyky
yqoha~k}ly &`eàckyky na cly bràmkily+ { `eàckyky loha~k}ly &ay~`nçy~ki` na kemaraeik`+#
ilgl cly iàciqcly na cly ilamkikae~ay na na~argke`ikùe &_ ?+ l na ilrrac`ikùe &_+#
ilgl ae ac i`yl na c` rabraykùe ckea`c ykgzca% Yke ago`rbl# ae ay~a i`yl# ya zqana
i`ciqc`r ac ilamkikae~a na ilrrac`ikùe baear`c { ilamkikae~ay na ilrrac`ikùe z`rik`cay#
ay naikr# ae mlrg` `eàclb` ` ilgl ya f`ia ile cly ilamkikae~ay οl# ο0# a~i%
Iq`enl f`oc`gly na rabraykùe ckea`c gøc~kzca ~aeagly c`y ykbqkae~ay
yk~q`ikleay8
0% Glnacl na zrkgar lrnae ile nly }`rk`ocay rabraylr`y l kenazaenkae~ay%
?% Glnacl na zrkgar lrnae ile gày na nly }`rk`ocay kenazaenkae~ay%
Glnacl na rabraykùe gøc~kzca baear`ckx`nl
Iq`enl ay~a glnacl baear`c ay ckea`c ae cly ilamkikae~ay ya naelgke` glnacl na
rabraykùe gøc~kzca% Zlr ahagzcl# z`r` ac i`yl na d }`rk`ocay kenazaenkae~ay v0# v?#
v9#%%%# vd # ac zrlgankl ay~à n`nl zlr [sv0# v?# v9#%%%# vd { ya n` zlr ac glnacl na
rabraykùe gøc~kzca zloc`ikle`c8[ 7 ±[ s v0# v?# v9#%%%# vd 7 οl $ ο0v0 $ ο? v? $ %%%$ οd vd $ αd &6'?;+
Ay~a glnacl# ~`gokëe ya zqana avzray`r ile l~r` `el~`ikùe ilgl8
[ h 7 οl $ ο0 V 0h $ ο? V ? h $ ……% $ οd V dh $ α h &6'?;`+
Cly z`ràga~rly ο h# h 7 :# 0# ?# 9#%%# d ya ileliae ilgl ilamkikae~ay na rabraykùe
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 488/846
Nr% Fëi~lr Pqa}anl Qrç`y
6'9>
zloc`ikle`cay% Zlr ahagzcl# ac z`ràga~rl ο h razrayae~` ac i`gokl ayzar`nl ae c`
rayzqay~` [ # zlr qekn`n na i`gokl ae v h# iq`enl ~lnly cly nagày zrlely~ki`nlray vk
ya g`e~kaeae iley~`e~ay% @nagày# αk { ak yle cly arrlray `ca`~lrkly l rayknqly na zloc`ikùe { na c ay~`nçy~ki` `ylik`nly ile c` rayzqay~` [ k%
Ac glnacl na rabraykùe ckea`c gøc~kzca na c` gqay~r` pqa ay~kg` `c glnacl
zloc`ikle`c na `rrko` ay8
[ 7 ol $ o0 V 0 $ o?v? $ %%% $ od V d $ a &6'9:+
Nlena i`n` ilamkikae~a na rabraykùe z`rik`c οk ay ay~kg`nl zlr ok% Ay~l ya naoa `
pqë# i`n` ilamkikae~a z`rik`c οk gkna ac i`gokl ayzar`nl ae [ zlr qekn`n na i`gokl
ae v0# iq`enl v? ya g`e~kaea iley~`e~a# { ο? gkna ac i`gokl ayzar`nl ae [ zlr
qekn`n na i`gokl ae v? iq`enl v0 ya g`e~kaea iley~`e~a%
Ac glnacl na zrkgar lrnae ile nly }`rk`ocay kenazaenkae~ay ay8
[ k 7 οl $ ο0 V k0 $ ο? V k? $ α &6'90+
Nlena [ k# c` }`rk`oca nazaenkae~a pqa nael~` c` rayzqay~` ae c`y 'ëykg`y ~ae~`~k}`y4
V k0 { V k? yle c`y nly }`rk`ocay kenazaenkae~ay na c` 'ëykg` ~ae~`~k}`4 οl# ο0# ο? yle cly
ilamkikae~ay na rabraykùe {# α ay ac arrlr l rayknql%
Glnacl na rabraykùe gøc~kzca ile gày na nly }`rk`ocay kenazaenkae~ay
[ k 7 οl $ ο0 V k0 $ ο? V k? $ … $ ο z'0 V k#z'0 $ α &6'9?+
Iq`enl f`oc`gly na rabraykùe ckea`c gøc~kzca# ac zrkeikz`c loha~k}l ay c` lo~aeikùe
na c` aiq`ikùe na c` cçea` na rabraykùe gqay~r`c# z`r` zrankiikùe { ay~kg`ikùe# c`
iq`c agqc` ` c` aiq`ikùe zloc`ikle`c% Yke ago`rbl# `e~ay na zlnar qy`r ac glnacl
na rabraykùe i`ciqc`nl# ëy~a ya ~kaea pqa a}`cq`r# z`r` }ar pqë ~`e~` ilemk`okckn`n yaca zqan` n`r% C` a}`cq`ikùe l }`ckn`ikùe nac glnacl na rabraykùe ay~kg`nl ya f`ia
` ~r`}ëy na `eàckyky loha~k}ly { yqoha~k}ly# ae mlrg` `eàclb` ilgl ae c` rabraykùe
ckea`c ykgzca% Zlr ahagzcl# cly `eàckyky loha~k}ky~`y ya f`iae ` ~r`}ëy na mqeikleay
ay~`nçy~ki`y na kemaraeik`% Zly~arklrgae~a# z`r` pqa c` }`ckn`ikùe nac glnacl ya`
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 489/846
Nr% Fëi~lr Pqa}anl Qrç`y
6'91
ilgzca~`# ac zrliankgkae~l ya ilgzcagae~` qy`enl aemlpqay yqoha~k}ky~`y# ` ~r`}ëy
na `eàckyky na c`y bràmki`y na cly }`clray rayknq`cay% Yk c` }`ckn`ikùe el ay
y`~kym`i~lrk`# ya zrliana ile ragank`ikùe nac glnacl# {` ya` f`ikaenl~r`eymlrg`ikleay na cly ahay l zrlo`enl l~rly glnacly gày `zrlzk`nly# ilgl
iq`nrà~kily l iøokily# a~i%
@zcki`ikùe na `eàckyky yqoha~k}ly { loha~k}ly z`r` c` a}`cq`ikùe nac glnacl na
rabraykùe
Ilgl ya f` ay~`nl gaeikle`enl `e~arklrgae~a# ya yqbkarae nly g`ear`y na ra}ky`r
c` q~kckn`n nac glnacl lo~aeknl% Ay~`y g`ear`y yle8 &0+ `eàckyky na bràmki`y na
rayknqly {# &?+ zrqao`y ay~`nçy~ki`y na kemaraeik`%
Zlr ahagzcl# z`r` }`ckn`r ac glnacl na rabraykùe `zcki`enl `eàckyky
yqoha~k}ly# ay naikr# ` ~r`}ëy na cly bràmkily na cly rayknqly &ak+# ëy~ly ya nayirkoae
ilgl c`y nkmaraeik`y ae~ra cly zqe~ly { c` cçea` na rabraykùe% Ykaenl `yç# c`y
yqzlykikleay yle na pqa cly rayknqly naoae yar kenazaenkae~ay { elrg`cgae~a
nky~rkoqknly# ile zrlgankl kbq`c ` iarl { ile }`rk`ex`y iley~`e~ay% Gày
avzcçik~`gae~a# c`y nayirkzikleay na c`y yqzlykikleay yle8
0% Cly }`clray na c` }`rk`oca `ca`~lrk` ay~`nçy~ki` ak naoae ay~`r elrg`cgae~a
nky~rkoqknly% Z`r` clbr`r ay~l# ya br`mki`e cly rayknqly &irqnly l ay~`en`rkx`nly+ na
c` }`rk`oca nazaenkae~a ae mqeikùe na cly }`clray na x l elrg`cay ayzar`nly% Z`r`
pqa ya raøe` c` ilenkikùe na elrg`ckn`n na cly n`~ly# ~lnly cly zqe~ly naoae na
ay~`r nae~rl na c`y o`en`y na ilemk`ex` { naoae na ay~`r gq{ iari`ely ` c` cçea` na
rabraykùe% @nagày# yk cly ~ërgkely nac arrlr ak ay~àe elrg`cgae~a nky~rkoqknly# clyrayknq`cay ay~`en`rkx`nly l irqnly naoaràe ay~`r# `zrlvkg`n`gae~a# na `iqarnl
ile c`y rabc`y nac =6!# ;>! { ;;!% Ay~l pqkara naikr pqë# ac =6! na cly rayknqly
naoaràe ay~`r ae~ra x 7 °04 ac ;1! naoaràe ay~`r ae~ra x 7 °? {# mke`cgae~a# ac ;;!
na cly rayknqly naoaràe ay~`r ae~ra x 7 °9%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 490/846
Nr% Fëi~lr Pqa}anl Qrç`y
6'9=
?% Cly }`clray na c` }`rk`oca `ca`~lrk` ak naoae yar kenazaenkae~ay qel nac l~rl% El
naoa f`oar ilckek`ckn`n l ilrrac`ikùe ae yarka% Ay~l ya ra}ky` br`mki`enl cly
rayknqly &ay~`en`rkx`nly l irqnly+ ae mqeikùe na cly raebcleay% Yk el f`{#`zrlvkg`n`gae~a# cly gkygly rayknqly zlyk~k}ly { eab`~k}ly ae c` bràmki`#
ae~leiay# ac glnacl ckea`c i`ciqc`nl el ay ac `zrlzk`nl { ~aenràe pqa oqyi`rya
l~r`y `c~are`~k}`y &ilgl mqeikleay zlckelgk`cay# iq`nrà~ki`y# iøoki`y# a~i%+% @pqç
i`oa el~`r pqa c` yqzlykikùe na kenazaenaeik` ay c` gày kgzlr~`e~a pqa ya zqan`
}klc`r# zlrpqa ay c` o`ya z`r` c`y zrqao`y ay~`nçy~ki`y ilgl c` _ ?# ac arrlr na cl
ay~kg`nl &y n`nl zlr ac zrlbr`g` Gkek~`o+# @EL]@# a~i%
9% Cly }`clray na c` }`rk`oca `ca`~lrk` ak naoae na ~aear c` gkyg` }`rk`ex`% Ay~l ya
cc`g` flglyian`y~kikn`n% Ay~l ya zqana ra}ky`r }kyq`cgae~a br`mki`enl cly
rayknq`cay ay~`en`rkx`nly l el ay~`en`rkx`nly &irqnly+ ile~r` i`n` }`clr na c`y
}`rk`ocay kenazaenkae~ay & V k+% @pqç# eqa}`gae~a# ~kaea pqa f`oar c` gkyg`
i`e~kn`n na }`clray zlyk~k}ly { eab`~k}ly avzray`nly ae c` bràmki`% @pqç# yke
ago`rbl# avky~ae l~rly gë~lnly z`r` ra}ky`r zlr ac zrlocag` na
fa~arlyian`y~kikn`n pqa ya ra~lg`ràe ae ac i`zç~qcl na rabraykùe zlckelgk`c%
L~rly ke}ay~kb`nlray ay~`nçy~kily &Na}lra# ?:::+ yqbkarae iq`~rl bràmkily na
nk`beùy~kil yqoha~k}l# z`r` c` }`ckn`ikùe nac glnacl na rabraykùe gøc~kzca% Ay~ly
bràmkily na nk`beùy~kil yle8
0% Ac bràmkil na cly rayknqly ay~`en`rkx`nly {,l irqnly ae c` lrnae`n` }aryqy cly
}`clray na V k ae c` `oyiky`%
?% Ac bràmkil na cly rayknqly ay~`en`rkx`nly {,l irqnly ae c` lrnae`n` }aryqy cly}`clray zrlely~ki`nly &ae c` `oyiky`+ zlr ac zrlbr`g` na ilgzq~`nlr` qy`nl%
9% Ac bràmkil na cly }`clray zrlely~ki`nly ae c` lrnae`n` }aryqy cly }`clray na [ k ae
c` `oyiky`%
>% Bràmkil na elrg`ckn`n na cly rayknqly ay~`en`rkx`nly }aryqy cly zariae~kcay na x
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 491/846
Nr% Fëi~lr Pqa}anl Qrç`y
6'92
&}`clray na x+%
1% Fky~lbr`g`y%
@zcki`ikùe na `eàckyky loha~k}ly z`r` c` a}`cq`ikùe nac glnacl na rabraykùe
Zlr l~rl c`nl# ae iq`e~l `c aemlpqa loha~k}ky~` &ay~`nçy~ki` kemaraeik`c+ z`r` c`
}`ckn`ikùe nac glnacl na rabraykùe# ëy~a ay~à rac`ikle`nl ile ac qyl na ay~`nçy~ki`y
ilgl ac ilamkikae~a na na~argke`ikùe gøc~kzca _ ? &l r ?+# ac ilamkikae~a na
na~argke`ikùe `hqy~`nl _ ?`hqy~`n`# ac arrlr ay~àen`r na cl ay~kg`nl# y# ~`oc`y na
`eàckyky na }`rk`ex`# zrqao`y na ~ na Ay~qnk`e~a# ke~ar}`cly na ilemk`ex`# ac irk~arkl
na G`ccl na Iz# Z_AYY# a~i%
Na ay~` g`ear`# iq`enl ya f`oc` na ilamkikae~ay ae ac glnacl na rabraykùe
gøc~kzca# avky~ae iq`~rl ~kzly na ilamkikae~ay8
&0+ Ac ilamkikae~a na na~argke`ikùe gøc~kzca &_ ?+
&?+ Ac ilamkikae~a na ilrrac`ikùe gøc~kzca &_+
&9+ Ac ilamkikae~a na na~argke`ikùe `hqy~`nl &_ ?`hqy~`n`+
&>+ Ac ilamkikae~a z`rik`c na ilrrac`ikùe gøc~kzca & _kh%d +
Zlr ahagzcl# ac ilamkikae~a na na~argke`ikùe gøc~kzca _ ? ay# ~`c }ax# c`
gankn` ay~`nçy~ki` gày zlzqc`r qy`n` z`r` gankr# pqë ~`e okae aei`h` ac glnacl na
rabraykùe ae cly n`~ly na c` gqay~r`% Ae ra`ckn`n ac qyl na _ ? ay qe` ~ëieki` z`r`
gankr c` `naiq`ikùe na qe glnacl na rabraykùe ckea`c gøc~kzca% Ay~` ay~`nçy~ki` ya
zqana namkekr ilgl qe` zrlzlrikùe l ilgl qe zlriae~`ha% Ilgl zrlzlrikùe# yqy
}`clray }`rç`e na iarl ` qel% Zlr ahagzcl# yk ac }`clr na _ ? ay~à iari`el ` iarl# ay~l
kenki` pqa el f`{ qe` rac`ikùe ckea`c ae~ra [ { c`y V ¾y# gkae~r`y pqa# qe }`clr iari`el ` qel# kenki` qe` `hqy~a zarmai~l% Yke ago`rbl# ac }`clr na _ ? el naoa na
ke~arzra~`rya ckbar`gae~a# yke ac `zl{l nac arrlr ay~àen`r na cl ay~kg`nl &y+# ac
rayknq`c &Z_AYY+# ac irk~arkl na G`ccl &Iz+ l cly m`i~lray na }`rk`ikùe kemc`nly
&}`rk`eia kemc`~kle m`i~lry# ]KM+% @nagày c` }`ckn`ikùe nac glnacl naoa ay~`r
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 492/846
Nr% Fëi~lr Pqa}anl Qrç`y
6'96
`zl{`n` zlr cly `eàckyky na cly bràmkily yqoha~k}ly%
Na `iqarnl ` c` cùbki` nac zrlbr`g` na EIYY# cly ykbqkae~ay aeqeik`nly n`e
`cbqe`y i`ckmki`ikleay na c` ke~arzra~`ikùe na _ ?
%0% Ac }`clr na _ ? zqana keiragae~`rya `brab`enl gày }`rk`ocay kenazaenkae~ay#
zarl ay~l zqana i`qy`r qe `qgae~l ae ac arrlr nac iq`nr`nl gankl# ayzaik`cgae~a#
iq`enl c` gqay~r` ay zapqað`%
?% C` g`bek~qn na _ ? ay~à kemcqaeik`n` zlr ac r`ebl na i`n` }`rk`oca kenazaenkae~a%
_ ? `qgae~` ` gankn` pqa ac r`ebl na c`y V ¾y `qgae~` { }kia}ary`%
9% Ac }`clr na _ ? el gkna c` g`bek~qn na c`y zaenkae~ay%
>% C` g`bek~qn na _ ? el gkna c` `z~k~qn nac glnacl ckea`c4 gkna c` mqarx` ckea`c nac
ilgzleae~a nac glnacl%
1% Qe }`clr br`ena na _ ? el eaiay`rk`gae~a ykbekmki` qe` zrankiikùe br`ena% Cl
lzqay~l ~`gokëe ay ilrrai~l% ^lnl ay~l ~kaea pqa yar ilgzcagae~`nl l
ilrrlolr`nl zlr l~r`y mqeikleay ay~`nçy~ki`y { zlr ac `eàckyky bràmkil yqoha~k}l%
=% Ac }`clr na _ ? ay `c~`gae~a yaeykoca `c eøgarl na loyar}`ikleay% Ae~ra gày
br`ena ya` ac ~`g`ðl na c` gqay~r`# gày `c~l yarà ac }`clr na _ ?% Gày `nac`e~a# f`{
cl pqa ya cc`g` ac }`clr `hqy~`nl nac ilamkikae~a na na~argke`ikùe gøc~kzca `hqy~`nl
&_ ?`hqy~`n`+% Ay~a ilamkikae~a na na~argke`ikùe gøc~kzca `hqy~`nl _ ?`hqy~`n` ay qe`
}arykùe `hqy~`n` na _ ? c` iq`c oqyi` ragl}ar c` nky~lrykùe i`qy`n` zlr qe ~`g`ðl
na gqay~r` zapqaðl% Kbq`cgae~a# ~`gokëe f`{ cl pqa ya cc`g` Z_AYY &zranki~an
yqg lm ypq`ray+ pqa ya qy` z`r` }`ckn`r ac glnacl na rabraykùe ae ~ërgkely na
zrankiikùe% @pqç# ae~ra gày zapqaðl ya` ac }`clr na Z_AYY# gahlr yarà ac glnacli`enkn`~l%
Ae mlrg` `eàclb`# ~`gokëe f`{ cl pqa ya cc`g` ac ilamkikae~a na ilrrac`ikùe
gøc~kzca _% Ay~a ilamkikae~a _ gkna c` mqarx` na c` rac`ikùe ckea`c ae~ra c` }`rk`oca
nazaenkae~a [ { c`y }`rk`ocay kenazaenkae~ay V 0# V ? # V 9 #…# V d % Ae ile~r`y~a ile ac
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 493/846
Nr% Fëi~lr Pqa}anl Qrç`y
6'9;
ilamkikae~a na ilrrac`ikùe ckea`c ykgzca# ac r`ebl na ay~a ilamkikae~a na ilrrac`ikùe
gøc~kzca ay na : ≮ _ ≮ 0% Ay~l ya naoa ` pqa _ el kenki` c` zaenkae~a na c`
aiq`ikùe na rabraykùe naoknl ` pqa el ay zlykoca kenki`r cly ykbely na ~lnly clyilamkikae~ay na rabraykùe pqa rac`ikle`e c` }`rk`oca nazaenkae~a [ ` c`y }`rk`ocay
kenazaenkae~a V k% @yç ilgl ae ac i`yl na c` ilrrac`ikùe ckea`c# c` gankikùe na _ ? ay
gày màikc na ke~arzra~`r pqa ac ilamkikae~a na ilrrac`ikùe gøc~kzca# _%
L~rl ~kzl na ilrrac`ikùe rac`ikle`nl ile rabraykùe { ilrrac`ikùe gøc~kzca ay
cl pqa ya cc`g` ilamkikae~a z`rik`c na ilrrac`ikùe gøc~kzca% Ay~a ilamkikae~a gkna c`
mqarx` na c` rac`ikùe ckea`c ae~ra c` }`rk`oca nazaenkae~a [ { c`y }`rk`ocay
kenazaenkae~ay V 0 # V ? # V 9 #…# V d % Ay~a ilamkikae~a ya zqana avzray`r ilgl _ kh%d ac
iq`c ay ac ay~kg`nlr nac ilamkikae~a na ilrrac`ikùe gøc~kzca zloc`ikle`c πkh%d % _ kh%d ya
zqana qy`r z`r` }ar c` rac`ikùe i`qy`c ae~ra [ { qe` na c`y }`rk`ocay kenazaenkae~ay#
g`e~aekaenl c`y nagày iley~`e~ay% Ay~a ilamkikae~a# ~`gokëe ya zqana qy`r z`r` }ar
c` rac`ikùe ae~ra nly }`rk`ocay kenazaenkae~ay%
Gày `nac`e~a# nae~rl na c` i`~ablrç` na `eàckyky loha~k}ly na ay~`nçy~ki`
kemaraeik`c rac`ikle`nly ile rabraykùe gøc~kzca# ~aeagly cl pqa ya cc`g` `eàckyky na
}`rk`ex` &@EL]@+ nkyiq~knl ae i`zç~qcly `e~arklray% Ae mlrg` `eàclb` ilgl ac
qyl na _ ?# ay~a `eàckyky ay qe gë~lnl ilgzcagae~`rkl z`r` ra}ky`r c`y yqzlykikleay
nac glnacl na rabraykùe% C` ilemk`okckn`n na cly rayqc~`nly nac @EL]@ ay~à
g`eilgqe`n` ` c` yqzlykikùe na pqa cly rayknq`cay ay~àe elrg`cgae~a
nky~rkoqknly% Ac qyl na @EL]@ zrqao` cly zrlgankly zloc`ikle`cay nlena ya
`e`ckx` c` }`rk`ikùe ~l~`c% @EL]@ a}`cø` c` q~kckn`n nac glnacl na rabraykùe zrlo`enl c` fkzù~ayky eqc` na pqa ~lnly cly ilamkikae~ay &οk+ na c` aiq`ikùe na
rabraykùe &zaenkae~ay+ yle kbq`c ` iarl% Cly ilgzleae~ay nac `eàckyky na }`rk`ex` l
na @EL]@# yle z`raiknly ` cly nac `eàckyky na }`rk`ex` ykgzca avzcki`nly ae
i`zç~qcly `e~arklray% Cly ilgzleae~ay yle c` mqae~a na }`rk`ikùe# cly br`nly na
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 494/846
Nr% Fëi~lr Pqa}anl Qrç`y
6'>:
ckoar~`n# c` yqg` na cly iq`nr`nly# ac iq`nr`nl nac zrlgankl# c` zrqao` na M { ac
ek}ac na zrlo`okckn`n% Zlr ahagzcl# c` mqae~a na }`rk`ikùe razrayae~` c`y z`r~kikleay
na c` }`rk`ikùe ae [ % F`{ iq`~rl mqae~ay na }`rk`ikùe ay naikr# ac ke~ariaz~l# acglnacl# ac rayknql l arrlr {# ac ~l~`c `hqy~`nl% C` zrqao` na kemaraeik` ile c`
ay~`nçy~ki` M ya qy` z`r` zrlo`r c` fkzù~ayky na ~ln`y c`y οk 7 :%
Gày kgzlr~`e~a ~ln`}ç`# ay ac iàciqcl nac ek}ac na zrlo`okckn`n z% Ac }`clr na
z ay c` zrlo`okckn`n na lo~aear qe ay~`nçy~kil na zrqao`# `c gaely ~`e
ile~r`nki~lrkl l gày av~ragl z`r` Fl8# ilgl ac }`clr loyar}`nl pqa ya lo~q}l#
`yqgkaenl pqa Fl8 ay }arn`nar`% Yk ac }`clr na z ay gaelr pqë# nkb`gly ΰ 7 :%:1#
c` fkzù~ayky eqc` ya raif`x`4 na l~r` g`ear` ya ra~kaea% Ae~ra gày zapqaðl ya` ac
}`clr na z# gaely irankokckn`n ~aenrà c` fkzù~ayky eqc`%
L~rly ay~`nçy~kily loha~k}ky~`y z`r` }`ckn`r ac glnacl na rabraykùe yle c`y
zrqao`y kenk}knq`cay na ~ na ay~qnk`e~a z`r` zrlo`r c` fkzù~ayky na pqa ο0# ο?# ο9# οd
yle kbq`cay ` iarl% @nagày ya zqanae qy`r cly ke~ar}`cly na ilemk`ex`% Zlr
ahagzcl# ae rabraykùe gøc~kzca ac }`clr na ~ na ay~qnk`e~a ya qy` z`r` zrlo`r c`
fkzù~ayky na pqa qel na cly ilamkikae~ay ay kbq`c ` iarl# nayzqëy na ragl}ar c`
kemcqaeik` na cly l~rly% Cly ke}ay~kb`nlray Z`mmaeoarbar a~ `c% &0;62+ n`e c` mqeikùe
z`r` ac ke~ar}`cl na ilemk`ex` z`r` οk% Yke ago`rbl# yk ya ileicq{a pqa ο0 l οd el
yle kbq`c ` iarl ay~l# el eaiay`rk`gae~a# nkia pqa ac glnacl na rabraykùe ay ø~kc
z`r` zrankiikùe% Ae }arn`n# z`r` na~argke`r yk ac glnacl ay `zrlzk`nl# ae cqb`r na
zrlo`r pqa ο0 7 : { ο? 7 :# yaz`r`n`gae~a &qy`enl c` zrqao` na ~+# ya qy` qe`
zrqao` ilehqe~` ilgl ac `eàckyky na }`rk`ex` &@EL]@+% Na iq`cpqkar g`ear`# c` zrqao` na fkzù~ayky okc`~ar`c z`r` zrlo`r cly ilamkikae~ay kenk}knq`cay οk ya qy` ac
ykbqkae~a mlrg`~l n`nl ae c` ~`oc` na `o`hl%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 495/846
Nr% Fëi~lr Pqa}anl Qrç`y
6'>0
^@OC@ 6%0:% ^`oc` gly~r`enl cly gai`ekygly z`r` qe` zrqao` na fkzù~ayky
okc`~ar`c z`r` cly ilamkikae~ay kenk}knq`cay οk keicqknly ae ac glnacl na rabraykùe
gøc~kzca% &Ac`olr`ikùe zrlzk`+Fkzù~ayky eqc`8 Fl8οk 7 :# fkzù~ayky `c~are`~k}`8 F08οk ≪ :
]`clr nac ay~`nçy~kil8 ~ 7 ok , y ok
_abc` na naikykùe8
_aif`x`r Fl8 yk ~ 5 ~ΰ,?4e'&d$0+ l okae yk ~ 3 '~ΰ,?4e'&d$0+%
El raif`x`r Fl8 yk ~ΰ,?4e'&d$0+ ≮ ~ ≮ ~ΰ,?4e'&d$0+
Nlena8 οk yle cly ilamkikae~ay na rabraykùe kenk}knq`cay%ok 7 ay~kg`nlray na οk
yok 7 arrlray ay~àen`r
ΰ 7 ek}ac na ykbekmki`eik` naya`nl
e 7 eøgarl na loyar}`ikleay
d 7 eøgarl na }`rk`ocay kenazaenkae~ay
~ 7 mqeikùe ay~`nçy~ki` na ~ na Ay~qnk`e~a
Ahagzcly `zcki`enl c` rabraykùe { ilrrac`ikùe gøc~kzca
Ahagzcl )=% Ae c` `nylrikùe na ~karr` { yankgae~l# c` g`bek~qn na c` `iqgqc`ikùe
ae mlrg` ilenaey`n` na cly zrlnqi~ly pqçgkily ae c` yqzarmkika ay qe`
i`r`i~arçy~ki` kgzlr~`e~a pqa kemcq{a ae c` amkikaeik` na keyai~kikn`y { }`rkly l~rly
zrlnqi~ly pqçgkily% Ac `r~çiqcl –@nylrz~kle lm Zflyzf`~a# @ryae`~a#
Ga~f`ea`ryle`~a `en I`iln{c`~a o{ C`da `en Y~ra`g Yankgae~y8 Ilgz`rkyle k~f
Ylkcy‗ &H% lm Ae}krle% Pq`c%# 0;6># zz% >;;'1:>+ zrayae~` cly ykbqkae~ay n`~ly ae c`~`oc` na `o`hl% @pqç ya ~lg` [ ilgl c` }`rk`oca nazaenkae~a# c` iq`c nael~` ac
çenkia na `nylrikùe na mlym`~l# V 0 ay qe` na c`y }`rk`ocay kenazaenkae~ay
nael~`enl c` i`e~kn`n na fkarrl av~r`çoca {# V ? ay l~r` na c`y }`rk`ocay
kenazaenkae~ay nael~`enl c` i`e~kn`n na `cqgkekl av~r`çoca% &Na}lra# ?:::+
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 496/846
Nr% Fëi~lr Pqa}anl Qrç`y
6'>?
^@OC@ 6%00% ^`oc` gly~r`enl cly n`~ly nac ahagzcl% \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
Loyar}`ikùe V 0 &Fkarrl av~r`çoca+ V ? &@cqgkekl av~r`çoca+ [ &Çenkia na `nylrikùe+ \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ 0 =0 09 >? 021 ?0 069 000 ?> 0>> 0?> ?9 061 09: => ?== 029 96 ?=2 0=; 99 ?06 0=; =0 9:
; 0=: 9; ?60: ?>> 20 9=00 ?12 00? =10? 999 66 =?09 0;; 1> >:
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
&Mqae~a8 Na}lra# ?:::+
F`iar cly iàciqcly zar~keae~ay%
Ylcqikùe8
Qy`enl qe z`pqa~a na ilgzq~`nlr` n`8 ol 7 '2%910# nay}k`ikùe ay~àen`r 7 9%>61#
o0 7 :%00?29# nay}k`ikùe ay~àen`r 7 :%:?;=;# o? 7 :%9>;::# y 7 :%:2090
C` aiq`ikùe na c` cçea` na rabraykùe ckea`c gøc~kzca ay8
[ 7 '2%910 $ &:%00?29+& V 0+ $ &:%9>;::+& V ?+
Aeyabqkn`# z`r` }ar# pqë ~`e ilemk`oca ay ac glnacl na rabraykùe i`ciqc`nl#
zrkgarl zrlianagly ` amai~q`r ac `eàckyky yqoha~k}l# ay naikr# ac `eàckyky na c`y
bràmki`y na cly rayknqly%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 497/846
Nr% Fëi~lr Pqa}anl Qrç`y
6'>9
Mkbqr` 6%0: Mkbqr` gly~r`enl c`y bràmki`y na cly rayknqly ay~`en`rkx`nly }aryqy
}`clray ayzar`nly na x &0+4 bràmki` gly~r`enl ac rayknql ay~`en`rkx`nl }aryqy c`}`rk`oca kenazaenkae~a V 0 &?+4 bràmki` gly~r`enl ac rayknql ay~`en`rkx`nl }aryqy c`
}`rk`oca kenazaenkae~a V ? &9+4 bràmki` gly~r`enl ac rayknql ay~`en`rkx`nl }aryqy ac
}`clr na [ zrlely~ki`nl &>+ {# mke`cgae~a# bràmki` na [ zrlely~ki`n` }aryqy
`nylrikùe &1+% &Ac`olr`ikùe zrlzk`+
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 498/846
Nr% Fëi~lr Pqa}anl Qrç`y
6'>>
Mkbqr` 6%00 Ay~` bràmki` gqay~r` qe aemlpqa qe zlil nkmarae~a `c na c` mkbqr`
`e~arklr# ay naikr qy`enl cly rayknqly el ay~`en`rkx`nly ae ile~r`y~a ile c` mkbqr`
6%0: pqa qy` cly rayknqly ay~`en`rkx`nly% Bràmki` gly~r`enl c` zrqao` na
elrg`ckn`n &0+% Bràmki` gly~r`enl c` zrqao` na kenazaenaeik` na rayknqly }aryqyraebcleay &?+% Bràmki` gly~r`enl cly rayknqly }aryqy }`clray zrlely~ki`nly &9+%
Bràmki` gly~r`enl cly rayknqly }aryqy }`rk`oca kenazaenkae~a na fkarrl &>+% Bràmki`
gly~r`enl cly rayknqly }aryqy }`rk`oca kenazaenkae~a `cqgkekl &1+% &Ac`olr`ikùe
zrlzk`+
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 499/846
Nr% Fëi~lr Pqa}anl Qrç`y
6'>1
Ac }`clr nac ilamkikae~a na na~argke`ikùe gøc~kzca ay8 _ ? 7 :%;>6:
Ac ilamkikae~a na na~argke`ikùe `hqy~`nl ay8 _ ?`hqy~`n` 7 :%;96:
Ac ilamkikae~a na ilrrac`ikùe gøc~kzca ay8 _ 7 :%;29=Cly ilamkikae~ay z`rik`cay ya zqanae ay~kg`r yk ya naya` y`oar c` rac`ikùe
ae~ra ac çenkia na `nylrikùe { ac `cqgkekl av~r`çoca# zlekaenl c` }`rk`oca
kenazaenkae~a# fkarrl iley~`e~a% ^`gokëe# yk ya naya`r` y`oar c` rac`ikùe ae~ra ac
çenkia na `nylrikùe { ac fkarrl av~r`çoca# ya zlenrç` c` }`rk`oca `cqgkekl iley~`e~a%
Ykgkc`rgae~a# yk ya naya`r` y`oar c` rac`ikùe ae~ra c`y }`rk`ocay `cqgkekl { c`
}`rk`oca nac fkarrl# ya zlenrç` c` }`rk`oca çenkia na `nylrikùe mkh`%
^@OC@ 6%0?% ^`oc` gly~r`enl cly ilamkikae~ay na rabraykùe# }`clray na ~ na
Ay~qnk`e~a# ek}acay na z { naikykleay ~lg`n`y ae Fl8 &Ac`olr`ikùe zrlzk`+
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ ]`rk`oca Ilamkikae~a ]`clr na ~ Ek}ac Naikykùekenazaenkae~a na rabraykùe na z &1!+
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ Ke~ariaz~l '2%91:== '?%0:;> :%:=00 @iaz~`r Fkarrl :%00?29 9%2;=; :%::91 _aif`x`r @cqgkekl :%9>;:: >%6;>> :%:::= _aif`x`r
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
^@OC@ 6%09% ^`oc` na `eàckyky na }`rk`ex`% &Ac`olr`ikùe zrlzk`+ \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ Mqae~a na b%c% Yqg` na cly Iq`nr`nl Mi`ci% ]`clr Zlnar na]`rk`ikùe iq`nr`nly gankl na z c` zrqao`
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ Ke~ariaz~l 0 0016:%90 0016:%90
_abraykùe ? 9?1;%;: 02=>%;1 ;?%:9 :%::: 0%::::Arrlr 0: 0;0%2; 0;%06
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ ^l~`c 0? 92?0%=; 90:%0>
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 500/846
Nr% Fëi~lr Pqa}anl Qrç`y
6'>=
^@OC@ 6%0>% ^`oc` gly~r`enl ac razlr~a na rayknqly% &Ac`olr`ikùe zrlzk`+ \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ _aebcùe ]`clr ]`clr _ayknql Arrlr ay~àen`r
`i~q`c zrlely~ki`nl \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ 0 > >%:=9: '=%9:1? 1%::22? 06 0;%2:== '0%2:== >%;1009 0> 09%1962 :%>=0? >%2:11> 06 0>%=11? 9%9>>2 >%=6=?1 ?= ?;%=>:= '9%=>:= 1%0:10= ?= ?1%>0>0 :%1616 >%1;;=2 ?0 ?9%?06? '?%?06? >%=>666 9: 9?%;;:? '?%;;:? >%==?9; ?6 ?>%?;2= 9%2:?> >%1=20
0: 9= >>%;91? '6%;91? >%2:0?00 =1 =:%2:;2 >%?;:? 1%>?1:0? =? =:%;:0> 0%:;6= 1%>0;109 >: 99%;?;? =%:2:2 >%1=>;
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
^@OC@ 6%01% ^@OC@ gly~r`enl cly ke~ar}`cly na ilemk`ex` z`r` ay~a zrlocag`%&Ac`olr`ikùe zrlzk`+
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ ]`rk`oca Cçgk~a kemarklr &;1!+ Cçgk~a yqzarklr &;1!+kenazaenkae~a
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ Ke~ariaz~l '01%00>; :%>092Fkarrl & V 0+ :%:>=2 :%026;@cqgkekl & V ?+ :%0;:0 :%1:2;
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 501/846
Nr% Fëi~lr Pqa}anl Qrç`y
6'>2
^@OC@ 6%0=% ^`oc` gly~r`enl c` ay~`nçy~ki` nayirkz~k}`% &Ac`olr`ikùe zrlzk`+ \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ ]`rk`oca Ile~al Zrlgankl Nay}k`ikùe ]`clr ]`clr
ay~àen`r gçekgl gàvkgl \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ Fkarrl & V 0+ 09 022%90 2:%0: =0 999@cqgkekl & V ?+ 09 >;%90 ?;%0; 09 00?Çenkia na &[ + 09 ?;%61 02%=0 > =1`nylrikùe
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
Ileicqykleay8 Ac glnacl na rabraykùe lo~aeknl ay }àcknl z`r` zrankiikùe {
ay~kg`ikùe% Cly n`~ly aei`h`e okae ile qe glnacl ckea`c gøc~kzca% Ay~` ile~aeikùeay~à o`y`n` ae ac `eàckyky yqoha~k}l na c`y bràmki`y na cly rayknqly% Zlr ahagzcl# ae
c` mkbqr` 6%0: { 6%00 c` zrqao` na elrg`ckn`n ay oqae`# zlrpqa ~lnly cly zqe~ly
ay~àe nae~rl na c`y o`en`y# { gq{ iari`ely ` c` cçea` na rabraykùe% @nagày# cly
zqe~ly ay~àe na `iqarnl ile c` rabc` nac =6!# ;1! { ;;!# ay naikr# ac =6! na cly
zqe~ly ay~àe nae~rl na x 7 °0# ac ;1! ay~àe nae~rl na x 7 °?# a~i% Ae c` mkbqr` 6%00
na cly rayknqly }aryqy cly raebcleay# ay~l y`~kym`ia c` yqzlykikùe na kenazaenaeik`#
zlrpqa f`{ ac gkygl eøgarl na rayknqly zlyk~k}ly { eab`~k}ly% @nagày# c`y
bràmki`y na cly rayknqly }aryqy c`y }`rk`ocay kenazaenkae~ay el }klc`e c` yqzlykikùe
na el ckea`ckn`n# zlrpqa el f`{ ~aenaeik`y namkekn`y% Mke`cgae~a# c` bràmki` na
rayknqly }aryqy }`clray zrlely~ki`nly ay~àe na `iqarnl ile c` yqzlykikùe na
}`rk`ex`y kbq`cay &flglyian`y~kikn`n+%
Ae iq`e~l ` cly `eàckyky loha~k}ky~`y# ay naikr# qy`enl zrqao`y ay~`nçy~ki`y#
eqa}`gae~a# zrayqzleae qe oqae `hqy~a nac glnacl na rabraykùe ay~kg`nl% Ay~l ya
naoa ` pqë# ac }`clr nac ilamkikae~a na na~argke`ikùe gøc~kzca _ ? ay~à gq{ iari`el
` qel% @nagày# ac }`clr na _ 7 :%;29= kenki` gq{ oqae` ilrrac`ikùe ae~ra c`
}`rk`oca nazaenkae~a { c`y }`rk`ocay kenazaenkae~ay% Ile rayzai~l ` c` ~`oc` nac
`eàckyky na }`rk`ex`# ac }`clr na M ay gqifl gaelr pqa ac }`clr irç~kil { ay~l ay~à
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 502/846
Nr% Fëi~lr Pqa}anl Qrç`y
6'>6
nagly~r`nl zlr ac }`clr na c` zrlo`okckn`n z ac iq`c ay gqifl gq{ ykbekmki`e~a%
C`y zrqao`y na ~ na ay~qnk`e~a# ~`gokëe yle gq{ `iaz~`ocay { nagqay~r`e pqa c`y
zaenkae~ay na οk el yle kbq`cay ` iarl% Cly ke~ar}`cly na ilemk`ex` n`e rayqc~`nlyykgkc`ray { yqbkarae pqa ac glnacl na rabraykùe ay oqae zrlely~ki`nlr% Ya zqanae
yabqkr f`ikaenl zrqao`y na fkzù~ayky z`r` ~lnly cly z`ràga~rly zloc`ikle`cay {# yke
cqb`r ` nqn`y# ëy~`y ~`gokëe `zl{`rç`e c` ile~aeikùe na pqa# ac glnacl na
rabraykùe# ay `zcki`oca%
Ahagzcl )2% Ileyknar`r cly n`~ly na c` ~`oc` na `o`hl% Qy`enl ac zrlbr`g` na
ilgzq~`nlr` Gkek~`o lo~aear ac glnacl na rabraykùe gày `zrlzk`nl# ay naikr# qe
glnacl gøc~kzca ckea`c &Glnacl 0+4 glnacl ile ~r`eymlrg`ikùe ae ac aha }ar~ki`c
&Glnacl ?+ { qe glnacl ile ~r`eymlrg`ikleay na cly ahay flrkxle~`cay { nac aha
}ar~ki`c &Glnacl 9+%
^@OC@ 6%02% ^`oc` gly~r`enl cly n`~ly ok}`rk`nly na rabraykùe% &Ac`olr`ikùe
zrlzk`+
V 0 s > > > = 9 = 9 ?
V ? s 9 > 9 > ? > ? ?
[ s 9 ? 2 = 1 = 2 >
Ylcqikùe8
@o`hl ya n`e cly rayqc~`nly na cly ~ray glnacly% @c hqxb`r zlr cly rayqc~`nly# ya ca
zkna `c cai~lr pqa naikn` iq`c glnacl ay ac gày `zrlzk`nl%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 503/846
Nr% Fëi~lr Pqa}anl Qrç`y
6'>;
^@OC@ 6%06% _ayqc~`nly gly~r`enl ac rayqgae na cly ~ray glnacly%&Ac`olr`ikùe zrlzk`+
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
_abrayykle @e`c{yky8 [ }aryqy V 0 # V ? &Glnacl 0+^fa rabrayykle apq`~kle ky8 [ 7 =%:: $ ?%::V0 ’ 9%::V?
Zranki~lr Ilam YA Ilam ^ ZIley~`e~ =%:::: 0%6:9 9%99 :%:?0V0 ?%:::: :%22>= ?%16 :%:>;V? '9%:::: 0%069 '?%1> :%:1?
y 7 0%>0> _'Yp 7 16%9! _'Yp&`nh+ 7 >0%2!Z_AYY 7 :%0?2> _'Yp&zran+ 7 10%=?!
@e`c{yky lm ]`rk`eia
Ylqria NM YY GY M Z_abrayykle ? 0>%::: 2%::: 9%1: :%00?_ayknq`c Arrlr 1 0:%::: ?%:::^l~`c 2 ?>%:::
_abrayykle @e`c{yky8 Clb [ }aryqy V 0 # V ? &Glnacl ?+
^fa rabrayykle apq`~kle ky8 Clb [ 7 :%60: $ 7%??1V0 ’ :%9>6V?
Zranki~lr Ilam YA ilam ^ ZIley~`e~ :%60:0 :%0=?? >%;; :%::>V0 :%??>6 :%:=;2 9%?9 :%:?9V? ':%9>2; :%0:=1 '9%?2 :%:??
y 7 :%0?2? _'Yp 7 =;%9! _'Yp&`nh+ 7 12%:!Z_AYY 7 :%0?2> _'Yp&zran+ 7 10%=?!
@e`c{yky lm ]`rk`eia
Ylqria NM YY GY M Z_abrayykle ? :%06?> :%:;0? 1%=9 :%:1?_ayknq`c Arrlr 1 :%:6:; :%:0=?^l~`c 2 :%?=9>
_abrayykle @e`c{yky8 Clb [ }y Clb V 0# Clb V ? &Glnacl 9+
^fa rabrayykle apq`~kle ky8 Clb [ 7 :%1;1 $ 0%69 Clb V0 ’ ?%0= Clb V?
Zranki~lr Ilam YA Ilam ^ ZIley~`e~ :%1;>; :%?:;1 ?%6> :%:9=Clb V0 0%69>? :%2?66 ?%1? :%:19Clb V? '?%0129 :%699? '?%1; :%:>;
y 7 :%0>69 _'Yp 7 16%?! _'Yp&`nh+ 7 >0%1!Z_AYY 7 :%9::1 _'Yp&zran+ 7 :%::!
@e`c{yky lm ]`rk`eia
Ylqria NM YY GY M Z_abrayykle 9 :%0199 :%:2=2 9%>6 :%009_ayknq`c Arrlr 1 :%00:: :%:??:^l~`c 2 :%?=9>
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 504/846
Nr% Fëi~lr Pqa}anl Qrç`y
6'1:
^@OC@ 6%0;% ^`oc` gly~r`enl cly n`~ly lrkbke`cay nac zrlocag`% &Ac`olr`ikùe zrlzk`+\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
I0 I? I9 I> I1 I=
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\[ V0 V? Clb [ Clb V 0 Clb V ? \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
0 9 > 9 :%>220?0 :%=:?:=: :%>220?0\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
? ? > > :%9:0:9: :%=:?:=: :%=:?:=:\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
9 2 > 9 :%6>1:;6 :%=:?:=: :%>220?0\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
> = = > :%226010 :%226010 :%:=:?:=\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
1 1 9 ? :%=;6;2: :%>220?0 :%9:0:9:\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
= = = > :%226010 :%226010 :%=:?:=:\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
2 2 9 ? :%6>1:;6 :%>220?0 :%9:0:9:\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
6 > ? ? :%=:?:=: :%9:0:9: :%9:0:9:\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
Ahagzcl )6% Ae ay~qnkly na pqçgki` `e`cç~ki`# ac qyl nac `eàckyky na mcqlrayiaeik`
na r`{ly V ya qy` ilgl qe` farr`gkae~` z`r` ay~kg`r cly zlriae~`hay na cly
kebrankae~ay na gqif`y gaxic`y% @ gaeqnl# c` ay~kg`ikùe na c`y ileiae~r`ikleay
nazaena ae c` f`okckn`n z`r` `hqy~`r glnacly na rabraykùe% Ae qe` ke}ay~kb`ikùe
ke~k~qc`n` –Ilrrai~kley mlr G`~rkv Ammai~y ke V'r`{y mcqlrayiae~ @e`ckyky Qykeb
Gqc~kzca _abrayykle Ga~flny‗# zqocki`nl zlr @e`c{~ki`c Ifagky~r{ &]lc% 92#
0;=1+ gaxic`y ile~kaenl > kebrankae~ay & V k+ mqarle zraz`r`n`y% C`y
ileiae~r`ikleay na cly ilgzleae~ay }`rk`rle ae c`y gaxic`y z`r` zrlnqikr ~kzly
ay~àen`ray na i`ckor`ikùe &[ k+% &R`czlca# 0;;?# z% >?0+% Cly n`~ly na ay~a zrlocag`
ya n`e `o`hl%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 505/846
Nr% Fëi~lr Pqa}anl Qrç`y
6'10
^@OC@ 6%?:% ^`oc` gly~r`enl cly n`~ly nac zrlocag` na `rrko`%
[ k V 0 V ? V 9 V >
:%110> 0%0?>: :%6;6: :%6?0; :%;;:=:%>>?= :%;?61 :%662? :%;9:6 :%;;>>:%1=90 0%0?0> :%6:9: :%2==6 0%0??0:%1=?> 0%0=91 :%62:= :%;?2? :%;69?:%>1:1 :%;>01 :%6:=> :%;:?= 0%00?2:%1?;: 0%:20? :%6>:> :%6==? 0%:69=:%>2:? :%;1=0 :%6290 :%6?:= 0%:?;::%1::0 0%:06= :%6>90 :%69>= 0%:1;0
:%>>?1 :%;:9; :%690> :%21;= 0%:;;>&Mqae~a8 R`czlca a~ `c% 0;;?+
&`+ @hqy~`r qe glnacl ckea`c na rabraykùe gøc~kzca ` cly n`~ly na c` ~`oc`%
Aeyabqkn`# ay~kg`r c`y ileiae~r`ikleay nac kebrankae~a @ z`r` qe` gaxic` iq{`
~`y` na ke~aeykn`nay na r`{ly'V ya`e# rayzai~k}`gae~a# V 0 7 0%0:# V ? 7 :%;::# V 9 7
:%6:: { V > 7 :%;;1%
Ylcqikùe8
&`+ Qy`enl qe z`pqa~a na ilgzq~`nlr` { `yqgkaenl qe glnacl na rabraykùe ckea`cgøc~kzca ya lo~kaea c` aiq`ikùe na rabraykùe%
[ 7 ':%9::> $ :%1962 V 0 $ :%022: V ? ’ :%:2:> V 9 $ :%01:= V >
Yqy~k~q{aenl c`y }`rk`ocay kenazaenkae~ay# ya lo~kaea ac }`clr na c` rayzqay~` [ # ay
naikr8
[ 7 ':%9::> $ :%196&0%0:+ $ :%022:&:%;:+ ’ :%:2:>&:%6:+ $ :%01:=&:%;;1+
7 :%1:Ahagzcl );% Gle~blgar{ { Zaid &0;;?+ nayirkoae ac qyl na qe glnacl na
rabraykùe z`r` rac`ikle`r c` i`e~kn`n na ~kagzl pqa rapqkara qe }aenanlr z`r` n`r
yar}kikl ` qe` gàpqke` avzaenanlr` na `r~çiqcly { ac eøgarl na agz`pqay
ile~aeknly ae c` gàpqke` { c` nky~`eik` nac }afçiqcl &zkay+ na yar}kikl nac yk~kl
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 506/846
Nr% Fëi~lr Pqa}anl Qrç`y
6'1?
nlena ya aeiqae~r` c` gàpqke`% Ay~a glnacl na rabraykùe gøc~kzca mqa q~kckx`nl
z`r` nkyað`r c` rq~`# cly flr`rkly { c` y`ckn` na cly }afçiqcly% C` ~`oc` na `o`hl
gqay~r` ?1 loyar}`ikleay nac ~kagzl na yqgkeky~rl# eøgarl na agz`pqay { c`nky~`eik`# nac }afçiqcl%
^@OC@ 6%?0% ^`oc` gly~r`enl cly n`~ly na yqgkeky~rl%
El% na loyar}`ikùe ^kagzl na yqgkeky~rl El% na ae}`yay Nky~`eik` nac }afçiqcl
0 ;%>1 ? 1:? ?>%>1 6 00:9 90%21 00 0?:> 91%:: 0: 11:
1 ?1%:? 6 ?;1= 0=%6= > ?::2 0>%96 ? 9216 ;%=: ? 921; ?>%91 ; 0::
0: ?2%1: 6 9::00 02%:6 > >0?0? 92%:: 00 >::09 >0%;1 0? 1::0> 00%== ? 9=:01 ?0%=1 > ?:10= 02%6; > >::02 =;%:: ?: =::06 0:%9: 0 1610; 9>%;9 0: 1>:?: >=%1; 01 ?1:?0 >>%66 01 ?;:?? 1>%0? 0= 10:?9 1=%?9 02 1;:
?> ??%09 = 0::?1 ?0%01 1 >::
&Mqae~a8 Gle~blgar{ a~ `c% 0;;?+
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 507/846
Nr% Fëi~lr Pqa}anl Qrç`y
6'19
Z`r` ay~a zrlocag` i`ciqc`r cly ykbqkae~ay aeqeik`nly8
&`+ Ac glnacl na rabraykùe ckea`c gøc~kzca zloc`ikle`c%
&o+ Ac glnacl na rabraykùe ckea`c gøc~kzca na c` gqay~r` pqa ay~kg` `c glnacl zloc`ikle`c%
&i+ Zranaikr ac ~kagzl na yqgkeky~rl z`r` z`ray na }`clray na c`y }`rk`ocay na
rabraykùe# eøgarl na agz`pqay & v0+ { nky~`eik` & v?+# iq`enl v0 7 0 agz`pqa { c`
nky~`eik` ay kbq`c ` v? 7 ?1 zkay%
&n+ A}`cq`r ac glnacl na rabraykùe lo~aeknl qy`enl ~ëieki`y loha~k}ky~`y {
yqha~k}ky~`y# ilgl c`y nayirk~`y ae ay~a i`zç~qcl%
Nkyiq~kr ac r`xle`gkae~l pqa ya ykbqa ae c` }`ckn`ikùe yqoha~k}` na cly bràmkily%
Ylcqikùe8
&`+ Ac glnacl na rabraykùe gøc~kzca# z`r` ? }`rk`ocay kenazaenkae~ay ay8
±[ s v0#v?s 7 οl $ ο0 v0 $ ο? v? $ α
&o+ Ac ilrrayzlenkae~a glnacl na rabraykùe ckea`c gøc~kzca gqay~r`c ay8
[ 7 ol $ o0 V 0 $ o? V ? $ a
Nlena8
[ 7 ~kagzl na yqgkeky~rl
V 0 7 el na ae}`yay
V ? 7 nky~`eik` nac }afçiqcl
Ac glnacl na rabraykùe na c` gqay~r` ay8
[ 7 0%2> $ ?%26 & V 0+ $ :%:09 & V ?+
&i+ Z`r` zranaikr ac ~kagzl na yqgkeky~rl &[ + ae rac`ikùe ile ac eøgarl na ae}`yay#iq`enl V 0 7 0 { ile c` nky~`eik` nac }afçiqcl# iq`enl V ? 7 ?1 zkay ya lo~kaea
yqy~k~q{aenl cly }`clray ae c` aiq`ikùe na rabraykùe# ay naikr8
[ 7 0%2> $ ?%26&0+ $ :%:09&?1+ 7 >%61
&n+ Cly rayqc~`nly loha~k}ky~`y ay~`nçy~kily yle8 _ ? 7 ;6%0!4 _ ?`hqy~`n` 7 ;2%;!4 y 7
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 508/846
Nr% Fëi~lr Pqa}anl Qrç`y
6'1>
?%9?4 Z_AYY 7 01;%6;%
^@OC@ 6%??% ^`oc` gly~r`enl cly }`clray na ^ { na Z% &Ac`olr`ikùe zrlzk`+%
Zranki~lr Ilamkikae~a YA ilamkikae~a ^ Z
Iley~`e~a 0%2>9 0%011 0%10 :%0>1
El% na ae}`yay ?%2;: :%:;? 9:%:; :%:::
Nky~`eik` nac }afçiqcl :%:09 :%::9 >%99 :%:::
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
^@OC@ 6%?9% ^`oc` na `eàckyky na }`rk`ex`% &Ac`olr`ikùe zrlzk`+
Mqae~a na b%c% YY GY M z]`rk`ikùeNaoknl ` c` ? 1;6>%6 ?;;?%> 111%? :%:::_abraykùeArrlr ?? 006%= 1%>
^l~`c ?>
Z`r` c` }`ckn`ikùe yqoha~k}` nac glnacl na rabraykùe# `e`ckx`enl c`y bràmki`y
na cly rayknqly ay~`en`rkx`nly# naoae avky~kr# `zrlvkg`n`gae~a# ac gkygl eøgarl
na rayknqly zlyk~k}ly { eab`~k}ly% @nagày# ae c` zrqao` na elrg`ckn`n# ~lnly cly
zqe~ly naoae ay~`r nae~rl na c`y o`en`y na ilemk`ex`% Ac ay~qnk`e~a naoarà f`iar
cly nk`beùy~kily yqoha~k}ly z`r` ilgzcagae~`r c` ramraen`ikùe l ilemk`okckn`n nac
glnacl na rabraykùe%
Zrliankgkae~l na rabraykùe gøc~kzca qy`enl ac zrlbr`g` Gkek~`o
Zrliankgkae~l80% Krya `8 Y~`~ ↟ _abrayykle ↟ _abrayykle
?% Ae c` }ae~`e` na –_abrayykle‗ `z`raiae c`y ae~r`n`y na c` }`rk`oca
nazaenaenkae~a &[ + { na c`y }`rk`ocay kenazaenkae~ay V 0# V ?# ae yqy ilcqge`y
rayzai~k}`y rac`ikle`n`y ile ac zrlocag`
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 509/846
Nr% Fëi~lr Pqa}anl Qrç`y
6'11
9% Ae c` }ae~`ekcc` na –_ayzleya‗ &na ay~` }ae~`e` na _abrayykle+ ae~r`r c` }`rk`oca
nazaenkae~a {# ae c` }ae~`ekcc` na –Zranki~lry#‗ ae~r`r c`y }`rk`ocay kenazaenkae~ay
&pqa ya ilzk`rle ae c`y ilcqge`y nac zrlbr`g`+%>% Nao`hl na ay~` }ae~` na _abrayykle ay~àe c`y }ae~`ekcc`y na –Br`zfy# Lz~kley#
_ayqc~y { Y~lr`ba‗% Zlr ahagzcl yk ya naya` qy`r Br`zfy ya zqanae yacaiikle`r cly
rayknq`cay rabqc`ray l cly ay~`en`rkx`nly% Ae c` }ae~`ekcc` na –Lz~kle rayknq`c
zcl~y‗# zqe~a`r c`y bràmki`y na c`y iq`~rl lzikleay# z`r` ac `eàckyky yqoha~k}ky~`%
1% Ae c` }ae~`e` na –_abrayykle'Lz~kley‗ zqe~a`r c`y mqeikleay naya`n`y# }%b%#
}`rk`eia –Kemc`~kle m`i~lry# Nqroke'R`~yle y~`~ky~kiy# Z_AYY‗# a~i%
=% Ae c` }ae~`e` na –_abrayykle'_ayqc~y‗ zqe~a`r c`y mqeikleay naya`n`y na c`y
iq`~rl aecky~`n`y# }%b%# –Ke `nnk~kle na yapqae~k`c yqg…%%‗
Ahagzcl )0:% Ay~a ay qe ahagzcl nac ckorl @zzckan Y~`~ky~kiy8 @eàckyky lm ]`rk`eia
`en _abraykle na cly `q~lray Nqee { Ic`rd% Ay~` ay qe` ke}ay~kb`ikùe rac`ikle`n`
ile c` ~agzar`~qr`# ~lg`n` ilgl c` }`rk`oca na rayzqay~`# ae mqeikùe na }`rk`ocay
rabraylr`y ilgl c` `c~k~qn# clebk~qn { c`~k~qn% C` ~`oc` na `o`hl gqay~r` cly
rayqc~`nly% Qy`enl ac zrlbr`g` Gkek~`o8
&`+ Aeile~r`r ac glnacl na rabraykùe gày `zrlzk`nl
&o+ ]`ckn`r ac glnacl qy`enl ga~lnly ay~`nçy~kily# ay naikr# ay~kg`enl ac
ilamkikae~a na na~argke`ikùe gøc~kzca _ ?# _ ? `hqy~`n`# y# Z_AYY# ~`oc` na @EL]@#
{ bràmki`y yqoha~k}ky~`y# ilgl rayknqly }aryqy ùrnaeay# rayknqly }aryqy }`clray
`hqy~`nly { zrqao`y na elrg`ckn`n%
&i+ F`iar ilgae~`rkly `iari` na cly rayqc~`nly
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 510/846
Nr% Fëi~lr Pqa}anl Qrç`y
6'1=
^@OC@ 6%?>% ^`oc` gly~r`enl cly }`clray na c` ~agzar`~qr` ae lM &[ +# @c~k~qn ae
zkay & V 0+# Clebk~qn ae br`nly & V ?+ { C`~k~qn ae br`nly & V 9+%
^agzar`~qr` &[ + @c~k~qn & V 0+ Clebk~qn & V ?+ C`~k~qn & V 9+
11%2 0:69 00? 99
92%6 >12 6= 96
1=%> 90? 006 9>
10%: 9:1 ;: 9?
9>%1 1??0 0:1 >:
9>%: ?6>? 00= >>
9=%2 6:2 ;> >099%> >?=: 00? >0
9?%= 601 69 >:
>;%0 9;?: 0:= 9?
>=%= 0:1> 6> 9>
9=%9 >9;2 0?: 9;
06%? 69: ;9 >1
9=%2 >=1 ;: 9;
09%9 00=? ;? >2
9:%0 262 6? >0
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
Ylcqikùe8
&`+ Ya `yyqga qe glnacl na rabraykùe ckea`c&o+ C` q~kckn`n nac glnacl ya n` zlr cly }`clray na _ ?# y# Z_AYY# a~i% gly~r`nly
zlr c`y Mkbqr`y 6%0? &`+# &o+ { &i+%
_abrayykle @e`c{yky8 &[ + ^agzar`~qr` }aryqy & V 0+ @c~k~qn# & V ? + Clebk~qn & V 9+
^fa rabrayykle apq`~kle ky8 &[+ ^agzar`~qr` 7 ;;%? ' :%::096 &V0+ @c~k~qn $ :%?;; &V?+Clebk~qn ' ?%?; &V9+ C`~k~qn
Zranki~lr Ilam YA Ilam ^ Z ]KMIley~`e~ ;;%?> 0:%2; ;%?: :%:::&V0+ @c~k~qn ':%::0926: :%:::1;=6 '?%90 :%:>: 0%2&V?+ Clebk~qn :%?;622 :%:229= 9%6= :%::? 0%2&V9+ C`~k~qn '?%?;:: :%022; '0?%62 :%::: 0%:
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 511/846
Nr% Fëi~lr Pqa}anl Qrç`y
6'12
y 7 9%0?0==# _'Yp 7 ;>%=!# _'Yp&`nh+ 7 ;9%?!# Z_AYY 7 ?0>%611# _'Yp&zran+ 7 ;:%:6!
@e`c{yky lm ]`rk`eia ^`oca
Ylqria NM YY GY M M irç~ki` Z
_abrayykle 9 ?:>6%1> =6?%61 2:%:2 M%:149#0? 7 9%>; 333 :%::0_ayknq`c Arrlr 0? 00=%;> ;%2>^l~`c 01 ?0=1%>6
Nqroke'R`~yle y~`~ky~ki 7 0%1996>
&`+ &o+ &i+
Loyar} ~kleLrnar
Y
~ ` e n ` r n k x a n _ a y k n q ` c
0=010>090?000:;62=1>9?0
?
0
:
'0
'?
_ayknq cy ]aryqy ~fa Lrnar lm ~fa N ~&rayzleya ky &[+ agzar ~qr +
Mk~~an] cqa
Y
~ ` e n ` r n k x a n _ a y k n q ` c
=:1:>:9:?:
?
0
:
'0
'?
_ayknq cy ]aryqy ~fa Mk~~an ] cqay&rayzleya ky &[+ agzar ~qr +
Y~ en rnkxan_ayknq c
Z a r i a e ~
9?0:'0'?'9
;;
;1
;:
6:
2:
=:
1:
>:
9:
?:
0:
1
0
Elrg c Zrlo okck~{ Zcl~ lm ~fa _ayknq cy&rayzleya ky &[+ agzar ~qr +
Mkbqr` 6%0?% C` Mkbqr` &`+ gqay~r` cly rayknq`cay }y% ùrnaeay4 c` mkbqr` &o+ gqay~r`
cly rayknq`cay }y% cly }`clray `hqy~`nly { c` mkbqr` &i+ n` c` zrqao` na elrg`ckn`n%
&i+ Ae ileicqykùe# na `iqarnl ` cly }`clray nac ilamkikae~a na na~argke`ikle _ ? 7
;1%=! f`{ qe oqae `hqy~a nac glnacl% Ac }`clr na M na c` @EL]@ raif`x` c`
fkzù~ayky na kbq`cn`n na zrlgankly &na `c~k~qn# clebk~qn { c`~k~qn+# ile qe }`clr na
z gq{ ykbekmki`~k}l% C`y zrqao`y na ^ yle ykbekmki`e~ay# cl pqa yqbkara pqa el f`{
zrlocag`y na gqc~kilckea`ckn`n% @eàclb`gae~a# cly }`clray o`hly na ]KM yqbkarae
kenki`e pqa el f`{ zrlocag`y na gqc~kilckea`ckn`n% Ay~l# `qe`nl# ` cly ykbely na
c`y }`rk`ocay rabraylr`y na c` aiq`ikùe na rabraykùe# cly iq`cay yk ay~àe na `iqarnl `
qe` cùbki` ` zly~arklrk% Ac }`clr na c` mqeikùe na Nqroke'R`~yle Y~`~ky~ki l na
ilrrac`ikùe ae yarka kbq`c ` 0%1996> kenki` pqa el f`{ zrlocag`y na
`q~lilrrac`ikùe &`qepqa `pqç# ay~l ya zqana kbelr`r zlrpqa ac zrlocag` el
ke}lcqir` yarkay na ~kagzl+% Ae iq`e~l ` c` Mkbqr` 6%0? c` bràmki` &`+ gqay~r` cly
rayknq`cay }aryqy cly ùrnaeay# ae c` iq`c f`{ `ca`~lrkan`n na cly n`~ly%
@eàclb`gae~a# c` bràmki` na rayknq`cay }y% }`clray `hqy~`nly &o+ kenki` `c`~alrkn`n
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 512/846
Nr% Fëi~lr Pqa}anl Qrç`y
6'16
l kenazaenaeik` na cly n`~ly# yke zrlocag`y na fa~arlyian`y~kikn`n &arrlray na
}`rk`ex`y el iley~`e~ay+# a~i% Mke`cgae~a# c` bràmki` na c` zrqao` na elrg`ckn`n &i+
kenki` pqa cly n`~ly ay~àe elrg`cgae~a nky~rkoqknly &zlrpqa ~lnly cly zqe~ly ay~àenae~rl na c`y o`en`y na ilemk`ex`+# `qepqa ile yaybl zlyk~k}l%
El~`8 ¼Iraa qy~an pqë# a}ae~q`cgae~a# ac i`cae~`gkae~l bclo`c# naoknl ` c`y
agkykleay na b`yay na ke}are`narl# baear`nly zlr agkykleay }afkiqc`ray a
kenqy~rk`cay }` ` glnkmki`r c`y ~agzar`~qr`y pqa }`e ae mqeikùe na c` c`~k~qn<
Ahagzcl )00% Ay~a ay qe ahagzcl fkzù~a~kil gly~r`enl c` rac`ikùe ae~ra c`y
ileiae~r`ikleay na lxlel `r~kmkik`c# ` ek}ac nac yqacl &zzg+ { c`y ~agzar`~qr`y &lM+%
Ay~a aharikikl ay~à aei`gke`nl ` i`ciqc`r# g`eq`cgae~a# cly rayknq`cay { na f`iar
qe` bràmki` gly~r`enl cly rayknq`cay irqnly% Cly n`~ly ya n`e ae c` ~`oc` na `o`hl%
^@OC@ 6%?1 gly~r`enl cly n`~ly na ay~a zrlocag`% \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ Ileiae~r`ikleay na L9 &{+s 21 6: 6= ;> ;; 0:2
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ ^agzar`~qr` &lM+ &v+ s =1 20 2; 61 ;9 0::
F`iar cly ykbqkae~ay iàciqcly8
&`+ I`ciqc`r ac glnacl na rabraykùe { gankr yq `naiq`ikùe ay~kg`enl _ ?# _ ?&`hqy~%+#
y# Z_AYY { c` ay~`nçy~ki` Nqroke'R`~yle
&o+ F`iar qe` ~`oc` gly~r`enl ac }`clr na c` nay}k`ikùe ae~ra cly n`~ly { ac `hqy~a#
ay naikr# na cly rayknq`cay rabqc`ray l irqnly ak 7 {k ’ { k%
&i+ F`iar qe` bràmki` na L9 &{+ { ~agzar`~qr`y &v+ gly~r`enl cly rayknq`cay irqnly
Ylcqikùe8&`+ Qy`enl qe z`pqa~a na ilgzq~`nlr` n` c` aiq`ikùe na rabraykùe8
Ileiae~r`ikùe na lxlel & { + 7 01%> $ :%;:; ^agzar`~qr` &v+
y 7 0%0:0# _ ? 7 ;;%9!# _ ?&`hqy~%+ 7 ;;%?!# Z_AYY 7 ;%>?692# ay~`nçy~ki` Nqroke'R`~yle 7 9%99
&o+ C` ^@OC@ 6%?= gqay~r` cly }`clray `hqy~`nly & { k+# cly rayknq`cay { YYA%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 513/846
Nr% Fëi~lr Pqa}anl Qrç`y
6'1;
^@OC@ 6%?=% ^`oc` gly~r`enl cly n`~ly nac zrlocag`% \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
_ayknq`c irqnl Yqg` na cly iq`nr`nly nac arrlr k vk {k { k 7 01%>> $ :%;:; v ak 7 {k ' { k YYA 7 &{k ' { k+
?
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ 0 =1 21 { 0 7 2>%19 21 ’ 2>%19 7 :%>6 :%?9:>
? 20 6: { ? 7 2;%;6 6: ’ 2;%;6 7 :%:? :%:::>
9 2; 6= { 9 7 62%?1 6= ’ 62%?1 7 '0%?1 0%1=?1
> 61 ;> { > 7 ;?%20 ;> ’ ;?%20 7 0%9: 0%=;::
1 ;9 ;; { 1 7 ;;%;6 ;; ’ ;;%;6 7 ':%;6 :%;=:>
= 0:: 0:2 { = 7 0:=%9> 0:2 ’ 0:=%9> 7 :%== :%>91=
Ω&{k ' { k+? 7 >%62;9
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
&i+ C` Mkbqr` 6%09 na `o`hl gqay~r` c`y ileiae~r`ikleay na lxlel ae mqeikùe na c`y~agzar`~qr`y ile cly }`clray na cly rayknq`cay ak%
6%09% Mkbqr` gly~r`enl c` gankikle na i`n` qel na cly }`clray rayknq`cay ile c`
cçea` na rabraykùe% &Ac`olr`ikùe zrlzk`+%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 514/846
Nr% Fëi~lr Pqa}anl Qrç`y
6'=:
Aharikikly I`zç~qcl 6
6%0% Cly n`~ly na `o`hl gqay~r`e c`y agkykleay na ùvknly na ek~rùbael &ELv+
zrl}aekae~ay na i`cnar`y na zc e~`y acëi~rki`y%
^`oc` gly~r`enl cly n`~ly z`r` ac zrlocag`% &Ac`olr`ikùe zrlzk`+
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ GO~q,fr'm~? & V + s0:: 0?1 0?1 01: 01: ?:: ?:: ?1: ?1: 9:: 9:: 91: >:: >::
ELv &[ + s01: 0>: 06: ?0: 0;: 9?: ?6: >:: >9: >>: 9;: =:: =0: 12:
&`+ I`ciqc`r c` aiq`ikùe na rabraykùe na c` gqay~r` pqa ay~kg` ` c` }arn`nar` aiq`ikùe
zloc`ikle`c% &[ 7 '?>%? $ 0%1; V +
&o+ I`ciqc`r ac ilamkikae~a na ilrrac`ikùe _ ? { _ pqa ay~kg`e ` π% &_ ? 7 :%;1+
&i+ ¼Iqàc ay c` ay~kg`ikùe ayzar`n` na c` agkykùe na ELv iq`enl c` ~`y` na ckoar`ikùe
ay na ??1 GO~q,fr'm~?< &999%=2+
&n+ Qy`r ac zrlbr`g` na ilgzq~`nlr` na Gkek~`o { `e`ckx`r c`y bràmki`y na cly
rayknqly z`r` c` zrqao` na elrg`ckn`n { na cly rayknqly ae mqeikùe nac }`clr na V % &Ac
cai~lr cl naoarà f`iar+
6%?% Ay~a ay qe ahagzcl nac ckorl na Ke~rlnqiikùe `c @e`ckyky na _abraykùe Ckea`c na
Gleblgar{ a~ `c% &?::0+# nlena f`oc` na qe ahagzcl rac`ikle`nl ile c`y
ileiae~r`ikleay na lxlel na naoknl `c i`clr% @yç# N`}knyle &–Qzn`~a le Lxlea
^raeay ke I`ckmlrek`‛y Ylq~f ILY@^ O`yke‗# @kr `en R`y~a# >9# ??=# 0;;9+ ay~qnkl
c`y ileiae~r`ikleay na lxlel ae c` iqaei` `ëra` na c` ily~` yqr na I`ckmlrek`# nqr`e~acly `ðly 0;2= ` 0;;0% Ya iraa pqa c` i`e~kn`n na nç`y ae pqa c`y ileiae~r`ikleay na
lxlel mqarle g`{lray pqa :%?: zzg nazaena nac çenkia ga~aralcùbkil ay~`ikle`c#
pqa ay ac zrlgankl ay~`ikle`c na c` ~agzar`~qr` ile 61: gkcko`ray% C` ykbqkae~a
kemlrg`ikùe gqay~r` cly n`~ly%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 515/846
Nr% Fëi~lr Pqa}anl Qrç`y
6'=0
^`oc` gly~r`enl cly n`~ly nac zrlocag`% \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
@ðl El% na Nç`y & {+ Çenkia ga~alrlcùbkil
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ 0;2= ;0 0=%20;22 0:1 02%00;26 0:= 06%?0;2; 0:6 06%00;6: 66 02%?0;60 ;0 06%?0;6? 16 0=%:0;69 6? 02%?0;6> 60 06%:0;61 =1 02%?0;6= =0 0=%;0;62 >6 02%00;66 =0 06%?0;6; >9 02%90;;: 99 02%10;;0 9= 0=%=
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ Mqae~a8 Gle~blgar{ a~ `c% ?::0
&`+ Ay~kg`r c` aiq`ikùe na rabraykùe
&o+ ¼Pqë ~`e~` ilemk`okckn`n ya ca zqana n`r `c glnacl yacaiikle`nl< Qy`r aemlpqay
ay~`nçy~kily { bràmkily z`r` hqy~kmki`r ay~` zrabqe~`%
6%9% Ae qe ay~qnkl `brçilc`# z`r` }ar cly amai~ly na cly i`gokly ickgà~kily bclo`cay
rac`ikle`nl# ile cly z`~rleay zcq}k`cay `c~ar`nly naoknl `c i`cae~`gkae~l bclo`c# zlr
c`y agkykleay na IL?# ya ay~qnkù c` zraikzk~`ikùe zcq}k`c `eq`c { ac raenkgkae~l na c`
ilyaif` na br`gçea`y% C` ~`oc` na `o`hl n` cly n`~ly%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 516/846
Nr% Fëi~lr Pqa}anl Qrç`y
6'=?
^`oc` gly~r`enl cly n`~ly nac zrlocag`% &Ac`olr`ikùe zrlzk`+
Zraikzk~`ikùe zcq}k`c _aenkgkae~l na c` ilyaif`
ae zqcb`n`y & V + ae ckor`y zlr `ira &[ +2%0? 0:92
=9%1> 96:
>2%96 >0=
>1%;? >?2
6%=6 =0;
1:%6= 966>>%6= 9?0
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
]ar iqàc glnacl na rabraykùe aei`h` gahlr ae cly n`~ly# `c hqxb`r zlr c`y ay~`nçy~ki`y
{ zlr cly `eàckyky bràmkily# ay naikr# qy`enl qe` `zrlvkg`ikùe ckea`c# qe` clb`rç~gki`
{ qe` `zrlvkg`ikùe na mqeikùe na zl~aeik` na c` mlrg` na Ce &[ + 7 Ce&`+ $ o&Ce V +%
&`+ Qy`r qe` `zrlvkg`ikùe ckea`c ilgl [ 7 ` $ oV {# `nagày# i`ciqc`r ac ilamkikae~a
na na~argke`ikùe _ ?% &[ 7 66:%> ’ ;%= & V+# _ ? 7 :%=;;+
&o+ Qy`r qe` `zrlvkg`ikùe clb`rç~gki` ilgl [ 7 ` $ o Ce & V+ { `nagày# i`ciqc`r ac
}`clr na _ ?% &[ 7 0990%:6 ’ 112%:9 Cb V +
&i+ Qy`enl qe` `zrlvkg`ikùe na mqeikùe na zl~aeik` na c` mlrg` na Ce &[ + 7 Ce &`+
$ o &Ce V + {# `nagày# i`ciqc`r _ ? &_ ? 7 :%6?;+
6%>% Ae qe ay~qnkl na pqçgki` `e`cç~ki`# ae c` ~`oc` na `o`hl ya n` c` rac`ikùe ae~ra c`~agzar`~qr` { c` glc`rkn`n &ae glcay zlr ck~rl+ na qe` yqy~`eik`% Z`r` ay~l f`iar cly
ykbqkae~ay8
&`+ Ay~kg`r ac glnacl na rabraykùe gày `zrlzk`nl o`y`nl ae `eàckyky ay~`nçy~kily na
_ ?# _ ?`hqy~`n`# Z_AYY# y# { Iz { ae `eàckyky bràmkily yqoha~k}ly na cly }`clray
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 517/846
Nr% Fëi~lr Pqa}anl Qrç`y
6'=9
rayknq`cay%
^`oc` gly~r`enl c` kemlrg`ikùe rapqarkn`% \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ ^agzar`~qr` lI s >%9 >%1 >%6 1%1 1%2 1%; =%> =%2 2%1 2%;
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ Glc`rkn`n s 0?%0 0?%1 0?%; 09%: 09%0 0>%: 0>%? 0>%6 01%: 01%1
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
6%1% Ac `cqgkekl ay ac ~ariar acagae~l gày `oqen`e~a pqa liqrra ae gkear`cay# rli`y
{ o`rrly% Ac `cqgkekl ya zqana `e`ckx`r ile ac gë~lnl na `oylrikùe `~ùgki`
ayzai~rlgë~rki` &gë~lnl @+# ac iq`c ay~à avae~l na ke~armaraeik`y ilgl mcqlrqrly {
mlym`~ly% Ac `cqgkekl ~`gokëe ya zqana `e`ckx`r zlr gankl nac gë~lnl na
i`clrkga~rç` na ik`eqrl na Arklirlgl _ &gë~lnl O+# ac iq`c ay gày ykgzca pqa ac
`e~arklr% C` ~`oc` na `o`hl gqay~r` cly rayqc~`nly na cly `eàckyky &ae gb,C+ na cly nly
gë~lnly qy`nly% F`iar cly ykbqkae~ay iàciqcly qy`enl ac zrlbr`g` na ilgzq~`nlr`
na Gkek~`o l Y@Y%
&`+ I`ciqc`r a ke~arzra~`r ac ilamkikae~a na na~argke`ikùe _ ? { ac ilamkikae~a na
ilrrac`ikùe _% &_ ?
7 :%;;??# _ 7 :%;;=0+^`oc` gly~r`enl cly n`~ly nac ahagzcl% &Ac`olr`ikùe zrlzk`+
Gë~lnl @ s 1 = = 6 0: 0: 00 00
Gë~lnl O s 6 ; ; 00 09 09 0> 0>
6%=% Ac oarkckl &Oa+ { yqy ilgzqay~ly yle av~rag`n`gae~a }aeaelyly { i`z`iay na
i`qy`r c` gqar~a ae ileiae~r`ikleay `c~`y% C` kef`c`ikùe nac Oa i`qy` qe` yark`
`maiikùe cc`g`n` oarkcklyky% Ac oarkckl ~`gokëe zqana i`qy`r narg`~k~ky# ilehqe~k}k~ky#
eaqgleç` `bqn` { oarkcklyky zqcgle`r irùeki`% Ay~a acagae~l pqçgkil ya qy` ae cly
ra`i~lray `~ùgkily# `}kleay# ilfa~ay { ae ilgoqy~kocay z`r` gçykcay% F`{ nly
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 518/846
Nr% Fëi~lr Pqa}anl Qrç`y
6'=>
gë~lnly z`r` ac `eàckyky &ae ±b,C+ nac oarkckl# ay naikr# ac gë~lnl ayzai~rlgë~rkil na
`oylrikùe `~ùgki` &gë~lnl 0+ { ac gë~lnl `cqgkele &gë~lnl ?+% Cly rayqc~`nly na
cly `eàckyky na cly nly gë~lnly ya n`e ae c` ~`oc` na `o`hl% F`iar cly ykbqkae~ayiàciqcly8
&`+ F`iar qe ay~qnkl ay~`nçy~kil loha~k}ky~`# ay naikr# ay~kg`enl cly }`clray na _ ?#
_ ?`hqy~`n`# Z_AYY# y { ~`oc`y na @EL]@% Ilgzcagae~`r ac ay~qnkl f`ikaenl `eàckyky
yqoha~k}ky~`y%
^`oc` gly~r`enl cly rayqc~`nly na cly gë~lnly 0 { ? z`r` c` gankikùe nac oarkckl%
&Ac`olr`ikùe zrlzk`+
Gë~lnl 0 s : 9 > 1 ; 0? 01 02 ?: ?:
Gë~lnl ? s 0 2 00 0; ?> 90 90 91 >0 >0
6%2% Ae ke}ay~kb`ikleay na ~lvkilclbç` avky~ae ay~qnkly pqa f`e nagly~r`nl pqa c`
zrlo`okckn`n na pqë# qe mqg`nlr na >: `ðly na an`n# pqkae f` yknl mqg`nlr cly
øc~kgly 0: `ðly ile~r`kb` ac iàeiar zqcgle`r ae cly zrùvkgly ?: `ðly ay `c~`
&`yqgkaenl pqa ile~keøa mqg`enl `c gkygl rk~gl+% Ay~` rac`ikùe }` ae mqeikùe nac
eøgarl zrlgankl na ikb`rrkccly pqa mqg`% @yqgkr qe glnacl na rabraykùe ckea`c% C`
~`oc` na `o`hl zrayae~` cly n`~ly na ay~` ke}ay~kb`ikùe na ~lvkilclbç`%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 519/846
Nr% Fëi~lr Pqa}anl Qrç`y
6'=1
^`oc` gly~r`enl cly n`~ly nac zrlocag`% &Ac`olr`ikùe zrlzk`+
Eøgarl na ikb`rrkccly Zrlo`okckn`n na iàeiar
mqg`nly zlr nç` zqcgle`r 1 %0::
0: %009?: %??19: %9::>: %>1:1: %1>:=: %2::6: %6=:
F`iar cly ykbqkae~ay iàciqcly8
&`+ Knae~kmki`r c` }`rk`oca nazaenkae~a { c` }`rk`oca kenazaenkae~a%
&o+ Nayirkokr c` aiq`ikùe na rabraykùe pqa gahlr aei`ha ae cly n`~ly% &[ 7 :%:;60 ’
:%::::?& V + $ :%:::9&V ?++
&i+ I`ciqc`r _ ?# _ ?`hqy~`n`# y# { Z_AYY% &_ ? 7 :%;;=# _ ?`hqy~`n` 7 :%;;1 y 7 :%:0;#
Z_AYY 7 :%::96+
&n+ @e`ckx`r a ke~arzra~`r cly ilgzleae~ay na c` ~`oc` na @EL]@ ilgl Mi`ci%# Mirç~ki` { ac }`clr na z%
&a+ Nkyiq~kr c` rac`ikùe avky~ae~a ae~ra _ ?# y# Z_AYY# Mi`ci%# { ac }`clr na z%
&m+ ]`ckn`r ac glnacl na rabraykùe yqoha~k}`gae~a# ay naikr# `e`ckx`enl cly bràmkily
na cly rayknqly ay~`en`rkx`nly%
6%6% Ya ra`ckxù qe ay~qnkl na pqçgki` `gokae~`c { ya rabky~r`rle c`y i`e~kn`nay na
iclrqrl na ylnkl &E`Ic+# ac iq`c# iq`enl ya nkylc}kù ae 0:: br`gly na `bq` nay~kc`n`#
` nkmarae~ay ~agzar`~qr`y &lI+ nkl cly ykbqkae~ay rayqc~`nly8
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 520/846
Nr% Fëi~lr Pqa}anl Qrç`y
6'==
^`oc` gly~r`enl cly n`~ly nac zrlocag`% &Ac`olr`ikùe zrlzk`+ \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ ^agzar`~qr` & V + E`Ic nkyqac~l ae br`gly na `bq` &[ +
: 6 = 601 0? 0: 0>9: ?1 ?0 ?>>1 90 99 ?6=: >> 9; >?21 >6 10 >>
I`ciqc`r cly ykbqkae~ay aeqeik`nly8
&`+ Br`mki`r cly n`~ly%
&o+ Aeile~r`r c` cçea` na rabraykùe { zlearc` ae c` bràmki`%
&i+ Ay~kg`r c` i`e~kn`n na E`Ic pqa ya nkylc}arà ` qe` ~agzar`~qr` na 9:: D%
&n+ @ y`okaen`y na pqa# ` gankn` pqa `qgae~` c` ~agzar`~qr`# c` nkylcqikùe na c`y
yqy~`eik`y# ilgl c`y y`cay na ylnkl# `qgae~` zrlzlrikle`c `c keiragae~l na c`
~agzar`~qr`# ae~leiay# ykaenl `yç# }arkmki`r na pqa f`{ qe` ilrrac`ikùe i`yk zarmai~`
ae~ra `go`y }`rk`ocay%&a+ F`iar qe` zrqao` na fkzù~ayky z`r` ac ilamkikae~a na ilrrac`ikùe gqay~r`c _# z`r`
}arkmki`r pqa yk avky~a qe` `ylik`ikùe ckea`c ykbekmki`e~a ae~ra c`y nly }`rk`ocay%
Yqbaraeik`8 qy`r c` ay~`nçy~ki` na ~ na Ay~qnk`e~a n`n` `o`hl8
?0?
∟∟7
e
_~
_ile η 7 e ' ? br`nly na ckoar~`n%
&m+ ^aùrki`gae~a# c` nkylcqikùe na gqif`y y`cay }` ae mqeikùe nkrai~` ` c`
~agzar`~qr` {# ae ~alrç`# ac }`clr nac ilamkikae~a na na~argke`ikùe# _ ? naoarç` na yar
na 0%:% Ykaenl `yç# aecky~`r ? m`i~lray &ae ac c`olr`~lrkl na pqçgki`+ pqa zqnkar`e
`mai~`r c` nkylcqikùe na c`y y`cay { na el n`r qe }`clr gaelr pqa 0%:%
6%;% Ae qe ay~qnkl na ga~alrlclbç` ae~ra c` i`e~kn`n na ccq}k` { c` raglikùe na
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 521/846
Nr% Fëi~lr Pqa}anl Qrç`y
6'=2
ile~`gke`e~ay `~glymërkily# ya nkl c` ykbqkae~a kemlrg`ikùe8
^`oc` gly~r`enl cly n`~ly% &Ac`olr`ikùe zrlzk`+
Zraikzk~`ikùe & V + _aglikùe na z`r~çiqc`y &[ +&:%:0 ig%,nç`+ &±b,g9+
>%9 0?=>%1 0?01%; 00=1%= 006=%0 00>1%? 0069%6 09?
?%0 0>02%1 0:6
&`+ I`ciqc`r c` raglikùe na ile~`gke`e~ay &[ + iq`enl ac }`clr na c` zraikzk~`ikùe
zcq}k c ay na V 7 6%:% &0:?%>>+
&o+ ]`ckn`r ac glnacl na rabraykùe loha~k}` { yqoha~k}`gae~a%
6%0:% Ae qe ay~qnkl z`r` a}`cq`r c` i`z`ikn`n na cly yky~ag`y na mcqhl mraà~kil
&a~c`eny+# qy`nly z`r` c` nabr`n`ikùe na c` g`~ark` lrbàeki` na c`y `bq`y rayknq`cay
ya qyl ac z`ràga~rl na c` nag`en` oklpqçgki` na lvçbael &NOL+ { }`rkly l~rly
ilgzleae~ay pqçgkily% Ay~a ay~qnkl nkl ilgl rayqc~`nl cly ykbqkae~ay n`~ly% Ay~ly
rayqc~`nly ay~àe rac`ikle`nly ile c` i`rb` na g`y` na NOL &ae Db%,fai~àra`,nç`+# c`
iq`c ya qyù ilgl c` }`rk`oca kenazaenkae~a & V + {# c` nabr`n`ikùe na c` ileiae~r`ikùe
na g`y` i`rolely` na NOL1 &ae Db%,f`,nç`+# c` iq`c ya qyl ilgl c` }`rk`oca
nazaenkae~a &[ +% &Mqae~a na kemlrg`ikùe ay –Yqrm`ia Mcllr Ra~c`eny8 @ Zarmlrg`eiaA}`cq`~kle‗% R`~ar Ae}krle% _ay%# 0;;1# zz%?>>'?>2+%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 522/846
Nr% Fëi~lr Pqa}anl Qrç`y
6'=6
^`oc` gly~r`enl cly n`~ly nac zrlocag`%
& V + s 9 6 0: 00 09 0= ?2 9: 91 92 96 >> 0:9 0>?
&[ + s > 2 6 6 0: 00 0= ?= ?0 ; 90 9: 21 ;:
&Mqae~a8 R`~ar Ae}krle% _ay%# 0;;1+
I`ciqc`r cly ykbqkae~ay aeqeik`nly8
&`+ Br`mki`r cly n`~ly
&o+ Ay~`ocaiar ac glnacl na rabraykùe gày `zrlzk`nl z`r` ay~a zrlocag`% F`iar cly
gkygly iàciqcly pqa ac zrlocag` `e~arklr%
&i+ ]`ckn`r ac glnacl na rabraykùe yacaiikle`nl# loha~k}ky~`gae~a# qy`enl cly
ykbqkae~ay irk~arkly l nk`beùy~kily8
&0+ Iàciqcl nac ilamkikae~a na na~argke`ikùe _ ?
&?+ Iàciqcl nac ilamkikae~a na na~argke`ikùe `hqy~`nl# _ ?`hqy~`nl
&9+ Ac ilamkikae~a na ilrrac`ikùe _ &>+ C` ay~`nçy~ki` Z_AYY
&1+ Ac arrlr ay~àen`r na cl ay~kg`nl# y &Q~kc z`r` gankr c` q~kckn`n nac glnacl% Ya
yacaiikle` ac glnacl pqa ~aeb` ac }`clr na y gày zapqaðl+
&n+ A}`cq`r ac glnacl i`enkn`~l ` ~r`}ëy na cly ykbqkae~ay irk~arkly bràmkily8
&0+ Zrqao` na elrg`ckn`n
&?+ _ayknq`cay ae mqeikùe na cly lrnaeay
&9+ _ayknq`cay ae mqeikùe na cly }`clray `hqy~`nly
&a+ Qe` }ax pqa ya f`{` yacaiikle`nl ac glnacl gày `zrlzk`nl# i`ciqc`r c`
raglikùe nac NOL nayzqëy na pqa ac `bq` rayknq`c ya nabr`nù ae ac a~c`en iq`enl
c` i`rb` mqa na 1: Db%,f`,nç`%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 523/846
Nr% Fëi~lr Pqa}anl Qrç`y
6'=;
El~`8 Cly yky~ag`y na mcqhl mraà~kil &àra`y z`e~`ely`y+ ya qy`e ilgl yky~ag y na
~r`~`gkae~l e`~qr`c# zlrpqa ~kaeae c` i`z`ikn`n na nabr`n`r c`y ileiae~r`ikleay
i`rolely`y na NOL `i~q`enl ilgl ayzaika na c`bqe`y na lvkn`ikùe% Ae Gke`~k~càe{ Il`~x`il`cily# ]ar`irqx# ya qy`e ay~ly ~kzly na ~r`~`gkae~ly e`~qr`cay%
Ylcqikùe8
Na `iqarnl ` c` ~`oc` na `rrko` ya ca zkna `c cai~lr naiknkr# iqàc glnacl ay yqzarklr%
6%00% Ac ~av~l na N`ekac# R%R%# H`gay I% ^arracc Oqykeayy Y~`~ky~kiy &0;6;+# z% ?12
gaeikle`e ac zrlocag` nac ae}aeae`gkae~l nac b`e`nl }`iqel# zlr cly keyai~kikn`y
ac iq`c ay qe zrlocag` gq{ yarkl# zlrpqa cly zay~kikn`y ~kaeae c` m`iqc~`n na
`iqgqc`rya ae cly ~ahknly na cly `ekg`cay {# na `fç ya z`y`e ` `pqacc`y zaryle`y pqa
cly ileyqgae% @yç# ae `ņly raikae~ay# cly `gokae~`cky~`y ya f`e zraliqz`nl gqifl
zlr cly amai~ly# ae ac gankl `gokae~a# naoknl `c qyl kenkyirkgke`nl na keyai~kikn`y%
Ay }arn`n pqa cly keyai~kikn`y g`~`e cly keyai~ly# zarl ~`gokëe g`~`e ~lnl cl nagày%
Na ay~` g`ear`# cly keyai~kikn`y ile~`gke`e cly mrq~ly na c`y zc`e~`y# ` cly `ekg`cay {
~`gokëe ` c` bae~a% Cly ke}ay~kb`nlray Glqe~ { Lafga~ ay~qnk`rle cly amai~ly na cly
keyai~kikn`y ae c`y l}ah`y rac`ikle`n` ile c` `i~k}kn`n aexkgà~ki` ae ac iaraorl%
@nagày# na l~rly `eàckyky ay~`nçy~kily# ay~ly ikae~çmkily nark}`rle qe` cçea` na
rabraykùe pqa nayirkoa c`y rac`ikùe ae~ra c` `i~k}kn`n aexkgà~ki` ae ac iaraorl na c`y
l}ah`y &[ + { ac ~kagzl# ae flr`y# nayzqëy na pqa c`y l}ah`y f`y yknl avzqay~`y ` cly
keyai~kikn`y & V +% C` mqeikùe na c` cçea` na rabraykùe ay~kg`n` zlr ay~ly ikae~çmkily ya
n` `o`hl%
[ 7 ?2%9? $ 0%9= V O`y`enl ac irk~arkl ae ay~` aiq`ikùe# ay~kg`r cl ykbqkae~a8
&`+ Yk nayzqëy na pqa f`e z`y`nl 9: flr`y# iq`enl c`y l}ah`y f`e yknl avzqay~`y `
cly keyai~kikn`y# ¼Iqàc yarç` ac }`clr na c` `i~k}kn`n aexkgà~ki`< &=6%0?+
&o+ Yk ac ilamkikae~a na ilrrac`ikùe gqay~r`c ya n` ilgl _ 7 :%6= {# ac ilamkikae~a na
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 524/846
Nr% Fëi~lr Pqa}anl Qrç`y
6'2:
na~argke`ikùe ay _ ? 7 :%2> &ac pqa gkna c` mqarx` na c` rac`ikùe ckea`c ae~ra V a [ # ay
naikr# ac ! na `ylik`ikùe ae~ra c`y nly }`rk`ocay+# ae~leiay# f`iar qe` zrqao` na
fkzù~ayky ile Fl8π 7 :# ile~r` F08π ≪ : &pqa ay cl gkygl pqa naikr pqa el f`{`ylik`ikùe ckea`c ae~ra V a [ +% @yqgkr pqa ac ~`g`ðl na c` gqay~r` ay na e 7 0= { ac
ek}ac ykbekmki`e~a na ΰ 7 :%:1%
Z`r` ay~l# yabqkr c`y ykbqkae~ay yqbaraeik`y8
Qy`r c` nky~rkoqikùe na ~ ile η 7 e ’ ? br`nly na ckoar~`n { qy`r c`y rabkleay irç~ki`y
n`n`y zlr ~T0'ΰ,?4 ηW%
6%0?% Ae ay~qnkly na pqçgki`# c` zraykùe na qe b`y pqa ilrrayzlena ` }`rkly
}lcøgaeay &na `iqarnl ` c` ca{ na cly b`yay na Ol{ca+ ya n` ae c` ~`oc` na `o`hl%
@yqgkr pqa ac }lcqgae nac b`y ay & V + { c` zraykùe ay &[ +% F`iar cly ykbqkae~ay
iàciqcly8
&`+ F`iar qe` bràmki` ile cly n`~ly%
&o+ Ay~kg`r c` cçea` na rabraykùe na c` gqay~r`%
&i+ Ay~kg`r ac ilamkikae~a na na~argke`ikùe _ ? { ac ilamkikae~a na ilrrac`ikùe _%
Ke~arzra~`r cly rayqc~`nly%
&n+ Zranaikr c` zraykùe nac b`y# iq`enl ac }lcqgae ay %::0 g9
&a+ Zranaikr c` zraykùe nac b`y# ae ckor`y zlr zqcb`n` iq`nr`n` &coy,ke?+ {# ae
`~gùymar`y &`~g+# iq`enl ac }lcqgae nac b`y ay na :%:1?6 iq`r~ly &%:1 C+%
&m+ Ae ~alrç`# naoknl ` pqa c` rac`ikùe ae~ra ac }lcqgae nac b`y { c` zraykùe ay
ke}ary`gae~a zrlzlrikle`c# ac ilamkikae~a na ilrrac`ikùe naoarç` yar na _ 7 '0%:% Yke
ago`rbl# yk _ nkmkrkar` nac }`clr na '0%:# aecky~`r 9 m`i~lray pqa zqnkar`e ke~ar}aekr z`r` avzcki`r ay~` yk~q`ikùe%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 525/846
Nr% Fëi~lr Pqa}anl Qrç`y
6'20
^`oc` gly~r`enl cly }lcøgaeay { c`y zraykleay nac b`y% &Ac`olr`ikùe zrlzk`+
]lcqgae ae ig9 s 1:%: =:%: 2:%: ;:%: 0::%:
Zraykùe ae Db%,ig? s =>%2 10%9 >:%1 ?1%; 2%6
Yqbaraeik`y8 Ya n`e cly ykbqkae~ay m`i~lray na ile}arykùe8 0 `~g 7 0>%2 coy,ke? 7 2=:
~lrr 7 0%:==6 db,ig?4 0 ig? 7 :%0= ke?4 0%:1=2 iq`r~ly 7 0 C4 0 zqcb`n` iq`nr`n` 7
=%?1 ig?4 0 g9 7 0::: C% 7 0:= ig9%
6%09% Ya ilcaiikleù qe` gqay~r` na 99 i`yly na qe` nayi`rb` na `bq`y rayknq`cay
gqekikz`cay% Ay~` gqay~r` ya `e`ckxù z`r` c` nag`en` oklpqçgki` na lvçbael na 1
nç`y &NOL1+# ae ckor`y zlr nç`# { c` nag`en` pqçgki` na lvçbael# NPL &ae ckor`y zlr
nç`+% C` ~`oc` na o`hl gqay~r` c kemlrg`ikùe rapqarkn`%
^`oc` gly~r`enl c`y gankikleay na NOL1 { NPL% &Ac`olr`ikùe zrlzk`+Nag`en` pqçgki` na lvçbael Nag`en` oklpqçgki` na lvçbael
&coy,nç`+ &coy,nç`+
>;> >6= ?0= ?:?>>> 11= ?:: ?>:1?6 =:: ?96 ?6:9;= >?6 0=> 06>19? >>: ?9: 0;>9:6 ?;0 00= 09>91: >;: 01: ?01>1= 1>1 0;: ?>=>>: 16? 0;: ?;?1>> 9=6 ?>6 022
9:; 96= 0?: 0;9196 >:: ??= 0=1>6: 9>2 ?:: 0=:1:: ?26 ??? 0?19;= 9:> 02= 092
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 526/846
Nr% Fëi~lr Pqa}anl Qrç`y
6'2?
F`iar cl ykbqkae~a8
&`+ ]ar yq avky~a qe` ilrrac`ikùe ykbekmki`e~a qy`enl cly }`clray _ nac NOL1 { ac
NPL% &_ 7 :%;=22# _ ?
7 :%;9=:+&o+ Ke~arzra~`r ac }`clr nac ilamkikae~a na ilrrac`ikùe _ { ac ilamkikae~a na
na~argke`ikùe _ ?% Qy`r ac zrlbr`g` Gkek~`o l AVIAC z`r` f`iar ac iàciqcl zanknl%
El~`% C` nag`en` oklpqçgki` na lvçbael na 1 nç`y &NOL1+ gkna c` ileiae~r`ikùe#
ae gb,C l ae ckor`y zlr nç` na c` g`~ark` i`rolely` nac `bq` rayknq`c% Na faifl ac
NOL gkna c` mr`iikùe oklnabr`n`oca nac nrae`ha# l nac `bq` rayknq`c kenqy~rk`c l
nlgëy~ki`# ae ~ërgkely nac i`rolel% Qyq`cgae~a# c`y qekn`nay yle ae gb,C% Yke
ago`rbl# ay~l ya naoa ` pqa# `e~arklrgae~a# ya qy`o` kenkyirkgke`n`gae~a c`y
qekn`nay na zzg { gb,C% Nayzqëy# ya }kl pqa# ile cly rayknqly ~ùvkily# c` br`}an`n
ayzaiçmki` ar` nkmarae~a ` c` na cly rayknqly i`rolelyly% Zlr ay~` r`xùe ay gahlr qy`r
c`y qekn`nay na gb,C% Zlr l~r` z`r~a# c` zrqao` nac NOL ay na 1 nç`y# z`r` a}k~`r c`
ek~rkmki`ikùe% Ae i`gokl# c` zrqao` na c` nag`en` pqçgki` na lvçbael gkna cly
ilgzqay~ly lrbàekily oklnabr`n`ocay { cly ilgzqay~ly lrbàekily ~ùvkily% Ay~lpqkara naikr pqa# c` nag`en` pqçgki` na lvçbael &NPL+ lvkn` c` i`e~kn`n na
g`~ark`cay ~l~`cay lvkn`ocay zrayae~ay ae ac `bq` rayknq`c { }`rç` ile c` ilgzlykikùe
nac `bq`# c` ~agzar`~qr`# ac zarklnl na ile~`i~l { l~rly m`i~lray gày%
6%0>% Ileyknar`r cly n`~ly na `o`hl rac`ikle`nly ile ac zayl nac }afçiqcl { ac
raenkgkae~l na b`ylcke`% Ac zayl nac `q~l ya n` ae ~leac`n`y {# ac raenkgkae~l nacilgoqy~koca ya n` ae gkcc`y b`cùe% Cly n`~ly ya n`e `o`hl%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 527/846
Nr% Fëi~lr Pqa}anl Qrç`y
6'29
^`oc` gly~r`enl cly n`~ly nac cly glnacly }afkiqc`ray ae mqeikùe nac zayl ae~leac`n`y &V+# { na gkcc`y zlr b`cùe &[+%
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
Glnacl }afkiqc`r Zayl &~leac`n`y+ Gkcc`y,b`cùe
Oqkid Ay~`~a R`ble >%9= 0=%;Mlrn Ilqe~r{ Ypqkra R`ble >%:1 01%1Ifa}{ G`ckoq R`ble 9%=0 0;%?Ifr{ycar Ca O`rle R`ble 9%;> 06%1^l{l~` Ilrle` ?%1= ?2%1N`~yqe 10: ?%9: ?2%?Nlnba Lgek ?%?9 9:%;@qnk 1::: ?%69 ?:%9]lc}l ;; BCA 9%0> 02%:
Y``o ;; BCA ?%6: ?0%=Zaqbl~ =;> YC 9%>0 0=%?Oqkid Iae~qr{ Yzaik`c 9%96 ?:%=Gariqr{ Xazf{r 9%:2 ?:%6Nlnba @yzae 9%=? 06%=@GI Ileilrn N,C 9%>0 06%0Ifa}{ I`zrkia Ic`yyki 9%6> 02%:Mlrn C^N 9%29 02%=Gariqr{ Br`en G`rpqky 9%;= 0=%1Mlrn Gqy~`eb ?%1; ?=%1G`xn` BCI 0%;6 9>%0Nlnba Ilc~ 0%;? 91%0]R Yikrliil 0%;; 90%1Flen` @iilrn CV ?%0> ?;%1Oqkid Yd{c`rd ?%=2 ?6%>Ifa}{ Ik~`~kle ?%=: ?6%6Lcnyglokca Lgab` ?%2: ?=%6Zc{glq~f Flrkxle ?%?: 9>%?N`~yqe ?0: ?%:? 90%6]R N`yfar ?%0; 9:%1N`~yqe 60: ?%6? ??%:OGR 9?0: ?%=: ?0%1]R _`ook~ 0%;9 90%;
&Mqae~a8 Zrlo`okckn`n { Ay~`nky~ki` @zcki`n`y ` c` Kebaekark % Gle~blgar{ a~ `c%
0;;=+
F`iar cly ykbqkae~ay iàciqcly qy`enl ac zrlbr`g` Gkek~`o%
&`+ Ay~kg`r c` cçea` na rabraykùe ae~ra c`y }`rk`ocay zayl nac }afçiqcl { ac
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 528/846
Nr% Fëi~lr Pqa}anl Qrç`y
6'2>
raenkgkae~l na b`ylcke`%
&o+ Ay~kg`r ac ilamkikae~a na ilrrac`ikùe na c` gqay~r` _ &cc`g`nl ~`gokëe
ilamkikae~a na ilrrac`ikùe na zrlnqi~l'glgae~l na Za`ryle+ { ac ilamkikae~a nana~argke`ikùe gqay~r`c _ ?%
&i+ F`iar qe` bràmki` pqa }`{` ae mqeikùe na [ { V # ~r`x`rc` ae c` bràmki` { ~`gokëe
~r`x`r c` cçea` flrkxle~`c qy`enl ac }`clr nac zrlgankl na [ %
&n+ F`iar qe` ~`oc` na @EL]@%
&a+ Ay~kg`r cly ke~ar}`cly na ilemk`ex` z`r` ΰ { ο c`y zrlo`okckn`nay
ilrrayzlenkae~ay z`r` i`n` qel na ëy~ly%
&m+ ¼Pqë l~rly m`i~lray ~aenrç`e pqa ileyknar`rya# z`r` pqa ac glnacl na rabraykùe
mqar` gày ilemk`oca<
6%01% Cly ga~`cay zay`nly ilgl ac Fb# Ir# Zo# a~i%# zqanae ke~armarkr ile ac
~r`~`gkae~l oklcùbkil ae c`y zc`e~`y gqekikz`cay na `bq`y rayknq`cay nlgëy~ki`y% Ae
ay~a ay~qnkl ya fkikarle gankikleay gaeyq`cay ae qe` zc`e~` glnacl na ~r`~`gkae~l
na c`y ileiae~r`ikleay na irlgl# Ir# ae gb,C# ~`e~l ae ac amcqae~a ilgl ae c`
ae~r`n`% Cly rayqc~`nly na c`y ileiae~r`ikleay na Ir ya n`e ae c` ~`oc` na `o`hl%
^`oc` gly~r`enl cly n`~ly% &Ac`olr`ikùe zrlzk`+Ae~r`n` & V + s ?1: ?;: ?2: 0:: 9:: >0: 00: 09: 00::&±b,C+Amcqae~a &[ + s 0; 0: 02 00 2: =: 06 9: 06:&±b,C+
F`iar cly ykbqkae~ay iàciqcly8
&`+ F`iar qe nk`br`g` na nkyzarykùe ae z`zac `rk~gë~kil%&o+ F`iar qe nk`br`g` ayz`riknl ae z`zac yagkclb`rk~gl { clb`rk~gl ilgzca~l
&~r`eymlrg`ikùe na cly ahay+%
&i+ I`ciqc`r cly glnacly na rabraykùe z`r` c`y z`r~ay &`+ { &o+%
&[ 7 ';%:= $ :%02 V 4 [ 7 '0%;= $ :%;2 V +
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 529/846
Nr% Fëi~lr Pqa}anl Qrç`y
6'21
&n+ Ay~kg`r [ iq`enl V 7 91: ae keikyly &`+ { &o+%
&a+ I`ciqc`r ac ilamkikae~a na ilrrac`ikùe z`r` &`+ { &o+% &_ ? 7 :%;>?1# _ 7 :%2>=;+
&m+ Ilgae~`r ylora cl `zrlzk`nl na [ { na _ ae i`n` i`yl%6%0=% Ae qe ay~qnkl na gkirloklclbç` `gokae~`c rac`ikle`nl ile ac iqc~k}l na qe`
gqay~r` na `bq` ya n`e cly ykbqkae~ay n`~ly%
^`oc` gly~r`enl cly n`~ly% &Ac`olr`ikùe zrlzk`+^kagzl ae nç`y nayna s 9 = ; 0? 01 06c` keliqc`ikùe & V +
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ El% na o`i~ark`y &[ + s 001#::: 0>#2:: ?9#;:: 91#=:: 12#;:: 6=#>::
F`iar cly ykbqkae~ay iàciqcly8
&`+ ^r`x`r qe` iqr}` Ce [ k }aryqy V k z`r` }ar pqë ~`e okae ya zqana `hqy~`r qe` iqr}`
avzleaeik`c ` cly n`~ly%
&o+ ^r`x`r qe` iqr}` [ k }aryqy V k z`r` }ar pqa ~`gokëe ya zqana `hqy~`r qe` cçea`
rai~` ` cly n`~ly%
&i+ Zlr ke~arzlc`ikùe# qy`enl `go`y bràmki`y ay~kg`r ac eøgarl na o`i~ark`y nayzqëy
na ?: nç`y% Iq`e~kmki`r c`y nkmaraeik`y ae `goly i`yly%
6%02% Ae ac ckorl na H% C Na}lra# Zrlo`okckn`n { Ay~`nçy~ki` z`r` Kebaekarç` {
Ikaeik`y ya n` qe` ke}ay~kb`ikùe rac`ikle`n` ile c` ~agzar`~qr` &lI+ { c`
zrlmqenkn`n na c` eka}a iqgqc`n` ae ac yqacl% Z`r` ay~l ya c` ~`oc` na `o`hl8
^`oc` gly~r`enl cly n`~ly nac zrlocag`% \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ ^agzar`~qr` &lM+ s '=? '>0 '9= ?= '99 '1= '1: '==
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ Zrlmqenkn`n na c` s ?0 09 0? 9 = ?? 0> 0;i`z` na eka}a
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ &Mqae~a8 Na}lra ?::0+ %
&`+ Knae~kmki`r c` }`rk`oca nazaenkae~a &[ + { c` }`rk`oca kenazaenkae~a & V +%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 530/846
Nr% Fëi~lr Pqa}anl Qrç`y
6'2=
&o+ Ay~kg`r qe glnacl na rabraykùe ckea`c% &[ 7 1%20 ’ :%?:?& V +# _ ? 7 :%2>0# y 7
9%21;# Z_AYY 7 >:;%:?# M 7 02%6# z 7 :%::=+
&i+ Ay~kg`r qe glnacl iq`nrà~kil% &[ 7 9%9 ’ :%:;>9& V + $ :%::?;& V ?
+# y 7 :%:0;# _ ?
7 :%;;=# Z_AYY 7 :%::96# M 7 0>%;6# z 7 %::6+
&n+ Ay~kg`r qe glnacl iøokil% &[ 7 ;%;= ’ :%09;& V + $ :%:06;& V ?+ $ :%:::??& V
9+#
_ ? 7 :%;0># y 7 ?%=1=# Z_AYY 7 6::2%21# M 7 0>%0># z 7 :%0>+
&a+ Na `iqarnl ` cly rayqc~`nly ay~`nçy~kily# ¼Iqàc na cly ~ray glnacly ay yqzarklr<
6%0;% C` ~`y` na mcqhl ae g9,gke ae qe gqay~ra`nlr na `c~l }lcqgae z`r` gankr c`
i`ckn`n nac `kra# ay naikr# z`r` z`r~çiqc`y `~glymërki`y# nazaena na c` i`çn` na
zraykùe# ae zqcb`n`y na `bq`# ` ~r`}ëy nac mkc~rl nac gqay~ra`nlr% Ykaenl `yç#
yqzùeb`ya pqa ya ilcaiikleù qe` gqay~r` na 01 }`clray na i`çn` na zraykùe { c` ~`y`
na mcqhl nac `kra ` ~r`}ëy nac mkc~rl nac yaeylr% Cly n`~ly ya n`e ae c` ~`oc` na `o`hl%
^`oc` gly~r`enl cly n`~ly z`r` ay~a zrlocag`% &Ac`olr`ikùe zrlzk`+^`y` na mcqhl nac `kra ile I`çn` na zraykùe nayzqëy nac`y z`r~çiqc`y &g9,gke+ `cbøe ~kagzl &zqcb`n`y na `bq`+
?%:: 1%:
0%;; =%:0%66 2%:0%2= 2%60%=6 6%>0%12 ;%=0%>= ;%;0%>: 0:%=0%9; 00%20%?: 0>%:
0%01 01%;0%:2 0;%:0%:0 ?>%:0%:: ?6%::%;1 91%:
&`+ I`ciqc`r ac glnacl na rabraykùe gqay~r`c pqa ay~kga ` c` }arn`nar` cçea`
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 531/846
Nr% Fëi~lr Pqa}anl Qrç`y
6'22
zloc`ikle c% Z`r` ay~l# knae~kmki`r# zrkgar`gae~a# c` }`rk`oca nazaenkae~a { c`
}`rk`oca rabraylr`% &[ 7 0%;1 ’ :%:9=> & V ++
&o+ ]`ckn`r ac glnacl na rabraykùe ay~kg`nl ae &`+ qy`enl aemlpqay yqoha~k}ly# aynaikr# ` ~r`}ëy na bràmki`y ile rayknqly ay~`en`rkx`nly }aryqy }`clray na i`çn` na
zraykùe% ^`gokëe f`iar l~r` bràmki` na rayknqly ay~`en`rkx`nly }aryqy cly raebcleay%
F`iar l~r` bràmki` gày ile cly }`clray rayknq`cay }aryqy cly }`clray na x z`r` c`
zrqao` na elrg ckn`n%
&i+ Ilgzcagae~`r c` }`ckn`ikùe nac glnacl na rabraykùe qy`enl gë~lnly ay~`nçy~kily
loha~k}ky~`y% Z`r` ay~l# ay~kg`r ac ilamkikae~a na na~argke`ikùe _ ?# ac arrlr ay~àen`r
na cl ay~kg`nl &y n`nl zlr ac Gkek~`o+ { Z_AYY% Qy`r ac zrlbr`g` Gkek~`o z`r`
ay~ly iàciqcly% &_ ? 7 2=%:!# y 7 :%06=;# Z_AYY 7 :%2>:1+
6%?0% Ya f`ia qe ay~qnkl ylora c` ileiae~r`ikùe na i`ngkl `~glymërkil# ae zzg# {k {
yq rac`ikùe ile V k 7 c` `c~qr` na cly gqay~ra`nlray { V ? 7 nky~`eik` na c` mqae~a
agkylr`% C` ~`oc` na `o`hl gqay~r` cly n`~ly% F`iar cly ykbqkae~ay iàciqcly8
&`+ @hqy~`r ac glnacl na rabraykùe pqa zqan` `hqy~`r ` cly n`~ly nac zrlocag` na c`
ileiae~r`ikùe na In% &[ 7 91:%;; ’ 0%?2 V 0 ’ :%01> V ?+
&o+ ]`ckn`r ac glnacl qy`enl aemlpqay na nk`beùy~kil na ay~`nçy~ki` na kemaraeik`
&loha~k}ky~`y+ { na `eàckyky bràmkil &yqoha~k}ky~`y+%
&i+ Qy`r ac glnacl na rabraykùe ckea`c gøc~kzca z`r` zranaikr ac c` ileiae~r`ikùe na
i`ngkl# iq`enl c` `c~qr` nac gqay~ra`nlr ay na V 0 7 ?1 ga~rly { c` nky~`eik` na c`
mqae~a agkylr`# ay V ? 7 610 ga~rly% &066%? zzg na In+
C` ~`oc` na `o`hl gqay~r` cly n`~ly rapqarknly zlr ay~a zrlocag`%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 532/846
Nr% Fëi~lr Pqa}anl Qrç`y
6'26
^`oc` gly~r`enl cly n`~ly% &Ac`olr`ikùe zrlzk`+
{ &ileiae~r`ikùe na In+ s 0;9 ?9: 02? ;0 009 0?1
V 0 &@c~qr` nac gqay~ra`nlr+ s 0%= 01%1 ??%: >9%: 99%: >:%:
V ? &Nky~`eik`+ s 610 60= 0:16 0?:0 0912 0001
6%??% Ac ~av~l @zzckan Y~`~ky~kiy8 @e`c{yky na ]`rk eia `en _abrayykle na cly
ke}ay~kb`nlray Lck}a Nqee { ]krbkek` Ic`rd# nkyiq~ae qe ahagzcl z`r` zranaikr ac
raenkgkae~l na c` ilyaif` na iao`n`# ae mqeikùe na c` zraikzk~`ikùe zcq}k`c V 0 { c`
~agzar`~qr V ?% Z`r` ay~l# f`iar cly ykbqkae~ay iàciqcly8&`+ Aecky~`r ac glnacl na rabraykùe ckea`c gøc~kzca pqa gahlr `hqy~a ` cly n`~ly%
&o+ Ay~kg`r c` aiq`ikùe na cly iq`nr`nly gçekgly pqa `hqy~a ac raenkgkae~l na ~rkbl
&[ + ` c` zraikzk~`ikùe zcq}k`c & V 0+ { c` ~agzar`~qr` & V ?+%
&i+ Zrlo`r c` fkzù~ayky na Fl8 ο? 7 : ile ΰ 7 :%:1%
&n+ Ay~kg`r ac ilamkikae~a na ilrrac`ikùe z`rik`c π?{%0 { zrlo`r Fl8π ?{%0 7 :
&a+ ]`ckn`r ac glnacl na rabraykùe nark}`nl z`r` }ar# pqë ~`e~` ilemk`okckn`n ya ca
zqana `irank~ r% F`iar ay~l# ` ~r`}ëy na hqkikly loha~k}ky~`y# ilgl cly nk`beùy~kily _ ?#
_ ?`hqy~`n`# _ ? zranaikn # y# Z_AYY { Iz% Ilgzcagae~`r c` a}`cq`ikùe nac glnacl qy`enl
~ëieki`y yqoha~k}ky~`y# ilgl cly `eàckyky na cly bràmkily na rayknq`cay ay~`en`rkx`nly {
ay~qnk`e~kx`nly# zrqao` na elrg`ckn`n# a~i%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 533/846
Nr% Fëi~lr Pqa}anl Qrç`y
6'2;
^`oc` gly~r`enl cly raenkgkae~ly na iao`n` ilgl }`rk`oca nazaenkae~a na c`
zraikzk~ ikùe zcq}k c { c` ~agzar`~qr`%
_aenkgkae~l na iao`n` & {k+ Zraikzk~`ikùe & v0+ ^agzar`~qr` & v?+&m`eab`y,`ira+ &zqcb`n`y+ &lM+
?0%: >1 1>%0?:%: >2 =0%=?0%: 99 1:%6?>%: 9; 1?%0?:%: 9: 1:%?0?%1 ?6 12%00;%: >0 11%2
?9%: >> 12%=?9%: 90 1:%00;%: ?; 96%:?0%: 9> 1=%?0?%: ?2 10%1?0%: >? 1>%0?2%: 91 >=%202%1 >9 =:%6?=%: 9; 1=%;00%: 90 =:%9
?>%: >? 1>%=?=%: >9 19%106%1 >2 =>%:01%1 ?1 >1%2
&Mqae~a8 Nqee a~ c% 0;2>% @zzckan Y~`~ky~kiy8 @e`c{yky na ]`rk`eia `en _abrayykle+
6%?9% Ac ~av~l @zzckan Y~ ~ky~kiy8 @e`c{yky na ]`rk eia `en _abrayykle na Nqee a~ `c%
0;2>+ f`ia qe ay~qnkl gënkil rac`ikle`nl ile ac i`gokl na c` faglbcloke` na c`
y`ebra na lzar`ikleay na c` bcàenqc` ~krlknay# ac iq`c ay~à rac`ikle`nl ile c` nqr`ikùena c` lzar`ikùe pqkrørbki` { ac i`gokl ae ac zlriae~`ha na c` faglbcloke` na c`
y`ebra% Cly n`~ly ya n`e ae c` ~`oc` na `o`hl%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 534/846
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 535/846
Nr% Fëi~lr Pqa}anl Qrç`y
6'60
zcày~kily ae zayl# ! na z`zac ae zayl# ! na o`yqr ae zayl { ! na fqgan`n na zayl%
^`oc` gly~r`enl cly n`~ly na ay~a zrlocag`% \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ Loy% ! Zcày~kil &v0+ ! Z`zac &v?+ ! O`yqr` &v9+ ! Fqgan`n Ile~aeknl aearbë~kil &{+ \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ 0 06%=; 01%=1 >1%:0 16%?0 ;>2? 0;%>9 ?9%10 9;%=; >9%=0 0>:29 0;%?> ?>%?9 >9%0= >=%=9 0>1?> ??%=> ??%?: 91%2= >1%61 01191 0=%1> ?9%1= >0%?: 11%0> ;6;= ?0%>> ?9%=1 91%1= >?%?> 00=?2 0;%19 ?>%>1 >:%06 >2%?: 0>==6 ?9%;2 0;%9; >>%00 >9%6? 0=1=; ?0%>1 ?9%6> 91%>0 10%:0 0?1>0: ?:%9> ?=%1: 9>%?0 >;%:= 099=00 02%:9 ?9%>= 9?%>1 19%?9 0:;20? ?0%:9 ?=%;; 96%0; 10%26 0?==09 ?:%>; 0;%62 >0%91 >=%=; 0>:00> ?:%>1 ?9%:0 >9%1; 19%12 0??901 06%60 ??%=? >?%?: 1?%;6 0?0=0= 06%?6 ?0%62 >0%1: >2%>> 099>02 ?0%>0 ?:%>2 >0%?: 1>%=6 001106 ?1%00 ??%1; 92%:? >6%2> 0>190; ?0%:> ?=%?2 96%== 19%?? 0?26?: 02%;; ?6%?? >>%06 19%02 0019?0 06%29 ?;%9; 9>%22 10%:= 0??1
?? 06%>; ?=%16 92%11 1:%== 0?92?9 ??%:6 ?>%66 92%:2 1:%2? 09?2?> 0>%?6 ?=%?2 91%6: >6%?> 0??;?1 02%2> ?9%=0 92%9= >;%;? 0?:1?= ?:%1> ?=%16 91%>: 19%16 0??0?2 06%?1 09%22 10%9? 10%96 0096?6 0;%:0 ?1%=? 9;%1> 1:%09 0?;1?; ?0%?1 ?:%=9 >:%2? >6%=2 09;?9: ?0%=? ??%20 9=%?? >6%0; 092? \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
Mqae~a8 H`{ C% Na}lra% Zrlo`okck~{ `en Y~`~ky~kiy mlr Aebkeaarkeb `en ~fa Yikaeiay
&?:::+
&`+ Lo~aear ac glnacl na rabraykùe { }`ckn`rcl `ilrnagae~a# ay naikr# qy`enl
nk`beùy~kily yqoha~k}ly { nayzqëy ilgzcagae~`r c` ~`ra` qy`enl nk`beùy~kily
loha~k}ly%
6%?1% ^rake~` gqay~r`y nac amcqae~a na qe` zc`e~` na ~r`~`gkae~l ya `e`ckx`rle z`r` c`
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 536/846
Nr% Fëi~lr Pqa}anl Qrç`y
6'6?
gankikùe nac NOL1 { c` nag`en` oklpqçgki` na lvçbael &NPL+% Cly n`~ly ya
gqay~r`e ae c` ~`oc` na `o`hl% F`iar cl ykbqkae~a8
&`+ I`ciqc`r ac zrlgankl# y { ac arrlr ay~àen`r nac NOL { nac NPL% & V NOL 7 >>:%=#y 7 ;9%06# arrlr ay~àen`r 7 02%:04 V NPL 7 0;>%># y 7 >1%9# arrlr ay~àen`r 7 6%?2+
&o+ Br`mki`r cly n`~ly ae z`zac na zrlo`okckn`n%
&i+ Na~argke`r ac NOL1 { ac NPL pqa ya avianarà ac 1:! na c`y }aiay% &Ac NOL1
avianarà 0;1 coy,nç` ac 1:! na c`y }aiay% Ac NPL avianarà >>: coy,nç` ac 1:! nac
~kagzl+
&n+ Na~argke`r ac NOL1 { ac NPL pqa ya avianarà ac ;:! nac ~kagzl%^`oc` gly~r`enl c`y ileiae~r`ikleay na NPL { na NOL1% &Ac`olr`ikùe zrlzk`+
NPL s >;> >;> 1?6 9;= 19? 9:6 91: >1= >>: 1>>&coy,nç`+ s 90: 196 >6: 1:: 9;= >6= 11= =:: >?6 >>:
s ?;0 >;: 1>= 16? 9=6 96= >:: 9>2 ?26 9:>NOL1 s ?0= ?:: ?96 0=> ?9: 00= 01: 0;: 0;: ?>6
&coy,nç`+ s 0?: ??= ?:: ??? 02= ?:? ?>: ?6: 06> 0;>s 09> ?01 ?>= ?;? 022 0;9 0=1 0=: 0?1 092
6%?=% Ac nkrai~lr na c` lmkike` na zaryle`c na qe` mkrg` iley~rqi~lr` naya` y`oar yk
c` nay~rax`# ae na~argke`nl ~kzl na ~r`o`hl# nae~rl na c` agzray`# zqana yar
zrlely~ki`n` qy`enl ilgl zrlely~ki`nlray c`y }`rk`ocay an`n { avzarkaeik` na cly
agzca`nly% C` ~`oc` na `o`hl n` c` kemlrg`ikùe na qe` gqay~r` `ca`~lrk` na 01
agzca`nly% &@n`z~`ikùe nac ckorl Oqykeayy Y~`~ky~kiy na N`ekac a~ `c% 0;6;# z% 122+%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 537/846
Nr% Fëi~lr Pqa}anl Qrç`y
6'69
^`oc` gly~r`enl cly n`~ly nac zrlocag`% \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ Ek}ac na & {+ Avzarkaeik` & v0+ An`n & v?+
Nay~rax`
01 : ?001 : 06?0 : ???6 0 ?>9: 0 ?191 0 ?1>: 0 ?=
91 ? 9>9: ? ?1>1 ? 961: 9 >>=: 9 10>1 > 9;=: > 1>1: 1 11
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ Mqae~a8 N`ekac a~ `c% 0;6;% Oqykeayy Y~`~ky~kiy
F`iar cly ykbqkae~ay iàciqcly8
&`+ Aeile~r`r c` aiq`ikùe na rabraykùe na cly iq`nr`nly gçekgly%
&o+ Ilgzq~`r _ ?{%0?%
&i+ Zrlo`r Fl8ο0 7 : { Fl8ο? 7 :% Nah`r pqa ΰ 7 :%:1 { i`ciqc`r ac }`clr na z z`r`
i`n` zrqao`%
&n+ Ilgzq~`r ac ;1! na ke~ar}`cl na ilemk`ex` z`r` ο?%
&a+ Nah`r pqa v0 7 ? { v? 7 ?1 { i`ciqc`r {%&m+ Aeile~r`r ac ke~ar}`cl na ;1! z`r` {%
6%?2% C` i`z`ikn`n na cly ailclbky~`y z`r` knae~kmki`r rabkleay na gàvkg` rkpqax`
na c`y zc`e~`y zlnrç` ~aear qe kgz`i~l ylora c` zrayar}`ikùe na c` nk}arykn`n
baeë~ki`% Ay~l ay qel na cly loha~k}ly na cly ailclbky~`y pqkaeay ay~àe zraliqz`nly
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 538/846
Nr% Fëi~lr Pqa}anl Qrç`y
6'6>
zlr ac gankl `gokae~a% Ac `r~çiqcl –Zranki~kle lm _`rk~kay mrlg F`ok~`~ ]`rk`ocay8
Il`y~`c Zc`ke Zc`e~y le El}` Yil~k`e C`dayflray‗ &Ailclb{# 0;;?# zz% 061?'061;+
qyù qe` gqay~r` na 92 c`bly { ya lo~q}l c` aiq`ikùe na rabraykùe na `o`hl% Ay~a zrlocag` ya y`iù nac ckorl nac ke}ay~kb`nlr H% C% Na}lra &?::0+%
{ 7 9%6; $ %:99 v0 $ %:?> v? $ %:?9 v9 ' %::6: v> ' %09 v1 ' %2?v=
Nlena8
{ 7 rkpqax` na ayzaikay na zc`e~`y
v0 7 àra` na c` iqaei`
v? 7 `eifl na c` zc`{`
v9 7 g`c nrae`nl &!+
v> 7 ilclr nac `bq`
v1 7 ! na `rae`
v= 7 `ci`ckekn`n%
Ac ay~qnkl razlr~ù qe ilamkikae~a na na~argke`ikùe gøc~kzca na _ ? 7 :%69% _a`ckx`r
qe` zrqao` na c` q~kckn`n nac glnacl na rabraykùe% Yqbaraeik`8 qy`r c` mqeikùe
ay~`nçy~ki`8 M 7 T_ ?,d W , T&0 ' _ ?+,&e ' &d $ 0++W# ile rabkùe na raif`xl z`r` qe`
zrqao` na ek}ac na M ≡ Mΰ#d#e'&d$0+# nlena d ay ac eøgarl na zrlely~ki`nlray qy`nly%
Qy`r c` ~`oc` na c` nky~rkoqikùe M% ]`clr`r c` q~kckn`n nac glnacl na `iqarnl `c
}`clr na c` zrlo`okckn`n z%
6%?6% Ay~a ay aharikikl pqa ke}lcqir` c` yacaiikùe na qe glnacl na rabraykùe ile ;
}`rk`ocay kenazaenkae~ay l zranki~lr`y# ay naikr# v0 # v? # v9 # v> # v1 # v= # v2 # v6 { v;%
O`y`enl ac irk~arkl ae cly nk`beùy~kily _ ?
# GYA { Iz &irk~arkl na G`ccl+# naikr iqàc glnacl na rabraykùe ay ac gày `zrlzk`nl% Ay~l ay# yacaiikle`enl cly gahlray
yqoilehqe~ly zlykocay% Cly n`~ly ya n`e `o`hl%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 539/846
Nr% Fëi~lr Pqa}anl Qrç`y
6'61
^`oc` gly~r`enl cly n`~ly% &Ac`olr`ikùe zrlzk`+ \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
Yqoilehqe~l na zranki~lray
0 ? 9 > 1 = 2 6 ;Irk~arkly \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
_ ?d s %91> %>19 %100 %11: %1=? %12: %12? %121 %121
GYAd s ??;1 0;>6 02>? 0=:2 01== 01>0 0191 019: 019?
Izd s 90> 029 6;%= 91%2 0;%; 00%: ;%> 6%? 0:%: \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
6%?;% Ae qe ay~qnkl na c`olr`~lrkl z`r` }ar c` rac`ikùe ae~ra cly yùcknly
yqyzaenknly { c`y ileiae~r`ikleay na NOL ya y`iù qe` gqay~r` ile cly n`~ly pqa
ya gqay~r`e ae c` ~`oc` na `o`hl%
^`oc` gly~r`enl cly n`~ly% &Ac`olr`ikùe zrlzk`+
Yùcknly yqyzaenknlys 06 2 0> 90 ?0 1 00 0= ?= ?;NOL1 s 11 02 9= 61 =? 06 99 >0 =9 62
&`+ F`iar qe` bràmki` pqa }`{` ae mqeikùe na c` }`rk`oca nazaenkae~a { na c`
}`rk`oca kenazaenkae~a%
&o+ Lo~aear ac glnacl na c` aiq`ikùe na rabraykùe { ~r`x`rc` ae c` bràmki`% &Yùcknly
yqyzaenknly [ 7 :%9? $ :%91? & V +
&i+ ]`ckn`r ac glnacl na rabraykùe loha~k}`gae~a# i`ciqc`enl ac ilamkikae~a na
na~argke`rle _ ?# y { Z_AYY% &_ ? 7 :%;=?# y 7 :%;12# y 7 0%61# Z_AYY 7 >?%96+
&n+ F`iar qe` ~`oc` na @EL]@ pqa keicq{` ac }`clr na M { z% &Ilgzca~`r c` ~`oc`
na `o`hl%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 540/846
Nr% Fëi~lr Pqa}anl Qrç`y
6'6=
^`oc` na @EL]@% &Ac`olr`ikùe zrlzk`+ \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ Mqae~a b%c% YY GY Mi`ci% M~`o% ]`clr z
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ Naoknl ` c` 0 =;>%0=rabraykùeArrlr ?2%>> 9%>9^l~`c ; 2?0%=:
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
&a+ F`iar qe nk`beùy~kil bràmkil z`r` }`ckn`r c` `q~ae~kikn`n nac glnacl na
rabraykùe yacaiikle`nl% Yqbaraeik`8 qy`r ac zrlbr`g` Gkek~`o%6%9:% ^rake~` i`yly nac amcqae~a na qe` zc`e~` na ~r`~`gkae~l ya `e`ckx`rle z`r` ac
NOL { ac NPL% Cly n`~ly ya gqay~r`e ae c` ~`oc` na `o`hl% F`iar cly ykbqkae~ay
iàciqcly8
^`oc` gly~r`enl cly n`~ly nac zrlocag`% &Ac`olr`ikùe zrlzk`+
NPL &coy,Nç`+s >;> >>> 1?6 9;= 19? 9:6 91: >1= >>: 1>> 90: 196
s >6: 1:: 9;= >6= 11= =:: >?6 >>: ?;0 >;: 1>= 16?s 9=6 96= >:: 9>2 ?26 9:>
NOL &coy,Nç` s ?0= ?:: ?96 0=> ?9: 00= 01: 0;: 0;: ?>6 0?: ??=
s ?:: ??? 02= ?:? ?>: ?6: 06> 0;> 09> ?01 ?>= ?;?
s 022 0;9 0=1 0=: 0?1 092
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
&`+ Na~argke`r _ ? { _% &_ ? 7 :%;91:# _ 7 :%;=2+
&o+ Br`mki`r cly n`~ly ae z`zac na zrlo`okckn`n { na~argke`r cl ykbqkae~a8
&0+ Na~argke`r cly }`clray na NOL { ac NPL pqa avianaràe ac 1:! { ac ;:! na
c`y }aiay% &0;1 coy,Nç` { >>: coy,Nç`+
&?+ Na~argke`r cly }`clray na NOL { nac NPL pqa ya clbr`ràe ac ;:! na c`y }aiay%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 541/846
Nr% Fëi~lr Pqa}anl Qrç`y
6'62
&9+ Na~argke`r cly }`clray na NOL { nac NPL pqa zqan`e yar avianknly ac 0:! na
c`y }aiay% &?=: coy,Nç` { 16: coy,Nç`+&>+ I`ciqc`r ac zrlgankl { c` nay}k`ikùe ay~àen`r nac NOL { nac NPL%
&i+ A}`cq`r ac glnacl na rabraykùe `zcki`enl aemlpqay yqoha~k}ky~`y# ay naikr# ile
bràmki`y na cly }`clray rayknq`cay ae mqeikùe na }`clray `hqy~`nly &z`r` c` zrqao` na
kenazaenaeik`+# zrqao`y na elrg`ckn`n# a~i%
6%90% Ay~a ay qe zrlocag` `n`z~`nl nac ckorl Ke~rlnqiikùe `c @eàckyky na _abraykùe
Ckea`c na cly `q~lray Gle~blgar{# Zaid { ]kekeb &?::0+% Ay~a zrl{ai~l ay~à
rac`ikle`nl ile qe ay~qnkl na aearbç` ylc`r ae ac ^aielcùbkil na Balrbk`# Ay~`nly
Qeknly% Ac zrl{ai~l ke}lcqir` n`~ly na zrqao`y na aearbç` ~ërgki` ile qe`
}`rk`oca nazaenkae~a &{+# pqa rac`ikle` `c mcqhl ~l~`c na i`clr &D`~~y+ { ikeil
}`rk`ocay kenazaenkae~ay pqa ay~àe rac`ikle`n`y ile c` keylc`ikùe &`~~y,g?+# c`
zlykikùe nac mlil ae nkraiikùe nac ay~a &ae zqcb`n`y+# c` zlykikùe nac mlil ae
nkraiikùe nac yqr &ae zqcb`n`y+# c` zlykikùe nac mlil ae nkraiikùe elr~a &ae
zqcb`n`y+ { c` flr` nac nç`% Z`r` ay~l# ay~kg`r cly ykbqkae~ay aeqeik`nly8
&`+ Zrlo`r ac glnacl na rabraykùe pqa gahlr `hqy~a ` cly n`~ly%
&o+ A}`cq`r ac glnacl na rabraykùe yacaiikle`nl# ay naikr# ` ~r`}ëy na irk~arkly
ay~`nçy~kily { ilgzcagae~`r c` naikykùe qy`enl bràmkily yqoha~k}ky~`y%
C` ~`oc` na `o`hl gqay~r` c` kemlrg`ikùe rapqarkn` z`r` ylcqikùe ~lnly cly
aeqeik`nly rapqarknly zlr ay~a zrlocag`%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 542/846
Nr% Fëi~lr Pqa}anl Qrç`y
6'66
^`oc` gly~r`enl cly n`~ly na c`y zrqao`y na aearbç` ylc`r ~ërgki`% \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
{ v0 v? v9 v> v1 \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
?20%6 269%91 99%19 >:%11 0=%== 09%?:
?=>%: 2>6%>1 9=%1: 9:%0; 0=%>= 0>%00
?96%6 =6>%>1 9>%== 92%90 02%== 01%=6
?9:%2 6?2%6: 99%09 9?%1? 02%1: 0:%19
?10%= 6=:%>1 91%21 99%20 0=%>: 00%::
?12%; 621%01 9>%>= 9>%0> 0=%?6 00%90
?=9%; ;:;%>1 9>%=: 9>%61 0=%:= 00%;=
?==%0 ;:1%11 91%96 91%6; 01%;9 0?%16
??;%0 21=%:: 91%61 99%19 0=%=: 0:%==
?9;%9 2=;%91 91%=6 99%2; 0=%>0 0:%61?16%: 2;9%1: 91%91 9>%2? 0=%02 00%>0
?12%= 6:0%=1 91%:> 91%?? 01%;? 00%;0
?=2%9 60;%=1 9>%:2 9=%1: 0=%:> 0?%61
?=2%: 6:6%11 9?%?: 92%=: 0=%0; 09%16
?1;%= 22>%;1 9>%9? 92%6; 0=%=? 0>%?0
?>:%> 200%61 90%:6 92%20 02%92 01%1=
??2%? =;>%61 91%29 92%:: 06%0? 01%69
0;=%: =96%0: 9>%00 9=%2= 06%19 0=%>0
?26%2 22>%11 9>%2; 9>%=? 01%1> 09%0:
?2?%9 212%;: 91%22 91%>: 01%2: 09%=9
?=2%> 219%91 9=%>> 91%;= 0=%>1 0>%10
?1>%1 2:>%2: 92%6? 9=%?= 02%=? 01%96
??>%2 ===%6: 91%:2 9=%9> 06%0? 0=%0:
060%1 1=6%11 91%?= 91%;: 0;%:1 0=%29
??2%1 =19%0: 91%1= 90%6> 0=%10 0:%16
?19%= 2:>%:1 91%29 99%0= 0=%:? 00%?6
?=9%: 2:;%=: 9=%>= 99%69 01%6; 00%;0
?=1%6 2?=%;: 9=%?= 9>%6; 01%69 0?%=1
?=9%6 =;2%01 92%?: 9=%?2 0=%20 0>%:= \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ { 7 Mcqhl ~l~`c na i`clr &d`~~y+4 v0 7 Keylc`ikùe &`~~y,g?+4 v? 7 Zlykikùe nac mlil ae nkraiikùe ay~a &zqcb`n`y+4 v9 7 Zlykikùe
nac mlil ae nkraiikùe yqr &zqcb`n`y+4 v> 7 Zlykikùe nac mlil ae nkraiikùe elr~a &zqcb`n`y+4 v1 7 Flr` nac nç`
Mqae~a8 Ke~rlnqiikùe `c @eàckyky na _abraykùe Ckea`c % Gle~blgar{ a~ `c% ?::0%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 543/846
Nr% Fëi~lr Pqa}anl Qrç`y
6'6;
6%9?% C` ke~aeikùe na ay~a aharikikl ay c` na f`iar qe` bràmki`# ile c` }`rk`oca na
rayzqay~` &[+ { ile iq`~rl }`rk`ocay rabrayk}`y &V0# V?# V9# V>+ qy`enl ac zrlbr`g`
Gkek~`o% Ykaenl `yç# na c` ilemkbqr`ikùe na cly zqe~ly ayz`riknly lo~aekn` zranaikr# pqë ~kzl na mqeikùe na rabraykùe ay~`nçy~ki` aei`h`rç` gahlr ae cly n`~ly%
@nagày# a}`cq`r ac glnacl na rabraykùe i`enkn`~l l yqzarklr qy`enl gë~lnly
ay~`nçy~kily { bràmkily% Yqbaraeik`8 z`r` f`iar c` bràmki` zankn`# qy`r ac zrlbr`g`
Gke~`o zrliankaenl na c` ykbqkae~a g`ear`8 Krya ` Br`zf ↟ Nr`m~yg`e Zcl~% Ay~l
cca}` `c raiq`nrl na –Nr`m~yg`e Zcl~‗% Aeyabqkn`# ae c` }ae~`ekcc` na –[ }`rk`oca‗
zlear c` }`rk`oca nazaenkae~a &[+ {# ae c` }ae~`ekcc` na –V }`rk`oca‗ zlear c`y
}`rk`ocay kenazaenkae~ay &V+%
^`oc` gly~r`enl cly n`~ly na ay~a zrlocag`% &Ac`olr`ikùe zrlzk`+% \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ ]`rk`oca na rayzqay~` &[+ ]`rk`oca rabrayk}` V0 ]`rk`oca rabrayk}` V? ]`rk`oca rabrayk}` V9 ]`rk`oca rabrayk}` V>
?91 ?: 0; 6= ;1
?90 ?2 02 61 ;:
?61 >: ?: 69 0:1
?2: 11 ?: 6? 69?;= =: ?: 62 ;:
90? =6 ?0 6; ;>
?;1 21 ?: 69 ;?
?;? 6: ?: 60 ;?
?=9 2: ?: 16 0:1
?20 1: 01 2; 0::
?69 >: 01 6: ;:
?1= 9: 01 2; 66
6%99% Ay~a ay qe zrlocag` `n`z~`nl nac ~av~l na H`{ C% Na}lra &?::0+# ae ac iq`c ya
n` c` kemlrg`ikùe rapqarkn` z`r` c` yacaiikùe nac glnacl na rabraykùe yqzarklr#
o`y`nl ae c` keicqykùe nac eøgarl na }`rk`ocay rabraylr`y# yacaiikle`nl ae~ra
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 544/846
Nr% Fëi~lr Pqa}anl Qrç`y
6';:
iq`~rl glnacly i`enkn`~ly% Ac ahagzcl ay~à rac`ikle`nl ile ac i`clr `iqgqc`nl nac
aenqraikgkae~l nac iagae~l ~lg`nl ilgl c` }`rk`oca nazaenkae~a# ae mqeikùe na
cly zranki~lray v0 7 ! na `cqgke`~l na ~rki`cikl# v? 7 ! na ykcki`~l na ~rki`cikl# v9 7! marr`~l na `cqgkekl { v> 7 ykcki`~l na nki`cikl% Ya n` qe ~`g`ðl na gqay~r` kbq`c
` 09 loyar}`ikleay {# nlena c` yqg` na cly iq`nr`nly nac ~l~`c ay kbq`c ` ?#201%2=%
Z`r` ay~l# ya zkna `c cai~lr ccae`r cly m`c~`e~ay na c` ~`oc` na `o`hl { naiknkr iq`c ay
ac glnacl yqzarklr pqa ~kaea ac eøgarl `naiq`nl na }`rk`ocay rabraylr`y%
^`oc` gly~r`enl c` kemlrg`ikùe% Ccae`r cly m`c~`e~ay% El% na rabraylray d _abraylr &ay+d YYAd _ ?d _ ?&`hqy~`n`+d Izd M&i`ci%+d
0 v> 66:%61 :%=2= :%=>2 096%?? v0# v? 16%:0 ?%29 v0# v?# v9 :%;6? :%62= 9%?> v0# v?# v9# v> :%;6? >%:
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 545/846
Nr% Fëi~lr Pqa}anl Qrç`y
6'0
I@ZK^QCL 6
_abraykùe ckea`c ykgzca { gøc~kzca
Yqzlykikleay nac glnacl na rabraykùe ckea`c%' Aiq`ikleay elrg`cay z`r`
i`ciqc`r ac ke~ariaz~l ae c` lrnae`n` ` { c` zaenkae~a o na c` iqr}` l cçea` na
rabraykùe%' Ilamkikae~a na na~argke`ikùe gøc~kzca _ ?
na c` gqay~r` pqa ay~kg`
` π?
ac ilamkikae~a na na~argke`ikùe zloc`ikle`c%' Ilamkikae~a na ilrrac`ikùe _
na c` gqay~r` pqa ay~kg` ` π# ac ilamkikae~a na ilrrac`ikùe zloc`ikle`c%'
Ke~ar}`cl na ilemk`ex` z`r` ac ilamkikae~a zloc`ikle`c ο ilgzleae~a na c`
cçea` na rabraykùe ±[sV 7 ΰ $ ο V # ay~kg`nl zlr o# c` zaenkae~a na c` cçea`%'
Ke~ar}`cl na ilemk`ex` z`r` ac z`ràga~rl zloc`ikle`c ΰ# ac ke~ariaz~l na c`
lrnae`n` na c` cçea` na rabraykùe ±[sV 7 ΰ $ ο V # iq{l ay~kg`nlr ay `%' Fkzù~ayky
eqc` na Fl8ο 7 οl ile~r` c`y fkzù~ayky `c~are`~k}`y na F08ο 3 0 { F?8ο 5 0%'
Fkzù~ayky eqc` na Fl8ΰ 7 ΰl ile~r` c`y fkzù~ayky `c~are`~k}`y na F08ΰ ≪ ΰl# F?8ΰ
5 ΰl# { na F98 ΰ 3 ΰl%' Ke~ar}`cl na ilemk`ex` z`r` ±[sV na c` cçea` na rabraykùe
zloc`ikle`c ay~kg`n` zlr [%' _abraykùe { ilrrac`ikùe gøc~kzca%' Gë~lnly z`r`
}`ckn`r ac glnacl na rabraykùe ckea`c ykgzca { gøc~kzca8 ` ~r`}ëy na ay~`nçy~ki` na
kemaraeik`y { ` ~r`}ëy nac `eàckyky bràmkil na cly rayknq`cay ay~`en`rkx`nly%
Zrliankgkae~l na rabraykùe gøc~kzca qy`enl ac zrlbr`g` Gkek~`o%'
Ac loha~k}l na ay~qnk`r rabraykùe ckea`c ykgzca ay z`r` lo~aear ac glnacl na
rabraykùe gày `zrlzk`nl# ay naikr# qe` aiq`ikùe na rabraykùe ckea`c ykgzca l
gøc~kzca z`r` mkeay na zrankiikùe { ay~kg`ikùe% Cly ilgzleae~ay na ay~` aiq`ikùe
na rabraykùe ckea`c# ile ylcl qe` }`rk`oca kenazaenkae~a# ~`gokëe cc`g`nl glnacl
ckea`c na zrkgar lrnae# yle c` }`rk`oca nazaenkae~a [¾ l mqeikùe na rayzqay~` {# c`
}`rk`oca kenazaenkae~a V % Ac glnacl na ay~` aiq`ikùe# pqa nayirkoa c` rac`ikùe na
c` }`rk`oca V ile c` }`rk`oca [ # ya cc`g` c` aiq`ikùe na rabraykùe na [ ylora V {# c`
bràmki` na ay~` mqeikùe# ya cc`g` c` iqr}` na rabraykùe%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 546/846
Nr% Fëi~lr Pqa}anl Qrç`y
6'?
Ac glnacl na rabraykùe ckea`c zloc`ikle`c pqa nayirkoa c` rac`ikùe ae~ra c`
rayzqay~` l }`rk`oca nazaenkae~a [ {# c` }`rk`oca kenazaenkae~a l rabraylr` V ay8
[ 7 οl $ ο0v0 $ α k 7 0# ?# …%# e &6'0+
Nlena8
[ 7 }`rk`oca nazaenkae~a zloc`ikle`c &~`gokëe ya qy` c` `el~`ikùe {+
οl 7 ke~ariaz~l ae c` lrnae`n`
ο0 7 zaenkae~a na c` cçea`
v0 7 }`rk`oca kenazaenkae~a
α 7 arrlr `ca`~lrkl ile zrlgankl na : { }`rk`ex` ω?
iley~`e~a% Ay~a }`clr na α ay c`nkmaraeik` ae~ra ac }`clr ~aùrkil na [ k { ac }`clr na [ i`ciqc`nl q loyar}`nl% C`y
ilenkikleay na α yle na pqa ay~a z`ràga~rl naoa ay~`r elrg`cgae~a nky~rkoqknl4 yqy
}`clray naoae na yar kenazaenkae~ay qel nac l~rl { c` }`rk`ex` na α ay ]`r&α+ 7 ω?α
e 7 eøgarl na & v# {+ z`ray na loyar}`ikleay
C` aiq`ikùe na c` cçea` na rabraykùe gqay~r`c pqa ay~kg` ` glnacl na rabraykùe
zloc`ikle`c &6'0+ na `rrko` ya n` ilgl8
[ 7 ` $ ov $ a &6'?+
Nlena8
[ 7 }`clr na c` }`rk`oca nazaenkae~a na c` gqay~r`
` 7 ke~ariaz~l ae c` lrnae`n`
o 7 zaenkae~a na c` cçea`
a 7 arrlr l rayknq`c na c` gqay~r` nael~`nl zlr ak 7 {k ' [ k% Ay~` ay~`nçy~ki` ay c`
ay~kg`nlr` nac z`ràga~rl α
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 547/846
Nr% Fëi~lr Pqa}anl Qrç`y
6'9
Yqzlykikleay nac glnacl na rabraykùe ckea`c
0% Cly }`clray na [ yle kenazaenkae~ay qel nac l~rl# ay naikr# el naoae na ay~`r
ilrrac`ikle`nly%?% C`y nky~rkoqikleay ilenkikle`cay na zrlo`okckn`n na [ n`nl V yle elrg`cay%
9% C` }`rk`ex` nac arrlr ay ω? { ay iley~`e~a%
>% Cly ilamkikae~ay οl { ο0 yle nayileliknly { naoae na ay~kg`rya%
Z`r` ay~kg`r c` aiq`ikùe na rabraykùe ckea`c ykgzca { gøc~kzca ya qy` cl pqa
ya cc`g` ac gë~lnl na cly iq`nr`nly gçekgly pqa `hqy~` cly n`~ly na c` gqay~r` `
c` cçea` na rabraykùe% Ay~` ay qe` na c`y ~ëieki`y gày qy`n`y ae ke}ay~kb`ikleay
ikae~çmki`y# z`r` aeile~r`r c` rac`ikùe ae~ra nly l gày }`rk`ocay pqa ay~àe
i`yq`cgae~a rac`ikle`n`y%
Ae ay~` yaiikùe }aragly ac zrlocag` na rabraykùe ckea`c na qe` }`rk`oca
nazaenkae~a &[ + l~r` kenazaenkae~a & V +# ile mkeay na zrankiikùe { ay~kg`ikùe% Yke
ago`rbl# qe` }ax pqa ya lo~kaea c` aiq`ikùe na rabraykùe ckea`c# ëy~` ya ~kaea pqa
a}`cq`r l }`ckn`r z`r` }ar pqë ~`e~` ilemk`okckn`n ya ca zqana zlear `c glnacl z`r`
qyly na zrankiikùe% Ay~l ya f`ia qy`enl aemlpqay loha~k}ly { yqoha~k}ly% Zlr
ahagzcl# ac aemlpqa loha~k}l ya f`ia f`ikaenl zrqao`y ay~`nçy~ki`y na kemaraeik`%
Ay~a aemlpqa ya ilgzcagae~` qy`enl aemlpqay yqoha~k}ly# ay naikr# `e`ckx`enl c`y
bràmki`y na cly rayknq`cay ay~`en`rkx`nly l el ay~`en`rkx`nly# ` ~r`}ëy na
keyzaiikleay }kyq`cay%
Zlr ahagzcl# c`y ilenkikleay l yqzlykikleay rapqarkn`y z`r` }`ckn`r ac
glnacl# yqoha~k}`gae~a# ya f`ia ` ~r`}ëy na cly `eàckyky na cly rayknqly irqnly lay~`en`rkx`nly &z`r` nkmaraeik`rcly na cly rayknqly ay~`en`rkx`nly+% Cly cc`g`nly
rayknqly ya namkeae ilgl c`y nkmaraeik`y ae~ra ac }`clr `i~q`c na [ { ac }`clr
zrlely~ki`nl na [ zlr ac glnacl na rabraykùe ay~kg`nl% Cly rayknqly ya nael~`e
zlr ak# ay~l ay# ak 7 [ k ’ [¾ k% Ae }arn`n# c`y bràmki`y na cly rayknqly n`e kemlrg`ikùe
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 548/846
Nr% Fëi~lr Pqa}anl Qrç`y
6'>
gq{ kgzlr~`e~a# `iari` na c` e`~qr`cax` { mqarx` na c` rac`ikùe ae~ra c`y }`rk`ocay%
C` mkbqr` na `o`hl gqay~r` cly rayknqly pqa yle c`y nkmaraeik`y ae~ra cly }`clray na
[ 0 # [ ? # [ 9 #…#[ d { cly }`clray loyar}`nly na [¾ 0 # [¾ ? # [¾ 9 #…#[¾ d na c` cçea` narabraykùe na c` gqay~r`% Zlr l~r` z`r~a# cly rayknqly ay~`en`rkx`nly ya lo~kaeae
nk}knkëenlcly zlr yqy rayzai~k}`y nay}k`ikleay ay~àen`ray%
Mkbqr`% 6%:% Bràmki` gly~r`enl cly rayknqly na qe ahagzcl% &Ac`olr`ikùe zrlzk`+
C`y yqzlykikleay na cly }`clray rayknq`cay yle8
&`+ Cly rayknq`cay ak ay~àe elrg`cgae~a nky~rkoqknly &αk ay~àe elrg`cgae~a
nky~rkoqknly+%
&o+ Cly rayknqly ~kaeae c` gkyg` }`rk`ex` &αk yle iley~`e~ay+%
&i+ Cly rayknq`cay ak el ay~àe ilrrac`ikle`nly# ay naikr# yle kenazaenkae~ay%
L~rl gë~lnl gaely zlzqc`r pqa ac `eàckyky na cly rayknqly# z`r` a}`cq`r c`
aiq`ikùe na rabraykùe ay ilgz`r`enl ac nk`br`g` ayz`riknl na cly zqe~ly# ilerayzai~l ` c` cçea` na rabraykùe# ile c` bràmki` na cly zqe~ly ile rayzai~l `c
zrlgankl na { % Ay~l ya naoa ` pqë# yke kgzlr~`r ac }`clr na V # ac zrlgankl {
ykagzra zarg`eaia iley~`e~a &cçea` flrkxle~`c ~r`x`n` ae ac nk`br`g` ayz`riknl na
c` bràmki`+% Na ay~` g`ear`# yk c` nkyzarykùe na cly zqe~ly ile rac`ikùe ` c` cçea` na
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 549/846
Nr% Fëi~lr Pqa}anl Qrç`y
6'1
rabraykùe ay gqifl gaelr# pqa c` nkyzarykùe na cly zqe~ly ile rayzai~l ` c` cçea`
flrkxle~`c na { # ae~leiay# ya zqana ileicqkr pqa c` aiq`ikùe na c` cçea` na
rabraykùe n` qe oqae `hqy~a z`r` cly n`~ly na c` gqay~r` &N`ekac a~ `c% 0;6;+%Ilgl ya nkhl `e~ay# ac aemlpqa loha~k}ky~` ay c` l~r` g`ear` pqa ya qy` z`r`
a}`cq`r ac glnacl na rabraykùe ckea`c# ay~l ay# ` ~r`}ëy na `eàckyky ay~`nçy~kily% Z`r`
ay~l# ya zqanae qy`r c`y ykbqkae~ay mqeikleay ay~`nçy~ki`y8
&`+ Ilamkikae~a na na~argke`ikùe ckea`c _ ? &l r ?+# ac ilamkikae~a na ilrrac`ikùe ckea`c
_# y { Z_AYY%
&o+ @eàckyky na }`rk`ex` ykgzca &@EL]@+# z`r` zrlo`r cly ilamkikae~ay nac glnaclna rabraykùe &ο+# z`r` π# a~i%
&i+ Ke~ar}`cly na ilemk`ex` z`r` π?# z`r` οl# οk# ± {sv# a~i%
^kzly na ilrrac`ikùe ckea`c
0% Ilrrac`ikùe ykgzca pqa ileyky~a na nly }`rk`ocay# qe` nazaenkae~a &[ + { c` l~r`
kenazaenkae~a & V +% Nae~rl na ay~` i`~ablrç` ~aeagly8
&`+ Ilrrac`ikùe nkrai~`% Ay~` ilrrac`ikùe ileyky~a ae ac keiragae~l ae qe` }`rk`oca c`
iq`c ay `ilgz`ð`n` zlr ac keiragae~l na l~r` }`rk`oca &ilrrac`ikùe zlyk~k}`+%
&o+ Ilrrac`ikùe ke}ary`% Ay~` ilrrac`ikùe ileyky~a ae ac keiragae~l na qe` }`rk`oca c`
iq`c ay `ilgz`ð`n` zlr ac keiragae~l na l~r` &ilrrac`ikùe eab`~k}`+%
&i+ Ilrrac`ikùe el ckea`c% Ae ay~` ilrrac`ikùe el f`{ ekebqe` `ylik`ikùe ae~ra c`y nly
}`rk`ocay%
?% Ilrrac`ikùe gøc~kzca% @pqç# f`{ gày na nly }`rk`ocay% Qe` }`rk`oca ay
nazaenkae~a &[ +# gkae~r`y pqa c`y l~r`y yle kenazaenkae~ay V 0 # V ? #…# V d # a~i%C`y mkbqr`y na `o`hl razrayae~`e }`rkly ~kzly na ilrrac`ikleay%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 550/846
Nr% Fëi~lr Pqa}anl Qrç`y
6'=
Mkb% 6%0% Nk`br`g`y ayz`riknly ile cçea`y na iq`nr`nly gçekgly% C` Mkbqr` &`+razrayae~` qe` cçea` rai~` ile V mkh`4 c` Mkbqr` &o+ razrayae~` cçea` el rai~` ile V mkh`4
c` Mkbqr` &i+ razrayae~` qe` nky~rkoqikùe `nhqe~` ile cçea` rai~`4 c` Mkbqr` &n+
razrayae~` qe` nky~rkoqikùe `nhqe~` ile cçea` el rai~`4 c` Mkbqr` &a+ razrayae~` qe
nk`br`g` nlena el f`{ `ylik`ikùe ae~ra c`y nly }`rk`oca {4 c` Mkbqr` &m+ razrayae~`
qe` rac`ikùe i`qy`c% C`y l~r`y nly bràmki`y razrayae~`e ilrrac`ikleay zarmai~`y%
&Ac`olr`ikùe zrlzk`+
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 551/846
Nr% Fëi~lr Pqa}anl Qrç`y
6'2
^kzly na iqr}`y gày ilgqeay
Mkbqr` 6%?% C` mkbqr` &`+ razrayae~` c` mqeikùe avzleaeik`c4 c` mkbqr` &o+ razrayae~` c`
mqeikùe na zl~aeik`# c` mkbqr` &i+ razrayae~` qe` mqeikùe raiçzrli` {# c` mkbqr` &n+
razrayae~` qe` mqeikùe fkzaroùcki`% &Ac`olr`ikùe zrlzk`+
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 552/846
Nr% Fëi~lr Pqa}anl Qrç`y
6'6
Aiq`ikleay elrg`cay z`r` i`ciqc`r ac ke~ariaz~l ae c` lrnae`n` ` { c` zaenkae~a
o na c` iqr}` l cçea` na rabraykùe
C`y }`rk`ocay ` { o ya lo~kaeae na c`y aiq`ikleay elrg`cay na `o`hl# ay naikr#raylc}këenlc`y ykgqc~àea`gae~a8
Ω[ 7 ` e $ o Ω V &6'9+
Ω V[ 7 ` Ω V $ o ΩV &6'>+
@c raylc}arya ykgqc~àea`gae~a n`e ac ke~ariaz~l# ` ae c` lrnae`n` {# c` zaenkae~a na
c` cçea`# o8
Ke~ariaz~l 7 ` 7 [ ’ o V &6'1+
Zaenkae~a 7 o 7 Te Ω V[ ’ &Ω V +&Ω[ +W , Te Ω V ? ’ &Ω V +? W &6'=+
7 Ω v{ , Ω v? &6'2+
Nlena8
Ω v{ { Ω v? ya n`e zlr c`y aiq`ikleay &6'6+ { &6';+ na `o`hl%
El~` 0% C`y ykbqkae~ay aiq`ikleay yle gq{ kgzlr~`e~ay%
Ω v? 7 Y vv 7 Ω V ? ’ &Ω V +? , e &6'6+
Ω v{ 7 Y v{ 7 Ω V[ ’ Ω V Ω[ , e &6';+
Ω {? 7 Y {{ 7 Ω[ ? ’ &Ω[ +? , e &6'0:+
El~` ?% Ay gq{ kgzlr~`e~a el~ r c`y nkmaraeik`y ae~ra ac qyl na c`y }`rk ocay
gkeøyiqc`y { c`y g`{øyiqc`y ae c`y aiq`ikleay na `rrko`%
Ilamkikae~a na na~argke`ikùe _ ?
na c` gqay~r` pqa ay~kg` ` π?
ac ilamkikae~a na
na~argke`ikùe zloc`ikle`c
Ac iàciqcl nac ilamkikae~a na na~argke`ikùe gøc~kzca _ ?
ay qe` zrqao` loha~k}ky~`na ay~`nçy~ki`% Ay~` ay qe` mqeikùe ay~`nçy~ki` gq{ kgzlr~`e~a# z`r` }`ckn`r ac
glnacl na rabraykùe ckea`c% Ay~a ilamkikae~a _ ? gkna c` zrlzlrikùe na }`rk`ikùe ae
c` }`rk`oca nazaenkae~a [ avzcki`n` zlr c` }`rk`oca kenazaenkae~a V % Cly }`clray na
_ ? }`rç`e na : ` 0% Zlr ahagzcl# qe }`clr iari`el ` : kenki` pqa el f`{ qe`
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 553/846
Nr% Fëi~lr Pqa}anl Qrç`y
6';
rac`ikùe ckea`c ae~ra [ { V # gkae~r`y pqa qe }`clr iari`el ` qel kenki` qe `hqy~a
ckea`c zarmai~l% @pqç# yke ago`rbl# ay eaiay`rkl `ic`r`r pqa# qe }`clr `c~l na _ ?# el
eaiay`rk`gae~a kenki` qe oqae `hqy~a nac glnacl na rabraykùe# ykel f`y~` pqa yaf`iae ~ln`y c`y zrqao`y loha~k}ky~`y { yqoha~k}`y% C` mqeikùe pqa i`ciqc` _ ? ay8
_ ? 7 &Ω v{+? , Ω v?Ω {? &6'00+
7 0 ’ YYa , YY~ &6'0?+
Nlena Ω v{# Ω v? { Ω {? ya n`e zlr c`y aiq`ikleay &6'6+# &6'6+ { &6'0:+ nayirk~`y z`r`
c` aiq`ikùe &6'00+% @nagày# z`r` c` aiq`ikùe &6'0?+ YYa ay c` yqg` na cly
iq`nr`nly nac arrlr l rayknq`c { YY~ ay c` yqg` na cly iq`nr`nly nac ~l~`c# gkygly
pqa ya nayirkoae ae ac mlrg`~l na c` ~`oc` na @EL]@%
^`gokëe f`{ ac cc`g`nl ilamkikae~a _ ? na na~argke`ikùe `hqy~`nl% Ay~` ay
qe` }arykùe `hqy~`n` na _ ?# ac iq`c oqyi` ragl}ar c` nky~lrykùe naokn` ` qe ~`g`ðl
na gqay~r` zapqaðl% Ya namkea ilgl8
_ ?`hqy~`n` 7 0 ’ T&0 ’ _ ?+ &e ’ 0+,&e ’ ?+W &6'09+
Nlena _ ? {` ya namkekù { e ay ac ~`g`ðl na c` gqay~r`
Ilamkikae~a na ilrrac`ikùe _ na c` gqay~r` pqa ay~kg` ` π# ac ilamkikae~a na
ilrrac`ikùe zloc`ikle`c
Ac ilamkikae~a na ilrrac`ikùe _# pqa ay~kg` ` π# ~`gokëe ya cc`g` ilamkikae~a na
ilrrac`ikùe na Za`ryle% Ay~a ilamkikae~a ay qe çenkia na c` mqarx` na c` `ylik`ikùe
ckea`c ae~ra c`y }`rk`ocay V a [ % Ac ilamkikae~a na ilrrac`ikùe _ ay8
_ 7
∐∐
∐
{ v
v{
??
&6'0>+
Nlena8 Ω v{# Ω v? { Ω {? ya n`e zlr c`y aiq`ikleay &6'6+# &6';+ { &6'0:+
El~`8 Ac ilamkikae~a na ilrrac`ikùe _ avzcki` ac br`nl na `ylik`ikùe ae~ra c`y
}`rk`ocay V a [ % Ay~a ilamkikae~a _ }`rç` na ’0 ` :# yk c` ilrrac`ikùe ay eab`~k}`# ay
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 554/846
Nr% Fëi~lr Pqa}anl Qrç`y
6'0:
naikr# ile zaenkae~a eab`~k}`% Zarl# yk c` ilrrac`ikùe ay zlyk~k}`# ae~leiay# _ }`rç`
na : ` 0% @yç# ` gankn` pqa _ ya `zrlvkg` ` °0# gahlr `ylik`ikùe f`orà ae~ra c`y
}`rk`ocay V a [ % Eù~aya pqa# ae i`yl na c` rabraykùe ckea`c gøc~kzca# ~aeagly cl pqaya cc`g`e ilamkikae~ay z`rik`cay na rabraykùe qy`nly z`r` gankr c` rac`ikùe ckea`c
ae~ra c` }`rk`oca nazaenkae~a { c` }`rk`oca kenazaenkae~a ayzaikmki`n`%
Ke~ar}`cl na ilemk`ex` z`r` ac ilamkikae~a zloc`ikle`c ο ilgzleae~a na c`
cçea` na rabraykùe ±[sV 7 ΰ $ ο V # ay~kg`nl zlr o# c` zaenkae~a na c` cçea`%
o ’ ~T0'ΰ,?4e'?W y , ∐ v? 3 ο 3 o $ ~T0'ΰ,?4e'?W y , ∐ v
? &6'01+
Nlena8o 7 Ω v{ , Ω v?
~T0'ΰ,?4e'?W 7 }`clr na c` nky~rkoqikùe na ~ na Ay~qnk`e~a
Ω v? 7 Ω V ? ’ &Ω V +? , e
y 7 ?
+&?
∟
∟∐ ∐e
v{o {&6'0=+
7 YYA,&e ’ ?+ 7 e
{ {∐ ∐∟ +&??
' &oΩ V[ ' Ω V Ω[ ,e+W , e'?
C` aiq`ikùe na c` }`rk`ex` ay8 y? 7 &Ω {? ’ oΩ v{+ , &e ’ ?+ &6'02+
ο 7 ilamkikae~a zloc`ikle`c na c` zaenkae~a na c` cçea`# ac iq`c ay ay~kg`nl zlr o 7
Ω v{ , Ω v? l ya` ac ilamkikae~a na c` cçea` na rabraykùe gqay~r`c%
Ke~ar}`cl na ilemk`ex` z`r` ac z`ràga~rl zloc`ikle`c ΰ# ac ke~ariaz~l na c`
lrnae`n` na c` cçea` na rabraykùe ±[sV 7 ΰ $ ο V # iq{l ay~kg`nlr ay `
&6'06+
Nlena8
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 555/846
Nr% Fëi~lr Pqa}anl Qrç`y
6'00
` {` ya namkekù `e~arklrgae~a
~T0'ΰ,?4e'?W 7 ` qe }`clr qy`enl c` nky~rkoqikùe na ~ na ay~qnk`e~a ile η 7 e ’ ? br`nly
na ckoar~`ny 7 na c` aiq`ikùe &6'0=+
Yvv 7 Ωv{ &na c` aiq`ikùe &6';++
Fkzù~ayky eqc` Fl8ο 7 οl ile~r` c`y fkzù~ayky `c~are`~k}`y F08ο 3 0 { F?8ο 5 0%
Z`r` ay~` zrqao` ~`gokëe ya qy` c` nky~rkoqikùe na ~ na Ay~qnk`e~a ile η 7 e ’ ?
br`nly na ckoar~`n# ay naikr8
~ 7 &o ’ οl+ , y,Ω v? &6'0;+
Nlena8
~ 7 c` ay~`nçy~ki` na c` nky~rkoqikùe na ~ na Ay~qnk`e~a
οl 7 qe }`clr n`nl
o 7 zaenkae~a na c` cçea`
Fkzù~ayky eqc` Fl8ΰ 7 ΰl ile~r` c`y fkzù~ayky `c~are`~k}`y F08ΰ ≪ ΰl# F?8ΰ 5 ΰl#
{ F98 ΰ 3 ΰl @pqç# eqa}`gae~a# ya qy` c` nky~rkoqikùe na ~ na Ay~qnk`e~a ile br`nly na ckoar~`n#
η 7 e ’ ?% Z`r` ay~l ya qy` c` mùrgqc` na `o`hl8
&6'?:+
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 556/846
Nr% Fëi~lr Pqa}anl Qrç`y
6'0?
Nlena8
ΰl 7 qe }`clr n`nl
y 7 {` namkekn` `e~arklrgae~a
` {` ya namkekù `e~arklrgae~a
Ke~ar}`cl na ilemk`ex` z`r` ±[sV na c` cçea` zloc`ikle`c ay~kg`n` zlr [
Ac ke~ar}`cl na ilemk`ex` z`r` ac }`clr na ±[sV ya f`ia ay qy`enl c` mùrgqc` &6'?0+ na
`o`hl8
[l‛’ ~Tΰ,?4 ηW ye0 $ &Vl ' V +?,Ωv? 3 ±[sV 3 [l‛$ ~Tΰ,?4 ηW y
e0 $ &Vl ' V +?,Ωv? &6'?0+
Nlena8
[ l‛ 7 ` $ o V l 7 }`clr na c` cçea` na rabraykùe ile qe }`clr na V l n`nl &6'??+
~Tΰ,?4e'?W 7 }`clr na c` nky~rkoqikùe na ~ ile qe ek}ac na ykbekmki`eik` na ΰ 7 %:1 l okae
:%:0 ile η 7 e ’ ? br`nly na ckoar~`n
` 7 {` namkekn` `e~arklrgae~a
y 7 {` namkekn` e~arklrgae~a
V l 7 qe }`clr n`nl
V 7 zrlgankl na c` gqay~r`
Fkzù~ayky eqc` Fl8ο 7 : ile~r` c`y fkzù~ayky `c~are`~k}`y F08ο 5 : { F?8ο 3 :
Z`r` f`iar ay~` zrqao` qy`gly c` nky~rkoqikùe na ~ na Ay~qnk`e~a ile η 7 e ’ ? br`nly
na ckoar~`n% C` mqeikùe ay~`nçy~ki` qy`n` z`r` ~`cay mkeay ay8~ 7 &o ’ ol+ , y , ∐ v
? &6'?9+
Nlena8
y 7 {` namkekn` `e~arklrgae~a
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 557/846
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 558/846
Nr% Fëi~lr Pqa}anl Qrç`y
6'0>
Zrqao`y na fkzù~ayky Fl8π 7 :# ile~r` c` fkzù~ayky `c~are`~k}`y F08π ≪ :# z`r` ac
ilamkikae~a na ilrrac`ikùe zloc`ikle`cπ ay~kg`nl zlr _% &Nqee a~ `c% 0;2>+
Z`r` ay~ly mkeay ya qy` c` ay~`nçy~ki` na ~ na Ay~qnk`e~a8
~ 7 ∘ η _ , _?
0∟ &6'?1+
Nlena8
_ 7 Ω v{ , ∐∐ { v?? &6'?=+
η 7 e ’ ? br`nly na ckoar~`n
@pqç# eqa}`gae~a# z`r` i`ciqc`r c`y rabkleay irç~ki`y ya qy` c` ~ na Ay~qnk`e~a# aynaikr# ~Tΰ,?4e'?W%
Ahagzcly na zrlocag`y qy`enl rabraykùe { ilrrac`ikùe ckea`c ykgzca
Ahagzcl )0% Ay~a zrlocag` ay~à rac`ikle`nl ile qe ay~qnkl `iari` na c` i`e~kn`n na
zraikzk~ ikùe zcq}k c { c` i`e~kn`n na ile~ gke`ikùe `~glymërki %
^@OC@ 6%:% ^`oc` gly~r`enl cly n`~ly% &Ac`olr`ikùe zrlzk`+ \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
Ccq}k` &:%:?=‗+ s 06 2 0> 90 ?0 1 00 0= ?= ?;_aglikùe na ile~`gke`ikùe s 11 02 9= 61 =? 06 99 >0 =9 62
F`iar c`y ykbqkae~ay ay~kg`ikleay8
&`+ Knae~kmki`r c` }`rk`oca nazaenkae~a { c` }`rk`oca kenazaenkae~a% F`iar qe` bràmki`
pqa }`{` ae mqeikùe na c` }`rk`oca nazaenkae~a [ # { c` }`rk`oca kenazaenkae~a V %
&o+ I`ciqc`r cly }`clray na c` ay~`nçy~ki` nayirkz~k}` na cly n`~ly%
&i+ Lo~aear c` aiq`ikùe na rabraykùe ckea`c ykgzca { ~r`x`rc` ae c` bràmki`%
&n+ ]`ckn`r c` ilemk`okckn`n nac glnacl na rabraykùe# ay naikr# ` ~r`}ëy na c` agkykùe
na qe hqkikl yqoha~k}l `e`ckx`enl cly }`clray na cly rayknqly ay~`en`rkx`nly# na c`
ykbqkae~a g`ear`8
0% F`iar qe` bràmki` pqa gqay~ra c` zrqao` na elrg`ckn`n%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 559/846
Nr% Fëi~lr Pqa}anl Qrç`y
6'01
?% F`iar qe` bràmki` ile cly rayknq`cay ay~`en`rkx`nly }aryqy }`clray `hqy~`nly na [ %
&Ac }`clr zranaiknl l `hqy~`nl na [ k ay ac }`clr na [ pqa ya ayzar`rç` iq`enl ya qy` c`
cçea` na rabraykùe% Ae l~r`y z`c`or`y# cly }`clray `hqy~`nly na [ 0# [ ?#%%# [ e ya lo~kaeaeyqy~k~q{aenl# yqiayk}`gae~a# v0# v?# %%# ve ae c` aiq`ikùe na c` cçea` na rabraykùe
ay~kg`n`8 [ k 7 οl $ ο0vk# %%# οl $ ο0ve%
9% F`iar qe fky~lbr`g` na rayknq`cay%
>% F`iar qe` br`mki` pqa gqay~ra cly rayknq`cay ay~`en`rkx`nly }aryqy raebcleay%
&a+ Ilgzcagae~`r c` a}`cq`ikùe nac glnacl ile kemaraeik`y ay~`nçy~ki`y# ilgl8
0% Iàciqcl nac ilamkikae~a na na~argke`ikùe _ ? { ac ilamkikae~a na ilrrac`ikùe _%
?% F`iar qe` ~`oc` na `eàckyky na }`rk`ex` &@EL]@+%
9% F`iar qe` ~`oc` ile cly ilamkikae~ay# cly arrlray ay~àen`ray# c`y zrqao`y na ~# cly
}`clray na z# { cly ke~ar}`cly na ilemk`ex` z`r` ac ke~ariaz~l { c` zaenkae~a%
Ylcqikùe8
&`+ C` }`rk`oca nazaenkae~a ay c` raglikùe na ile~`gke`e~ay &[+ { c` }`rk`oca
kenazaenkae~a ay c` i`e~kn`n na ccq}k` &V+% C` mkbqr` na `o`hl gqay~r` ay~` ylcqikùe8
Mkbqr` 6%9% Bràmki` gly~r`enl [ }aryqy V# ile qe` cçea` rai~` flrkxle~`c
ilrrayzlenkae~a `c }`clr nac zrlgankl na [ 7 >;%2:::% &Ac`olr`ikùe zrlzk`+
&o+ Cly }`clray na c` ay~`nçy~ki` nayirkz~k}` yle8
V 7 02%6:::# [ 7 >;%2:::% Cly }`clray gàvkgly { gçekgly na cly }`clray na [ yle
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 560/846
Nr% Fëi~lr Pqa}anl Qrç`y
6'0=
62%::: { 02%:::# rayzai~k}`gae~a% Cly }`clray gàvkgly { gçekgly na cly }`clray na
V yle 90%::: { 1%::::# rayzai~k}`gae~a% Iq`nr`nl gankl nac arrlr 7 y? {sv 7 ?=%==24
arrlr iq`nrà~kil gankl ay y {sv 7 1%0=>&i+ Qy`enl qe zrlbr`g` na ilgzq~`nlr` ya ay~kg`e cly }`clray nac ke~ariaz~l ae c`
lrnae`n` { c` zaenkae~a% Ay~ly yle8 ke~ariaz~l 7 ` 7 0%:?09# zaenkae~a na c` cçea` 7 o
7 ?%29>6% Yqy~k~q{aenl ay~ly }`clray n`e c` cçea` na rabraykùe gqay~r`c &gkyg` pqa ya
}a ae c` Mkbqr` 6%9+# n`%
[ 7 ` $ oV
[ 7 0%:?09 $ ?%29>6& V +
&n+ Z`r` ay~a keikyl c` Mkbqr` 6%> gqay~r` c` kemlrg`ikùe rapqarkn`%
_ayknq`c
Z a r i a e ~
0:1:'1'0:
;;
;:
1:
0:
0
Mk~~an ]`cqa
_ a y k n q ` c
6:=:>:?:
1
:
'1
'0:
_ayknq`c
M r a p q a e i {
>:'>'6
9
?
0
:
Loyar}`~kle Lrnar
_ a y k n q ` c
0:;62=1>9?0
1
:
'1
'0:
Elrg`c Zrlo`okck~{ Zcl~ lm ~fa _ayknq`cy _ayknq`cy ]aryqy ~fa Mk~~an ]`cqay
Fky~lbr`g lm ~fa _ayknq`cy _ayknq`cy ]aryqy ~fa Lrnar lm ~fa N`~`
_ayknq`c Zcl~y mlr _aglikle na ile~`gke`~eay &[+
Mkbqr` 6%>% Bràmki`y gly~r`enl c`y rayzqay~`y z`r` ac keikyl &n+%
Ilgl ya }a ae c` Mkbqr` 6%> c` mkbqr` yqzarklr kxpqkarn` gqay~r` c` zrqao` na
elrg`ckn`n ile ~lnly cly zqe~ly mlrg`enl qe` ckea` rai~`% Ay~l kenki` pqa c`
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 561/846
Nr% Fëi~lr Pqa}anl Qrç`y
6'02
nky~rkoqikùe na cly n`~ly ay elrg`c% Kbq`cgae~a# c` mkbqr` yqzarklr naraif` gqay~r`
cly rayknq`cay ae mqeikùe na cly }`clray `hqy~`nly na [% @pqç# f`{ `ca`~lrkan`n ae
c` nky~rkoqikùe na cly zqe~ly ile c` gkyg` i`e~kn`n na zqe~ly eab`~k}ly { zlyk~k}ly# cl pqa kenki` pqa el f`{ ilrrac`ikle na cly n`~ly% C` mkbqr` kemarklr
kxpqkarn` gqay~r` c` mraiqaeik` }aryqy cly rayknq`cay% Mke`cgae~a# c` mkbqr` kemarklr
naraif` gqay~r` cly rayknq`cay ae mqeikùe na cly ùrnaeay na c`y loyar}`ikleay%
@pqç# ae ay~` mkbqr` f`{ `ca`~lrkan`n { ac gkygl eqgarl na zqe~ly zlyk~k}ly {
eab`~k}ly# cl pqa yqbkara pqa el f`{ ilckea`ckn`n l ilrrac`ikle ae yarka na c`
kemlrg`ikùe yqgkeky~r`n`%
&a+ Z`r` ilgzcagae~`r ac ay~qnkl loha~k}ky~`# ay~l ya f`ia f`ikaenl zrqao`y
ay~`nçy~ki`y na kemaraeik`%
&0+ Ilgl ya nkhl `e~ay# ac ilamkikae~a na na~argke`ikùe _ ? ay qe aemlpqa loha~k}ky~`#
pqa ykr}a z`r` }`ckn`r ac glnacl na rabraykùe% Ay~a ilamkikae~a na na~argke`ikùe _ ?#
gkna c` mqarx` rac`~k}` na c` rac`ikùe ckea`c ae~ra V a [ &gkna c` zrlzlrikùe na
}`rk`ikùe ae [ pqa zqana yar avzcki`n` zlr c` }`rk`ikùe ae V + ay n`nl zlr c` aiq`ikùe
&6'00+ { zlr c`y aiq`ikleay &6'=+# &6'2+ { &6'6+# rayzai~k}`gae~a8
_ ? 7 :%;=?:
Ac iàciqcl nac ilamkikae~a na ilrrac`ikùe _ ay8
_ 7 ? _ 7 :%;6:6
&?+ Z`r` ac `eàckyky na }`rk`ex` &@EL]@+# pqa ~`gokëe ykr}a z`r` }`ckn`r ac glnacl
na rabraykùe# ay qe` mqeikùe ay~`nçy~ki` loha~k}ky~` pqa zrqao` c` fkzù~ayky eqc` na pqa
c` zaenkae~a ay kbq`c ` :% @pqç ya }arà pqa# qe }`clr br`ena na M kenki` pqa ac glnaclna rabraykùe yacaiikle`nl ay q~kc% Yke ago`rbl# ay eaiay`rkl `e`ckx`r ~lnly cly nagày
irk~arkly `e~ay na agk~kr qe hqkikl mke`c% C` ~`oc` na @EL]@ na `o`hl n` cly
rayqc~`nly%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 562/846
Nr% Fëi~lr Pqa}anl Qrç`y
6'06
^@OC@ 6%0% `oc` na eàckyky na }`rk`ex` &@EL]@+ z`r` ac ahagzcl%&Ac`olr`ikùe zrlzk`+
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
Mqae~a na Yqg` na cly b%c% Iq`nr`nl nac Mi`ci% M~`o% ]`clr na z}`rk`ikùe iq`nr`nly zrlgankl \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ Naoknl `c 1#9;=%22 0 1#9;=%22 ?:?%96 1%9? :%::::0~r`~`gkae~l_ayknq`c &arrlr+ ?09%99 6 ?=%=2
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ ^l~`c 1#=0:%0 ;
Ac }`clr na M~`o% ya y`i` ileyqc~`enl c` ~`oc` na c` nky~rkoqikùe na M# ay~l ay Mΰ4 η0# η?# aciq`c n` M%;140#6 7 1%9?% @pqç# naoknl ` pqa ac }`clr na Mi`ci% 7 ?:?%96 555 1%9?# ya
raif`x` c` fkzù~ayky yqy~ae~`n` na pqa Fl8ο0 7 : { ya keicke` zlr Fl8ο0 ≪ :% C`
ileicqykùe ay na pqa c` zaenkae~a na c` cçea` el ay kbq`c ` : q flrkxle~`c%
&9+ C` ~`oc` na `o`hl gqay~r` cly }`clray nac ke~ariaz~l ae c` lrnae`n`# ac br`nkae~a
na c` cçea` na rabraykùe# cly arrlray ay~àen`r# c` zrqao`y na fkzù~ayky qy`enl c` ~ na
ay~qnk`e~a# cly }`clray na c` zrlo`okckn`n z { cly ke~ar}`cly na ilemk`ex` &;1!+ z`r`
οl &ke~ariaz~l+ { ο0 &zaenkae~a+%
^@OC@ 6%?% ^`oc` gly~r`enl cly }`clray nac ke~ariaz~l# zaenkae~a# zrqao`y na ~ naAy~qnk`e~a# }`clr nac ek}ac na z { yqy ke~ar}`cly% &Ac`olr`ikùe zrlzk`+
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ Ilamkikae~a Arrlr Zrqao` ~ ]`clr z Cçgk~a Cçgk~a
ay~àen`r kemarklr yqzarklr \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ Ke~ariaz~l 0%:? 9%2; :%?2 :%2; '2%22? ;%2=
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\Zaenkae~a ?%29 :%0; 0>%?9 1%6v0:'2 ?%?; 9%06 \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
@pqç# eù~aya pqa ac ke~ar}`cl na ilemk`ex` z`r` ac ke~ariaz~l ay gq{ `gzckl { c`
fkzù~ayky el ya zqana raif`x`r# zqay~l pqa ac }`clr na ~ ay gq{ zapqaðl { ac }`clr na
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 563/846
Nr% Fëi~lr Pqa}anl Qrç`y
6'0;
z 7 :%2; ay br`ena% Ay~l ay `zl{`nl zlr ac }`clr na :%2; na z { zlr qe arrlr ay~àen`r
na 9%2;# rac`~k}`gae~a br`ena4 cl ile~r`rkl liqrra ile c`y zrqao`y ay~`nçy~ki`y na c`
zaenkae~a# iq{l }`clr na ~ ay br`ena { iq{l }`clr na z ay gq{ zapqaðl%Ahagzcl )?% Ae qe ay~qnkl na gkirloklclbç` `gokae~`c# ae gqay~r`y na `bq`# ya
nkarle cly ykbqkae~ay n`~ly na c` ~`oc` na `o`hl% Ay~ly n`~ly ya ramkarae `c iraikgkae~l
na qe` ilclek` na o`i~ark`y ae qe gankl na iqc~k}l%
^@OC@ 6%9% ^`oc` gly~r`enl cly n`~ly% &Ac`olr`ikùe zrlzk`+
^kagzl ae nç`y na s 9 = ; 0? 01 06keliqc`ikùe & V +
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ El% o`i~ark`y &[ + s 001#::: 0>2#::: 06;#::: ?91#=:: ?12#;:: ?6=#>::
F`iar cly ykbqkae~ay iàciqcly8
&`+ I`ciqc`r c` cçea` na rabraykùe%
&o+ I`ciqc`r ac ilamkikae~a na na~argke`ikùe _ ? { ac ilamkikae~a na ilrrac`ikùe _%
&i+ Ile c` aiq`ikùe na rabraykùe# ay~kg`r ac eøgarl na o`i~ark`y nayzqëy na ?: nç`y
&n+ Aeile~r`r cly ke~ar}`cly na ilemk`ex` z`r` ΰ { ο qy`enl ac z`pqa~a na AVIAC%&a+ Qy`r ac zrlbr`g` Gkek~`o { ay~kg`r cly }`clray rayknq`cay { `e`ckx`rcly
yqoha~k}`gae~a# z`r` ra}ky`r zlr c` i`ckn`n nac glnacl na rabraykùe%
Ylcqikùe8
&`+ C` aiq`ikùe na c` cçea` na rabraykùe ay8
[ 7 60#1?:%:: $ 00#22>%?; V
&o+ Ac ilamkikae~a na na~argke`ikùe ckea`c gøc~kzca _ ? ay kbq`c ` :%;66:% Ac ilamkikae~a
na ilrrac`ikùe _ ay kbq`c ` :%;;>:%
&i+ Iq`enl V 7 ?: nç`y# ac eøgarl na o`i~ark`y ay na8
[ 7 60#1?: $ 00#22>%?; &?:+ ≆ 902#::= o`i~ark`y
&n+ Ae iq`e~l ` cly ke~ar}`cly na ilemk`ex` z`r` ΰ { ο# ac zrlbr`g` na ilgzq~`nlr` na
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 564/846
Nr% Fëi~lr Pqa}anl Qrç`y
6'?:
AVIAC rrlh` cly ykbqkae~ay rayqc~`nly8
Ke~ar}`cl na ilemk`ex` na ;1! z`r` ΰ8 =0#?1;%>1 3 ΰ 3 0:0#26:%=4 }`clr na c`
zrlo`okckn`n z 7 :%:::>4 Ke~ar}`cl na ilemk`ex` na ;1! z`r` ο ay8 0::>:%0> 3 ο 3091:6%>9# ile qe }`clr na c` zrlo`okckn`n z 7 :%::::>=
&a+% C`y mkbqr`y na `o`hl gqay~r`e c`y bràmki`y pqa ~r`~`e na }`ckn`r ac glnacl na
rabraykùe ckea`c# ile nac eøgarl na o`i~ark`y ae mqeikùe nac ~kagzl na keiqo`ikùe%
Mkbqr` 6%1% Mkbqr`y gly~r`enl cly rayqc~`nly nac eøgarl na o`i~ark`y }aryqy ac ~kagzlna keiqo`ikùe% C` bràmki` &`+ gqay~r` c` rac`ikùe ae~ra [ { V # ile c` cçea` rai~` na [ 4
c` bràmki` &o+ gqay~r` cly rayknqly irqnly }aryqy V 4 c` bràmki` &i+ gqay~r` cly rayknqly
irqnly }aryqy cly raebcleay {# c` bràmki` &n+ gqay~r` cly rayknqly irqnly }aryqy
rayknqly rax`b`nly &Ac`olr`ikùe zrlzk`+%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 565/846
Nr% Fëi~lr Pqa}anl Qrç`y
6'?0
^ln`y ay~`y bràmki`y yqbkarae# yqoha~k}`gae~a# pqa ac glnacl na rabraykùe ckea`c ay
ilemk`oca% ¼Zlr pqë<
Ahagzcl )9% Ae qe ay~qnkl na `brkiqc~qr`# rac`ikle`nl ile c` ykagor` na `cblnùe# aeikar~l ay~`nl na c` Qekùe @garki`e`# c` zraikzk~`ikùe `eq`c { ac raenkgkae~l na c`
ilyaif` na `cblnùe yle ilgl ykbqa%
^@OC@ 6%>% ^`oc` gly~r`enl cly n`~ly% &Ac`olr`ikùe zrlzk`+
Zraikzk~`ikùe s 2%0? =9%1> >2%96 >1%;? 6%=6 1:%6= >>%>=ae zqcb`n`y
& V +
_aenkgkae~l na s 0:92 96: >0= >?2 =0; 966 9?0c` ilyaif` aeckor`y,`ira
&[ +
F`iar cly ykbqkae~ay iàciqcly8
&`+ I`ciqc`r cly }`clray nac ke~ariaz~l ` { c` zaenkae~a o%
&o+ Ayirkokr c` aiq`ikùe na c` cçea` na rabraykùe%
&i+ I`ciqc`r ac ilamkikae~a na na~argke`ikùe _
?
{ ac ilamkikae~a na ilrrac`ikùe _%&n+ Zranaikr ac raenkgkae~l na c` ilyaif` na `cblnùe# yk c` zraikzk~`ikùe ay na 9:
zqcb`n`y%
&a+ F`iar qe` ~`oc` na `eàckyky na }`rk`ex`%
Ylcqikùe8
&`+ Qy`enl qe z`pqa~a na ilgzq~`nlr` ilgl ac Aviac n`8
Ke~ariaz~l ae c` lrnae`n` 7 ` 7 66:%>:
Zaenkae~a na c` cçea` 7 o 7 ';%=0
&o+ Zlr cl ~`e~l# c` aiq`ikùe na c` cçea` na rabraykùe ay8
[ 7 66:%>: ’ ;%=0 & V +
&i+ Ac ilamkikae~a na na~argke`ikùe 7 _ ? 7 :%=;;0
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 566/846
Nr% Fëi~lr Pqa}anl Qrç`y
6'??
Ac ilamkikae~a na ilrrac`ikùe 7 _ 7 :%69=0
&n+ Iq`enl c` zraikzk~`ikùe na ccq}k` ay na 9: zqcb`n`y# ac raenkgkae~l na c`
ilyaif` ya i`ciqc` qy`enl ac glnacl na rabraykùe lo~aeknl# ay naikr yqy~k~q{aenl ac}`clr na V 7 9:% Na ay~` g`ear`# qy`enl c` aiq`ikùe na rabraykùe n`n` `rrko` {
yqy~k~q{aenl ac }`clr na V 7 9: ely n`8
[ 7 66:%> ’ ;%=0 &9:+ 7 1;?%0
&a+ C` ~`oc` na `eàckyky na }`rk`ex` n`n` zlr ac z`pqa~a Aviac ya n` `o`hl%
^@OC@ 6%1% ^`oc` na `eàckyky na }`rk`ex` &@EL]@+% &Ac`olr`ikùe zrlzk`+
Mqae~a na }`rk`ikùe b%c% YY GY Mi`ci% M~`o% ]`clr na zNaoknl ` c` _abraykùe 0 ?=:#=?6%? ?=:#=?6%? 00%=? 1%9? :%:0;
_ayknql 1 00?#0=1%1 ??#>99%00
^l~`c = 92?2;9%2
Ae ileicqykùe# `c ilgz`r`r ac }`clr na c` ay~`nçy~ki` i`ciqc`n` M ile ac }`clr irç~kil na
M ya raif`x` c` fkzù~ayky yqy~ae~`n` ile qe }`clr na z kbq`c ` :%:0;%
Ahagzcl )>% Ac ckorl @zzckan Y~`~ky~kiy8 @eàckyky lm ]`rk`eia `en _abrayykle na Nqee
{ Ic`rd &0;2>+ nayirkoa qe ay~qnkl na mçyki`# ay naikr# na ùz~ki`# nlena ya lo~q}karle
cly n`~ly na `o`hl pqa gqay~r`e cly nkàga~rly na c`y mkor`y ùz~ki`y &ae gkir`y+ ae
mqeikùe na c` mqarx` na rlgzkgkae~l na ëy~`y% Z`r` ay~a zrlocag` f`iar cly ykbqkae~ay
iàciqcly
&`+ F`iar ~lnly cly i`ciqcly zrackgke`ray { i`ciqc`r c` aiq`ikùe na c` cçea` na
rabraykùe gqay~r`c pqa ay~kg` ` c` aiq`ikùe na rabraykùe zloc`ikle`c ±[sV 7 ΰ $ ο V %&o+ Qy`enl qe z`pqa~a na ilgzq~`nlr`# aeile~r`r ac ke~ar}`cl na ilemk`ex` z`r` ac
ilamkikae~a na rabraykùe zloc`ikle`c ΰ &ke~ariaz~l ae [ +# pqa ay~kg` ` `%
&i+ Ae mlrg` `eàclb` pqa ile ae ac keikyl &o+# aeile~r`r ac ke~ar}`cl na ilemk`ex` z`r`
ac ilamkikae~a na rabraykùe ο &c` zaenkae~a na c` cçea`+ iq{l ay~kg`nlr ay o%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 567/846
Nr% Fëi~lr Pqa}anl Qrç`y
6'?9
&n+ Zrlo`r c` fkzù~ayky eqc` na Fl8ο 7 οl# ay naikr# ο 7 : ile~r` c` fkzù~ayky `c~are`~k}`
na F08ο 5 : { F?8ο 3 :% I`ciqc`r ac }`clr na c` zrlo`okckn`n z%
&a+ F`iar qe ke~ar}`cl na ilemk`ex` z`r` ±[sVl% &m+ I`ciqc`r cly irk~arkly a}`cq`nlray nac glnacl na rabraykle# }% b%# _ ?# Z_AYY { y%
&b+ F`iar qe` zrqao` na fkzù~ayky z`r` ac ilamkikae~a na ilrrac`ikùe zloc`ikle`c π%
&f+ Br`mki`r cly n`~ly { ~r`x`r c` aiq`ikùe na c` cçea` na rabraykùe ylora c` bràmki` {
~r`x`r c` cçea` flrkxle~`c ilrrayzlenkae~a `c }`clr nac zrlgankl [ %
&k+ Agk~kr qe hqkikl yqoha~k}l pqa `{qna ` }`ckn`r ac qyl nac glnacl na rabraykùe%
C` ~`oc` na `o`hl gqay~r` cly n`~ly%
^@OC@ 6%=% ^`oc` gly~r`enl ac nkàga~rl na mkor`y }y% mqarx` na rlgzkgkae~l%
Nkàga~rl na c` mkor` & V + Clb na c` mqarx` na rlgzkgkae~l &[ +
??%1 %0;?6%: %=??2%1 %10?1%1 %19??%: %?>
9:%1 %62?9%: %?1?1%: %?1?9%1 %92?2%: %9??0%1 %09??%: %91?;%: %19?:%1 %???2%: %=1
&Mqae~a8 Nqee a~ `c% 0;2>% @zzckan Y~`~ky~kiy8 @e`c{yky lm ]`rk`eia `en _abrayykle+
Ylcqikùe8
&`+ Cly iàciqcly zrackgke`ray yle8
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 568/846
Nr% Fëi~lr Pqa}anl Qrç`y
6'?>
e 7 01# ΩV 7 92>%1# &ΩV+?,e 7 ;#91:%:# Ω[ 7 =%:9# &Ω[+?,e 7 ?%>?# ΩV[ 7 016%?1# ΩV? 7 ;#>6?%21# Ω[ ? 7 9%:9# &ΩVΩ[+,e 7 ?#?16%?>,01 7 01:%11# V 7 ?>%;2# [ 7 :%>:?#
Ωv? 7 ΩV ? ’ &ΩV+?,e 7 ;#>6?%21 ’ ;#91:%: 7 09?%21# Ωv{ 7 ΩV[ ’ ΩVΩ[,e 7 016%?1 ’ 01:%11 7 2%2:# Ω{? 7 Ω[ ? ’ &Ω[+?,e 7 9%:9 ’ &=%:9+?,01 7 %=:2>
Z`r` i`ciqc`r c` cçea` na rabraykùe na c` gqay~r`# zrkgarl i`ciqc`gly g`eq`cgae~a#
cly ilamkikae~ay ` { o na c` cçea` na rabraykùe gqay~r`c pqa ay~kg`e ` ΰ { ο%
o 7 Ω v{,Ω v? 7 2%2:,09?%21
7 %:16
` 7 [ ’ o V 7 :%>:? ’ &:%:16+&?>%;2+ 7 '0%:>=Zlr cl ~`e~l# c` cçea` na rabraykùe gqay~r`c ay8
{ 7 ` $ o& V +
{ 7 '0%:>= $ :%:16& V +
&o+ Ac ke~ar}`cl na ilemk`ex` z`r` ΰ ay qy`enl c` mqeikùe &6'06+ l qy`enl qe z`pqa~a
na ilgzq~`nlr` ilgl Aviac zrliankaenl ilgl8 ^llcy ↟ N`~` eàckyky ↟ _abrayykle
{ LD% Aeyabqkn`# nayzqëy na pqa cly n`~ly ya ke~rlnqharle ae c`y ilcqge`y @ { O na
c` flh` na Aviac krya ` c` }ae~`ekcc` na –Kezq~ [ _`eba‗ { –Kezq~ V _`eba‗# cl pqa
baear` c` ^@OC@ 6%2 na `o`hl%
^@OC@ 6%2% ^`oc` gly~r`enl ac }`clr nac ke~ariaz~l# c` zaenkae~a# cly }`clray na ~ { z
{ cly ke~ar}`cly na ilemk`ex` z`r` ΰ { ο%
Zlr cl ~`e~l# ac ke~ar}`cl na ilemk`ex` z`r` ac ke~ariaz~l &ΰ+ ya caa na c` ~`oc` ilgl8
'0%12:= 3 ΰ 3 ':%1??>
&i+ Ae mlrg` `eàclb` ac ke~ar}`cl na ilemk`ex` z`r` ο ya caa na c` ^@OC@ 6%2 ilgl8
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 569/846
Nr% Fëi~lr Pqa}anl Qrç`y
6'?1
:%:266 5 ο 5 :%:920
&n+ Z`r` zrlo`r c` fkzù~ayky eqc` Fl8ο 7 οl ay naikr# ο 7 :# ile~r` F08ο 5 : { F?8ο 3 :
qy`gly c` nky~rkoqikùe na ~ na ay~qnk`e~a ile η 7 e ’ ? 7 01 ’ ? 7 09 b%c% C` mùrgqc`ay8 ~ 7 &o ’ οl+ , y,∘ Ω v?% Yqy~k~q{aenl ~lnly cly }`clray na οl 7 : { nagày }`clray ae
c` mùrgqc` na `rrko` n`8
~ 7 &:%:16 ’ :+ , :%0?, 09?%29
7 1%6
C`y rabkleay irç~ki`y yle8 ~ 7 ° ?%0=%
Ae ileicqykùe8 naoknl ` pqa ~i`ci%
7 1%6 5 ~~`o%
7 ?%0=# ya raif`x` c` fkzù~ayky eqc` na
Fl8ο 7 : { ya keicke` zlr F08ο 5 :% Ac }`clr na c` zrlo`okckn`n ya i`ciqc` qy`enl c`
mùrgqc` na ke~arzlc`ikùe &='0:+8 &δ? ’ δ0+,&~? ’ ~0+ 7 &δ? ’ V+,&~? ’ ~i`ci%+
Yqy~k~q{aenl cly }`clray `zrlzk`nly na c` ~`oc` na ~ ely n`8
&%::::0 ' %::::?+,&=%?62 ’ 1%=:2+ 7 &%::::0 ’ V+,&=%?62 ’ 1%6+
Cl pqa n` V 7 z 7 %::::?% Zarl ilgl c` zrqao` ay okc`~ar`c# cl gqc~kzcki`gly zlr ? {
n` z 7 %::::>% Ay~a }`clr `zl{`# gq{ ile~qenae~a# c` fkzù~ayky `c~are`~k}` na F08ο 5
:%
&a+ Ac ke~ar}`cl na ilemk`ex` z`r` c` }`rk`oca nazaenkae~a na c` cçea` na rabraykùe
zloc ikle`c# ±[sV ay~kg`n` zlr [ # ile ek}ac na ykbekmki`eik` na ΰ 7 :%:1# n`r }`rkly
}`clray V l% Z`r` f`iar ay~l# ya qy` c` mqeikùe na `o`hl8
[¾ l ' ~Tΰ,?4e'?W ye
0 $& V l ’ V +?,Ω v? 3 ±[sV 3 [¾ ù $ ~Tΰ,?4e'?W ye
0 $& V l ’ V +?,Ω v? &6'?6+
Nlena8 V 7 zrlgankl
~Tΰ,?4e'?W 7 }`clr na ~ ile η 7 e ’ ? b%c%
~T%:?1409W 7 ° ?%0=
V l 7 cly nkmarae~ay }`clray pqa ya ca nae ` V l z`r` iley~rqkr cly cçgk~ay l o`en`y na
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 570/846
Nr% Fëi~lr Pqa}anl Qrç`y
6'?=
ilemk`ex` z`r` ±[sV
@flr` okae# ile cly }`clray na8 ` 7 '0%:>2# V 7 ?>%;2# Ωv? 7 09?%29# y 7 :%0?# ~%:%?1409
7 ° ?%0= { `ykbeàenlca }`clray ` V l# nkb`gly na 0;# ?6# 9:%:# a~i%# ya zrliana na c`ykbqkae~a g`ear`8
Z`r` V l 7 0;%:4 [‛ l 7 '0%:>2 $ :%:16&0;%:+ 7 :%:11# a~i%
Aeyabqkn`# qy`enl c` mùrgqc` &6'?6+ { yqy~k~q{aenl cly }`clray# ay naikr# z`r` V l 7 0;
n`8
%:11’?%0=&:%0?+01
0 $&0;%:'?>%;2+?,09?%29 3 ±[s0; 3 %:11$?%0=&:%0?+01
0 $&0;%:'?>%;2+?,09?%29
Ac iq`c ya ykgzckmki` `8 :%991 5 ±[s0; 5 :%?;;
@yç ya zqana ile~keq`r n`enl nkmarae~ay }`clray na V l { yqy~k~q{ëenlcly# ilgl ya
fkxl `rrko`# z`r`# mke`cgae~a# f`iar c`y o`en`y na ilemk`ex` z`r` ±[sV %
&m+ Z`r` i`ciqc`r cly }`clray na _# _ ?# y { Z_AYY ya zqanae f`iar ile qe z`pqa~a na
ilgzq~`nlr`% Zlr ahagzcl# yk ya f`ia g`eq`cgae~a# ac ilamkikae~a _ ya i`ciqc` qy`enl
c` aiq`ikle &6'0>+# a~i% Na l~r` g`ear`# yk ya qy` ac Gke~`o zrlianar ilgl8
Y~`~ ↟ _abrayykle↟ _abrayykle
Ae c` }ae~`e` na –_ayzleya‗ zlear c` }`rk`oca nazaenkae~a# { ae c` }ae~`e` na
–Zranki~lry‗ zlear c` }`rk`oca kenazaenkae~a% ^`gokëe ya zqanae qy`r c`y }ae~`e`y na
–Br`zfy‗# –Lz~kley‗ { –_ayqc~y‗ z`r` lo~aear kemlrg`ikùe `nkikle`c% Zlr ahagzcl cly
}`clray na c`y ay~`nçy~ki`y loha~k}ky~`y na kemaraeik` n`n`y zlr ac zrlbr`g` yle8
_ ? 7 29%=!# _ 7 :%616# y 7 :%000?# Z_AYY 7 :%??:>% Zlr ahagzcl# ac }`clr na _ 7
:%612= kenki` kenki` qe` ilrrac`ikùe zlyk~k}` pqa }` na `iqarnl ile c` zaenkae~a
zlyk~k}` na c` iqr}` na %:16% Cly }`clray ~`e zapqaņly na y { na Z_AYY kenki`e qe
oqae hqy~a na cly n ~ly c glnacl na rabraykùe%
&b+ Z`r` c` zrqao` na fkzù~ayky Fl8π 7 :# ay naikr# z`r` ac ilamkikae~a na ilrrac`ikùe
zloc ikle`c# π ile ΰ 7 :%:1# ile~r` c` fkzù~ayky `c~are`~k}` na F08≪ :# ay~l ay# F?8π 5 :
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 571/846
Nr% Fëi~lr Pqa}anl Qrç`y
6'?2
{ F98π 3 : ya qy`e c`y ykbqkae~ay ay~`nçy~ki`y8
&0+ Qy`enl c` ay~`nçy~ki` na ~ na Ay~qnk`e~a &6'?1+8
~ 7 ?∟e _ , _?0∟
Nlena8
_ 7 {` namkekn`Z`r` i`ciqc`r c`y rabkleay irç~ki`y ya qy` c` nky~rkoqikùe na ~# ay naikr# ~Tΰ,?4e'?W 7 ~%:?1409 7° ?%0=Ae~leiay# qy`enl c` mùrgqc` na `o`hl { yqy~k~q{aenl cly }`clray n`8
_ 7Ω v{
,∐∐ { v
?? 7 2%2:0 , %=:2>+&09?%29+&: 7 :%6=
{ _ ? 7 :%29;=
@flr`# qy`enl c` ay~`nçy~ki` na `o`hl { yqy~k~q{aenl n`
~ 7 ?∟e _ , _?
0∟
~ 7 09 &:%6=+ , ?=:>% 7 =%:2
Yk ya naya` y`i`r ac }`clr na z ya oqyi` =%:2 ae c` ~`oc` na c` nky~rkoqikùe na ~ ile η 7
09 { ile ΰ 7 %:1# cl pqa n` %:?1 3 z 3 %:1%
&f+ Z`r` br`mki`r cly n`~ly `qe`nly ` c` aiq`ikùe na c` cçea` na rabraykùe ile qe` cçea`
flrkxle~`c ilrrayzlenkae~a `c }`clr nac zrlgankl [ ya f`ia qy`enl qe z`pqa~a na
ilgzq~`nlr`%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 572/846
Nr% Fëi~lr Pqa}anl Qrç`y
6'?6
Mkbqr` 6%=% Bràmki` gly~r`enl c` mqarx` na rlgzkgkae~l &clb0:+ ae mqeikùe nac
nkàga~rl na c` mkor`# ile c` aiq`ikùe na c` ckea` na rabraykùe [ 7 '0%:>= $ :%:16&V+ {
ile ac zrlgankl [ 7 :%>:?% &Ac`olr`ikùe zrlzk`+%&k+ Agk~kr qe hqkikl yqoha~k}l pqa `{qna ` }`ckn`r ac qyl nac glnacl na rabraykùe% Z`r`
rayzlenar ay~` zrabqe~` ya f`iae cly ykbqkae~ay bràmkily8
? > = 6 0: 0? 0>
'?
'0
:
0
?
Loyar}`~kle Lrnar
Y ~ ` e n ` r n k x a n _ a y k n q ` c
_ayknq`cy ]aryqy ~fa Lrnar lm ~fa N`~`
&rayzleya ky Clb mqar+
Mkbqr` 6%2`% Bràmki` gly~r`enl cly rayknqly ay~`en`rkx`nly }aryqy ac lrnae na c`
loyar}`ikùe% Ay~` ay qe` bràmki` pqa gqay~r` ~lnly cly rayknq`cay ae ac lrnae ae aciq`c cly n`~ly mqarle ilcaiikle`nly% @pqç f`{ ac gkygl eøgarl na n`~ly zlyk~k}ly
{ eab`~k}ly% Ay~` bràmki` ~`gokëe ykr}a z`r` aeile~r`r arrlray el `ca`~lrkly#
ayzaik`cgae~a# ae amai~ly rac`ikle`nly ile ac ~kagzl%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 573/846
Nr% Fëi~lr Pqa}anl Qrç`y
6'?;
:%2:%=:%1:%>:%9:%?:%0
?
0
:
'0
'?
Mk~~an ]`cqa
Y ~ ` e n ` r n k x a n _ a y k n q ` c
_ayknq`cy ]aryqy ~fa Mk~~an ]`cqay
&rayzleya ky Clb mqar+
Mkbqr` 6%2o% Ay~à bràmki` gqay~r` cly rayknq`cay }aryqy }`clray `hqy~`nly% Z`r` pqaac glnacl na rabraykùe ya` `iaz~`oca# ya rapqkara pqa8 cly zqe~ly ae c` bràmki` ya`e
`ca`~lrkly ae `goly c`nly na :4 el naoa f`oar yarkay na zqe~ly pqa `qgae~ae l
nkygkeq{`e4 el naoa f`oar zranlgke`eik` na rayknq`cay zlyk~k}ly l eab`~k}ly# ek
~`gzlil naoa f`oar z`~rleay na rayknq`cay pqa `qgae~ae ile }`clray `hqy~`nly
pqa `qgae~ae% Ilgl ya }a# ~ln`y ay~`y ilenkikleay ay~àe okae yqy~ae~`n`y%
'? '0 : 0 ?
'?
'0
:
0
?
E l r g ` c Y i l r a
Y~ en`rnkxan _ayknq c
Elrg`c Zrlo`okck~{ Zc l~ lm ~fa _ayknq`cy
&rayzleya ky Clb mqar+
Mkbqr` 6%2i% Bràmki` gly~r`enl c` zrqao` na elrg`ckn`n% Cly n`~ly naoae mlrg`r
qe` cçea` rai~` yk cly rayknq`cay ay~àe elrg`cgae~a nky~rkoqknly &yk~q`ikùe pqa
liqrra `pqç+% Na l~r` g`ear`# c` yqzlykikùe na elrg`ckn`n ya ke}àckn`%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 574/846
Nr% Fëi~lr Pqa}anl Qrç`y
6'9:
Ilgl ya loyar}` ae ay~`y bràmki`y# c` agkykle na qe hqkikl yqoha~k}l ay `iaz~`oca#
zlrpqa ac glnacl na rabraykùe yacaiikle`nl `hqy~ okae cly n`~ly% Ay~l ya naoa ` pqa#
ae c` Mkbqr` 6%2` f`{ `ca`~lrkn`n ae cly n`~ly# ay naikr# ile ac gkygl eøgarl na}`clray zlyk~k}ly { eab`~k}ly% @nagày# ae c` Mkbqr` 6%2o c` nayirkzikùe na ëy~`#
yqbkara qe glnacl na rabraykùe razrayae~`~k}l na c` kemlrg`ikùe n`n`% Yk~q`ikùe
ykgkc`r liqrra ile c` nayirkzikùe na c` Mkbqr` 6%2i%
Ahagzcl )1% Ae qe ay~qnkl na kebaekarç` nac `bq` rac`ikle`nl ile c`y ranqiikleay na
cly yùcknly yqyzaenknly# ae mqeikùe na c` nag`en` pqçgki` na lvçbael &NPL+# ya
y`iù qe` gqay~r` `ca`~lrk`# iq{ly n`~ly ya n`e ae c` ~`oc` na `o`hl% Z`r` cl ykbqkae~a8
&`+ Knae~kmki`r c` }`rk`oca nazaenkae~a { c` kenazaenkae~a { f`iar qe` bràmki` na NPL
}aryqy ranqiikùe na yùcknly%
&o+ I`ciqc`r c` aiq`ikùe na c` cçea` na rabraykùe%
&i+ F`iar qe` ~`oc` na `eàckyky na }`rk`ex` pqa keicq{` c` M irç~ki` { ac }`clr na z%
&n+ ]`ckn`r ac glnacl i`enkn`~l# ` ~r`}ëy na ay~`nçy~ki`y ilgl _ ?# Z_AYY# y { na c`
ay~`nçy~ki` na Nqroke'R`~yle &z`r` c` zrqao` na q~lilrrac`ikùe na rayknq`cay+%
&a+ A}`cq`r c` q~kckn`n nac glnacl ` ~r`}ëy na bràmkily yqoha~k}ly8
^@OC@ 6%6% ^`oc` gly~r`enl c`y gankikleay na yùcknly { c` nag`en` pqçgki` nalvçbael% &Ac`olr`ikùe zrlzk`+ \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
Yùcknly yqzaenknly NPL \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
9: ?; 99 92 ?1 9? ?; ?2 90 9= ?1 909: 9: 99 9: 91 90 ?; ?6 9? ?; 9: 9:
?; 9: 9> 9: 9= 9: ?6 ?; 9> ?; 9> ?;
9> 90 9= ?; 90 9: 99 9: 91 ?6 9: ?6?6 90 9= ?6 99 9? ?= 9: 9> ?6 9: 90?2 9? 9= ?2 90 9? ?2 9? 9> ?= ?; 90
Ylcqikùe8
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 575/846
Nr% Fëi~lr Pqa}anl Qrç`y
6'90
&`+ C` }`rk`oca nazaenkae~a ay NPL { c` }`rk`oca kenazaenkae~a ay ranqiikùe na
yùcknly yqyzaenknly% C` mkbqr` 6%6 na `o`hl gqay~r` c`y ileiae~r`ikleay na NPL
}aryqy ranqiikùe na yùcknly yqyzaenknly%
?2 9? 92
?1
9:
91
Ylc knly yqyzaenk nly &V+
N P L & [
+
Mkbqr` gly~r`enl c` br`mki` na NPL { ylcknly yqyzaenkly%
Mkbqr` 6%6% Bràmki` gly~r`enl ac NPL }aryqy ranqiikùe na yùcknly%
&Ac`olr`ikùe zrlzk`+
&o+ C` aiq`ikùe na c` cçea` na rabraykùe ay8
NPL &[+ 7 0%19 $ :%;:; V&yùcknly yqyzaenknly+
C` zaenkae~a ay kbq`c ` :%;:; { ac ke~ariaz~l ay 0%19
&i+ C` ~`oc` na `o`hl gqay~r` c` kemlrg`ikùe na @EL]@%
^@OC@ 6%;% ^`oc` na @EL]@ na yùcknly yqyzaenknly { NPL%Mqae~a na YY b%c% GY Mi`ci% Mirç~ki` ]`clr na z]`rk`ikùe
Ae~ra cly brqzly 9?%:: 0 9?%:: >%91 9%;6 :%:>_ayknq`c &arrlr+ 101%>> 2: 2%91^l~`c 1>=%>> 20
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
&n+ y 7 :%;:9; _ ? 7 66%6! _ ?&`hqy~`n`+ 7 66%1!Z_AYY 7 90%6;?6 _ ?&zranaikn`+ 7 62%09! Nqroke'R`~yle y~`~ky~ki 7 0%=2
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 576/846
Nr% Fëi~lr Pqa}anl Qrç`y
6'9?
@pqç# ac ilamkikae~a na na~argke`ikle _ ?# gkna# pqë ~`e okae ac glnacl na
rabraykùe `hqy~` cly n`~ly% @eàclb`gae~a# ac ay~`nçy~kil Z_AYY &yqg` na
iq`nr`nly na arrlr na zrankiikùe+ gkna c` i`ckn`n nac glnacl na rabraykùe% Aeiq`e~l ` c` ay~`nçy~ki` Nqroke'R`~yle# yk ay~à iari`e` ` ? el f`{ `q~lilrrac`ikleay
ae yarkay zlyk~k}`y l eab`~k}`y% C` }`rk`ikùe na cly n`~ly c` n` c` ay~`nçy~ki` y%
&a+ C` Mkbqr` 6%; n` c` kemlrg`ikùe yqoha~k}` z`r` c` a}`cq`ikùe nac glnacl%
&`+
?1 9: 91
'>
'9
'?
'0
:
0
?
Mk~~an ]`cqa
Y ~ ` e n ` r n k x a n _ a y k n q ` c
_ayknq`cy ]aryqy ~fa Mk~~an ]`cqay
&rayzleya ky NPL &[++
&o+
'> '9 '? '0 : 0 ?
'?
'0
:
0
?
E l r g ` c Y i l r a
Y~ en`rnkxan _ayknq c
Elrg`c Zrlo`okck~{ Zc l~ lm ~fa _ayknq`cy
&rayzleya ky NPL &[++
Mkbqr` 6%;% C` mkbqr` &`+ zrqao` zlr c` `q~lilrrac`ikùe l m`c~` na kenazaenaeik` na cly
n`~ly% @nagày# c` mkbqr` &o+ zrqao` zlr c` elrg`ckn`n na cly n`~ly%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 577/846
Nr% Fëi~lr Pqa}anl Qrç`y
6'99
_abraykùe { ilrrac`ikùe ckea`c gøc~kzca
Gqif`y `zcki`ikleay nac `eàckyky na rabraykùe ke}lcqir`e yk~q`ikleay nlena ya ~kaea
gày na qe` }`rk`oca kenazaenkae~a% Ae c` g`{lr z`r~a na cly zrlocag`y nake}ay~kb`ikùe ya eaiayk~`e }`rk`y }`rk`ocay kenazaenkae~ay z`r` }ar ac amai~l ae c`
}`rk`oca nazaenkae~a% C` }`rk`oca nazaenkae~a l na rayzqay~` &[ + zqana ay~`r
rac`ikle`n` ile gqif`y }`rk`ocay kenazaenkae~ay l rabraylr`y V 0# V ?# a~i%
Ae ac ay~qnkl na rabraykùe ckea`c gøc~kzca ya zqanae qy`r ac aemlpqa g`~rkik`c%
^`gokëe ya zqanae f`iar zrqao`y na fkzù~ayky# ke~ar}`cly na ilemk`ex`# `eàckyky
yqoha~k}ly &`eàckyky na cly bràmkily+ { `eàckyky loha~k}ly &ay~`nçy~ki` na kemaraeik`+#
ilgl cly iàciqcly na cly ilamkikae~ay na na~argke`ikùe &_ ?+ l na ilrrac`ikùe &_+#
ilgl ae ac i`yl na c` rabraykùe ckea`c ykgzca% Yke ago`rbl# ae ay~a i`yl# ya zqana
i`ciqc`r ac ilamkikae~a na ilrrac`ikùe baear`c { ilamkikae~ay na ilrrac`ikùe z`rik`cay#
ay naikr# ae mlrg` `eàclb` ` ilgl ya f`ia ile cly ilamkikae~ay οl# ο0# a~i%
Iq`enl f`oc`gly na rabraykùe ckea`c gøc~kzca ~aeagly c`y ykbqkae~ay
yk~q`ikleay8
0% Glnacl na zrkgar lrnae ile nly }`rk`ocay rabraylr`y l kenazaenkae~ay%
?% Glnacl na zrkgar lrnae ile gày na nly }`rk`ocay kenazaenkae~ay%
Glnacl na rabraykùe gøc~kzca baear`ckx`nl
Iq`enl ay~a glnacl baear`c ay ckea`c ae cly ilamkikae~ay ya naelgke` glnacl na
rabraykùe gøc~kzca% Zlr ahagzcl# z`r` ac i`yl na d }`rk`ocay kenazaenkae~ay v0# v?#
v9#%%%# vd # ac zrlgankl ay~à n`nl zlr [sv0# v?# v9#%%%# vd { ya n` zlr ac glnacl na
rabraykùe gøc~kzca zloc`ikle`c8[ 7 ±[ s v0# v?# v9#%%%# vd 7 οl $ ο0v0 $ ο? v? $ %%%$ οd vd $ αd &6'?;+
Ay~a glnacl# ~`gokëe ya zqana avzray`r ile l~r` `el~`ikùe ilgl8
[ h 7 οl $ ο0 V 0h $ ο? V ? h $ ……% $ οd V dh $ α h &6'?;`+
Cly z`ràga~rly ο h# h 7 :# 0# ?# 9#%%# d ya ileliae ilgl ilamkikae~ay na rabraykùe
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 578/846
Nr% Fëi~lr Pqa}anl Qrç`y
6'9>
zloc`ikle`cay% Zlr ahagzcl# ac z`ràga~rl ο h razrayae~` ac i`gokl ayzar`nl ae c`
rayzqay~` [ # zlr qekn`n na i`gokl ae v h# iq`enl ~lnly cly nagày zrlely~ki`nlray vk
ya g`e~kaeae iley~`e~ay% @nagày# αk { ak yle cly arrlray `ca`~lrkly l rayknqly na zloc`ikùe { na c ay~`nçy~ki` `ylik`nly ile c` rayzqay~` [ k%
Ac glnacl na rabraykùe ckea`c gøc~kzca na c` gqay~r` pqa ay~kg` `c glnacl
zloc`ikle`c na `rrko` ay8
[ 7 ol $ o0 V 0 $ o?v? $ %%% $ od V d $ a &6'9:+
Nlena i`n` ilamkikae~a na rabraykùe z`rik`c οk ay ay~kg`nl zlr ok% Ay~l ya naoa `
pqë# i`n` ilamkikae~a z`rik`c οk gkna ac i`gokl ayzar`nl ae [ zlr qekn`n na i`gokl
ae v0# iq`enl v? ya g`e~kaea iley~`e~a# { ο? gkna ac i`gokl ayzar`nl ae [ zlr
qekn`n na i`gokl ae v? iq`enl v0 ya g`e~kaea iley~`e~a%
Ac glnacl na zrkgar lrnae ile nly }`rk`ocay kenazaenkae~ay ay8
[ k 7 οl $ ο0 V k0 $ ο? V k? $ α &6'90+
Nlena [ k# c` }`rk`oca nazaenkae~a pqa nael~` c` rayzqay~` ae c`y 'ëykg`y ~ae~`~k}`y4
V k0 { V k? yle c`y nly }`rk`ocay kenazaenkae~ay na c` 'ëykg` ~ae~`~k}`4 οl# ο0# ο? yle cly
ilamkikae~ay na rabraykùe {# α ay ac arrlr l rayknql%
Glnacl na rabraykùe gøc~kzca ile gày na nly }`rk`ocay kenazaenkae~ay
[ k 7 οl $ ο0 V k0 $ ο? V k? $ … $ ο z'0 V k#z'0 $ α &6'9?+
Iq`enl f`oc`gly na rabraykùe ckea`c gøc~kzca# ac zrkeikz`c loha~k}l ay c` lo~aeikùe
na c` aiq`ikùe na c` cçea` na rabraykùe gqay~r`c# z`r` zrankiikùe { ay~kg`ikùe# c`
iq`c agqc` ` c` aiq`ikùe zloc`ikle`c% Yke ago`rbl# `e~ay na zlnar qy`r ac glnacl
na rabraykùe i`ciqc`nl# ëy~a ya ~kaea pqa a}`cq`r# z`r` }ar pqë ~`e~` ilemk`okckn`n yaca zqan` n`r% C` a}`cq`ikùe l }`ckn`ikùe nac glnacl na rabraykùe ay~kg`nl ya f`ia
` ~r`}ëy na `eàckyky loha~k}ly { yqoha~k}ly# ae mlrg` `eàclb` ilgl ae c` rabraykùe
ckea`c ykgzca% Zlr ahagzcl# cly `eàckyky loha~k}ky~`y ya f`iae ` ~r`}ëy na mqeikleay
ay~`nçy~ki`y na kemaraeik`% Zly~arklrgae~a# z`r` pqa c` }`ckn`ikùe nac glnacl ya`
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 579/846
Nr% Fëi~lr Pqa}anl Qrç`y
6'91
ilgzca~`# ac zrliankgkae~l ya ilgzcagae~` qy`enl aemlpqay yqoha~k}ky~`y# ` ~r`}ëy
na `eàckyky na c`y bràmki`y na cly }`clray rayknq`cay% Yk c` }`ckn`ikùe el ay
y`~kym`i~lrk`# ya zrliana ile ragank`ikùe nac glnacl# {` ya` f`ikaenl~r`eymlrg`ikleay na cly ahay l zrlo`enl l~rly glnacly gày `zrlzk`nly# ilgl
iq`nrà~kily l iøokily# a~i%
@zcki`ikùe na `eàckyky yqoha~k}ly { loha~k}ly z`r` c` a}`cq`ikùe nac glnacl na
rabraykùe
Ilgl ya f` ay~`nl gaeikle`enl `e~arklrgae~a# ya yqbkarae nly g`ear`y na ra}ky`r
c` q~kckn`n nac glnacl lo~aeknl% Ay~`y g`ear`y yle8 &0+ `eàckyky na bràmki`y na
rayknqly {# &?+ zrqao`y ay~`nçy~ki`y na kemaraeik`%
Zlr ahagzcl# z`r` }`ckn`r ac glnacl na rabraykùe `zcki`enl `eàckyky
yqoha~k}ly# ay naikr# ` ~r`}ëy na cly bràmkily na cly rayknqly &ak+# ëy~ly ya nayirkoae
ilgl c`y nkmaraeik`y ae~ra cly zqe~ly { c` cçea` na rabraykùe% Ykaenl `yç# c`y
yqzlykikleay yle na pqa cly rayknqly naoae yar kenazaenkae~ay { elrg`cgae~a
nky~rkoqknly# ile zrlgankl kbq`c ` iarl { ile }`rk`ex`y iley~`e~ay% Gày
avzcçik~`gae~a# c`y nayirkzikleay na c`y yqzlykikleay yle8
0% Cly }`clray na c` }`rk`oca `ca`~lrk` ay~`nçy~ki` ak naoae ay~`r elrg`cgae~a
nky~rkoqknly% Z`r` clbr`r ay~l# ya br`mki`e cly rayknqly &irqnly l ay~`en`rkx`nly+ na
c` }`rk`oca nazaenkae~a ae mqeikùe na cly }`clray na x l elrg`cay ayzar`nly% Z`r`
pqa ya raøe` c` ilenkikùe na elrg`ckn`n na cly n`~ly# ~lnly cly zqe~ly naoae na
ay~`r nae~rl na c`y o`en`y na ilemk`ex` { naoae na ay~`r gq{ iari`ely ` c` cçea` na
rabraykùe% @nagày# yk cly ~ërgkely nac arrlr ak ay~àe elrg`cgae~a nky~rkoqknly# clyrayknq`cay ay~`en`rkx`nly l irqnly naoaràe ay~`r# `zrlvkg`n`gae~a# na `iqarnl
ile c`y rabc`y nac =6!# ;>! { ;;!% Ay~l pqkara naikr pqë# ac =6! na cly rayknqly
naoaràe ay~`r ae~ra x 7 °04 ac ;1! naoaràe ay~`r ae~ra x 7 °? {# mke`cgae~a# ac ;;!
na cly rayknqly naoaràe ay~`r ae~ra x 7 °9%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 580/846
Nr% Fëi~lr Pqa}anl Qrç`y
6'9=
?% Cly }`clray na c` }`rk`oca `ca`~lrk` ak naoae yar kenazaenkae~ay qel nac l~rl% El
naoa f`oar ilckek`ckn`n l ilrrac`ikùe ae yarka% Ay~l ya ra}ky` br`mki`enl cly
rayknqly &ay~`en`rkx`nly l irqnly+ ae mqeikùe na cly raebcleay% Yk el f`{#`zrlvkg`n`gae~a# cly gkygly rayknqly zlyk~k}ly { eab`~k}ly ae c` bràmki`#
ae~leiay# ac glnacl ckea`c i`ciqc`nl el ay ac `zrlzk`nl { ~aenràe pqa oqyi`rya
l~r`y `c~are`~k}`y &ilgl mqeikleay zlckelgk`cay# iq`nrà~ki`y# iøoki`y# a~i%+% @pqç
i`oa el~`r pqa c` yqzlykikùe na kenazaenaeik` ay c` gày kgzlr~`e~a pqa ya zqan`
}klc`r# zlrpqa ay c` o`ya z`r` c`y zrqao`y ay~`nçy~ki`y ilgl c` _ ?# ac arrlr na cl
ay~kg`nl &y n`nl zlr ac zrlbr`g` Gkek~`o+# @EL]@# a~i%
9% Cly }`clray na c` }`rk`oca `ca`~lrk` ak naoae na ~aear c` gkyg` }`rk`ex`% Ay~l ya
cc`g` flglyian`y~kikn`n% Ay~l ya zqana ra}ky`r }kyq`cgae~a br`mki`enl cly
rayknq`cay ay~`en`rkx`nly l el ay~`en`rkx`nly &irqnly+ ile~r` i`n` }`clr na c`y
}`rk`ocay kenazaenkae~ay & V k+% @pqç# eqa}`gae~a# ~kaea pqa f`oar c` gkyg`
i`e~kn`n na }`clray zlyk~k}ly { eab`~k}ly avzray`nly ae c` bràmki`% @pqç# yke
ago`rbl# avky~ae l~rly gë~lnly z`r` ra}ky`r zlr ac zrlocag` na
fa~arlyian`y~kikn`n pqa ya ra~lg`ràe ae ac i`zç~qcl na rabraykùe zlckelgk`c%
L~rly ke}ay~kb`nlray ay~`nçy~kily &Na}lra# ?:::+ yqbkarae iq`~rl bràmkily na
nk`beùy~kil yqoha~k}l# z`r` c` }`ckn`ikùe nac glnacl na rabraykùe gøc~kzca% Ay~ly
bràmkily na nk`beùy~kil yle8
0% Ac bràmkil na cly rayknqly ay~`en`rkx`nly {,l irqnly ae c` lrnae`n` }aryqy cly
}`clray na V k ae c` `oyiky`%
?% Ac bràmkil na cly rayknqly ay~`en`rkx`nly {,l irqnly ae c` lrnae`n` }aryqy cly}`clray zrlely~ki`nly &ae c` `oyiky`+ zlr ac zrlbr`g` na ilgzq~`nlr` qy`nl%
9% Ac bràmkil na cly }`clray zrlely~ki`nly ae c` lrnae`n` }aryqy cly }`clray na [ k ae
c` `oyiky`%
>% Bràmkil na elrg`ckn`n na cly rayknqly ay~`en`rkx`nly }aryqy cly zariae~kcay na x
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 581/846
Nr% Fëi~lr Pqa}anl Qrç`y
6'92
&}`clray na x+%
1% Fky~lbr`g`y%
@zcki`ikùe na `eàckyky loha~k}ly z`r` c` a}`cq`ikùe nac glnacl na rabraykùe
Zlr l~rl c`nl# ae iq`e~l `c aemlpqa loha~k}ky~` &ay~`nçy~ki` kemaraeik`c+ z`r` c`
}`ckn`ikùe nac glnacl na rabraykùe# ëy~a ay~à rac`ikle`nl ile ac qyl na ay~`nçy~ki`y
ilgl ac ilamkikae~a na na~argke`ikùe gøc~kzca _ ? &l r ?+# ac ilamkikae~a na
na~argke`ikùe `hqy~`nl _ ?`hqy~`n`# ac arrlr ay~àen`r na cl ay~kg`nl# y# ~`oc`y na
`eàckyky na }`rk`ex`# zrqao`y na ~ na Ay~qnk`e~a# ke~ar}`cly na ilemk`ex`# ac irk~arkl
na G`ccl na Iz# Z_AYY# a~i%
Na ay~` g`ear`# iq`enl ya f`oc` na ilamkikae~ay ae ac glnacl na rabraykùe
gøc~kzca# avky~ae iq`~rl ~kzly na ilamkikae~ay8
&0+ Ac ilamkikae~a na na~argke`ikùe gøc~kzca &_ ?+
&?+ Ac ilamkikae~a na ilrrac`ikùe gøc~kzca &_+
&9+ Ac ilamkikae~a na na~argke`ikùe `hqy~`nl &_ ?`hqy~`n`+
&>+ Ac ilamkikae~a z`rik`c na ilrrac`ikùe gøc~kzca & _kh%d +
Zlr ahagzcl# ac ilamkikae~a na na~argke`ikùe gøc~kzca _ ? ay# ~`c }ax# c`
gankn` ay~`nçy~ki` gày zlzqc`r qy`n` z`r` gankr# pqë ~`e okae aei`h` ac glnacl na
rabraykùe ae cly n`~ly na c` gqay~r`% Ae ra`ckn`n ac qyl na _ ? ay qe` ~ëieki` z`r`
gankr c` `naiq`ikùe na qe glnacl na rabraykùe ckea`c gøc~kzca% Ay~` ay~`nçy~ki` ya
zqana namkekr ilgl qe` zrlzlrikùe l ilgl qe zlriae~`ha% Ilgl zrlzlrikùe# yqy
}`clray }`rç`e na iarl ` qel% Zlr ahagzcl# yk ac }`clr na _ ? ay~à iari`el ` iarl# ay~l
kenki` pqa el f`{ qe` rac`ikùe ckea`c ae~ra [ { c`y V ¾y# gkae~r`y pqa# qe }`clr iari`el ` qel# kenki` qe` `hqy~a zarmai~l% Yke ago`rbl# ac }`clr na _ ? el naoa na
ke~arzra~`rya ckbar`gae~a# yke ac `zl{l nac arrlr ay~àen`r na cl ay~kg`nl &y+# ac
rayknq`c &Z_AYY+# ac irk~arkl na G`ccl &Iz+ l cly m`i~lray na }`rk`ikùe kemc`nly
&}`rk`eia kemc`~kle m`i~lry# ]KM+% @nagày c` }`ckn`ikùe nac glnacl naoa ay~`r
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 582/846
Nr% Fëi~lr Pqa}anl Qrç`y
6'96
`zl{`n` zlr cly `eàckyky na cly bràmkily yqoha~k}ly%
Na `iqarnl ` c` cùbki` nac zrlbr`g` na EIYY# cly ykbqkae~ay aeqeik`nly n`e
`cbqe`y i`ckmki`ikleay na c` ke~arzra~`ikùe na _ ?
%0% Ac }`clr na _ ? zqana keiragae~`rya `brab`enl gày }`rk`ocay kenazaenkae~ay#
zarl ay~l zqana i`qy`r qe `qgae~l ae ac arrlr nac iq`nr`nl gankl# ayzaik`cgae~a#
iq`enl c` gqay~r` ay zapqað`%
?% C` g`bek~qn na _ ? ay~à kemcqaeik`n` zlr ac r`ebl na i`n` }`rk`oca kenazaenkae~a%
_ ? `qgae~` ` gankn` pqa ac r`ebl na c`y V ¾y `qgae~` { }kia}ary`%
9% Ac }`clr na _ ? el gkna c` g`bek~qn na c`y zaenkae~ay%
>% C` g`bek~qn na _ ? el gkna c` `z~k~qn nac glnacl ckea`c4 gkna c` mqarx` ckea`c nac
ilgzleae~a nac glnacl%
1% Qe }`clr br`ena na _ ? el eaiay`rk`gae~a ykbekmki` qe` zrankiikùe br`ena% Cl
lzqay~l ~`gokëe ay ilrrai~l% ^lnl ay~l ~kaea pqa yar ilgzcagae~`nl l
ilrrlolr`nl zlr l~r`y mqeikleay ay~`nçy~ki`y { zlr ac `eàckyky bràmkil yqoha~k}l%
=% Ac }`clr na _ ? ay `c~`gae~a yaeykoca `c eøgarl na loyar}`ikleay% Ae~ra gày
br`ena ya` ac ~`g`ðl na c` gqay~r`# gày `c~l yarà ac }`clr na _ ?% Gày `nac`e~a# f`{
cl pqa ya cc`g` ac }`clr `hqy~`nl nac ilamkikae~a na na~argke`ikùe gøc~kzca `hqy~`nl
&_ ?`hqy~`n`+% Ay~a ilamkikae~a na na~argke`ikùe gøc~kzca `hqy~`nl _ ?`hqy~`n` ay qe`
}arykùe `hqy~`n` na _ ? c` iq`c oqyi` ragl}ar c` nky~lrykùe i`qy`n` zlr qe ~`g`ðl
na gqay~r` zapqaðl% Kbq`cgae~a# ~`gokëe f`{ cl pqa ya cc`g` Z_AYY &zranki~an
yqg lm ypq`ray+ pqa ya qy` z`r` }`ckn`r ac glnacl na rabraykùe ae ~ërgkely na
zrankiikùe% @pqç# ae~ra gày zapqaðl ya` ac }`clr na Z_AYY# gahlr yarà ac glnacli`enkn`~l%
Ae mlrg` `eàclb`# ~`gokëe f`{ cl pqa ya cc`g` ac ilamkikae~a na ilrrac`ikùe
gøc~kzca _% Ay~a ilamkikae~a _ gkna c` mqarx` na c` rac`ikùe ckea`c ae~ra c` }`rk`oca
nazaenkae~a [ { c`y }`rk`ocay kenazaenkae~ay V 0# V ? # V 9 #…# V d % Ae ile~r`y~a ile ac
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 583/846
Nr% Fëi~lr Pqa}anl Qrç`y
6'9;
ilamkikae~a na ilrrac`ikùe ckea`c ykgzca# ac r`ebl na ay~a ilamkikae~a na ilrrac`ikùe
gøc~kzca ay na : ≮ _ ≮ 0% Ay~l ya naoa ` pqa _ el kenki` c` zaenkae~a na c`
aiq`ikùe na rabraykùe naoknl ` pqa el ay zlykoca kenki`r cly ykbely na ~lnly clyilamkikae~ay na rabraykùe pqa rac`ikle`e c` }`rk`oca nazaenkae~a [ ` c`y }`rk`ocay
kenazaenkae~a V k% @yç ilgl ae ac i`yl na c` ilrrac`ikùe ckea`c# c` gankikùe na _ ? ay
gày màikc na ke~arzra~`r pqa ac ilamkikae~a na ilrrac`ikùe gøc~kzca# _%
L~rl ~kzl na ilrrac`ikùe rac`ikle`nl ile rabraykùe { ilrrac`ikùe gøc~kzca ay
cl pqa ya cc`g` ilamkikae~a z`rik`c na ilrrac`ikùe gøc~kzca% Ay~a ilamkikae~a gkna c`
mqarx` na c` rac`ikùe ckea`c ae~ra c` }`rk`oca nazaenkae~a [ { c`y }`rk`ocay
kenazaenkae~ay V 0 # V ? # V 9 #…# V d % Ay~a ilamkikae~a ya zqana avzray`r ilgl _ kh%d ac
iq`c ay ac ay~kg`nlr nac ilamkikae~a na ilrrac`ikùe gøc~kzca zloc`ikle`c πkh%d % _ kh%d ya
zqana qy`r z`r` }ar c` rac`ikùe i`qy`c ae~ra [ { qe` na c`y }`rk`ocay kenazaenkae~ay#
g`e~aekaenl c`y nagày iley~`e~ay% Ay~a ilamkikae~a# ~`gokëe ya zqana qy`r z`r` }ar
c` rac`ikùe ae~ra nly }`rk`ocay kenazaenkae~ay%
Gày `nac`e~a# nae~rl na c` i`~ablrç` na `eàckyky loha~k}ly na ay~`nçy~ki`
kemaraeik`c rac`ikle`nly ile rabraykùe gøc~kzca# ~aeagly cl pqa ya cc`g` `eàckyky na
}`rk`ex` &@EL]@+ nkyiq~knl ae i`zç~qcly `e~arklray% Ae mlrg` `eàclb` ilgl ac
qyl na _ ?# ay~a `eàckyky ay qe gë~lnl ilgzcagae~`rkl z`r` ra}ky`r c`y yqzlykikleay
nac glnacl na rabraykùe% C` ilemk`okckn`n na cly rayqc~`nly nac @EL]@ ay~à
g`eilgqe`n` ` c` yqzlykikùe na pqa cly rayknq`cay ay~àe elrg`cgae~a
nky~rkoqknly% Ac qyl na @EL]@ zrqao` cly zrlgankly zloc`ikle`cay nlena ya
`e`ckx` c` }`rk`ikùe ~l~`c% @EL]@ a}`cø` c` q~kckn`n nac glnacl na rabraykùe zrlo`enl c` fkzù~ayky eqc` na pqa ~lnly cly ilamkikae~ay &οk+ na c` aiq`ikùe na
rabraykùe &zaenkae~ay+ yle kbq`c ` iarl% Cly ilgzleae~ay nac `eàckyky na }`rk`ex` l
na @EL]@# yle z`raiknly ` cly nac `eàckyky na }`rk`ex` ykgzca avzcki`nly ae
i`zç~qcly `e~arklray% Cly ilgzleae~ay yle c` mqae~a na }`rk`ikùe# cly br`nly na
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 584/846
Nr% Fëi~lr Pqa}anl Qrç`y
6'>:
ckoar~`n# c` yqg` na cly iq`nr`nly# ac iq`nr`nl nac zrlgankl# c` zrqao` na M { ac
ek}ac na zrlo`okckn`n% Zlr ahagzcl# c` mqae~a na }`rk`ikùe razrayae~` c`y z`r~kikleay
na c` }`rk`ikùe ae [ % F`{ iq`~rl mqae~ay na }`rk`ikùe ay naikr# ac ke~ariaz~l# acglnacl# ac rayknql l arrlr {# ac ~l~`c `hqy~`nl% C` zrqao` na kemaraeik` ile c`
ay~`nçy~ki` M ya qy` z`r` zrlo`r c` fkzù~ayky na ~ln`y c`y οk 7 :%
Gày kgzlr~`e~a ~ln`}ç`# ay ac iàciqcl nac ek}ac na zrlo`okckn`n z% Ac }`clr na
z ay c` zrlo`okckn`n na lo~aear qe ay~`nçy~kil na zrqao`# `c gaely ~`e
ile~r`nki~lrkl l gày av~ragl z`r` Fl8# ilgl ac }`clr loyar}`nl pqa ya lo~q}l#
`yqgkaenl pqa Fl8 ay }arn`nar`% Yk ac }`clr na z ay gaelr pqë# nkb`gly ΰ 7 :%:1#
c` fkzù~ayky eqc` ya raif`x`4 na l~r` g`ear` ya ra~kaea% Ae~ra gày zapqaðl ya` ac
}`clr na z# gaely irankokckn`n ~aenrà c` fkzù~ayky eqc`%
L~rly ay~`nçy~kily loha~k}ky~`y z`r` }`ckn`r ac glnacl na rabraykùe yle c`y
zrqao`y kenk}knq`cay na ~ na ay~qnk`e~a z`r` zrlo`r c` fkzù~ayky na pqa ο0# ο?# ο9# οd
yle kbq`cay ` iarl% @nagày ya zqanae qy`r cly ke~ar}`cly na ilemk`ex`% Zlr
ahagzcl# ae rabraykùe gøc~kzca ac }`clr na ~ na ay~qnk`e~a ya qy` z`r` zrlo`r c`
fkzù~ayky na pqa qel na cly ilamkikae~ay ay kbq`c ` iarl# nayzqëy na ragl}ar c`
kemcqaeik` na cly l~rly% Cly ke}ay~kb`nlray Z`mmaeoarbar a~ `c% &0;62+ n`e c` mqeikùe
z`r` ac ke~ar}`cl na ilemk`ex` z`r` οk% Yke ago`rbl# yk ya ileicq{a pqa ο0 l οd el
yle kbq`c ` iarl ay~l# el eaiay`rk`gae~a# nkia pqa ac glnacl na rabraykùe ay ø~kc
z`r` zrankiikùe% Ae }arn`n# z`r` na~argke`r yk ac glnacl ay `zrlzk`nl# ae cqb`r na
zrlo`r pqa ο0 7 : { ο? 7 :# yaz`r`n`gae~a &qy`enl c` zrqao` na ~+# ya qy` qe`
zrqao` ilehqe~` ilgl ac `eàckyky na }`rk`ex` &@EL]@+% Na iq`cpqkar g`ear`# c` zrqao` na fkzù~ayky okc`~ar`c z`r` zrlo`r cly ilamkikae~ay kenk}knq`cay οk ya qy` ac
ykbqkae~a mlrg`~l n`nl ae c` ~`oc` na `o`hl%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 585/846
Nr% Fëi~lr Pqa}anl Qrç`y
6'>0
^@OC@ 6%0:% ^`oc` gly~r`enl cly gai`ekygly z`r` qe` zrqao` na fkzù~ayky
okc`~ar`c z`r` cly ilamkikae~ay kenk}knq`cay οk keicqknly ae ac glnacl na rabraykùe
gøc~kzca% &Ac`olr`ikùe zrlzk`+Fkzù~ayky eqc`8 Fl8οk 7 :# fkzù~ayky `c~are`~k}`8 F08οk ≪ :
]`clr nac ay~`nçy~kil8 ~ 7 ok , y ok
_abc` na naikykùe8
_aif`x`r Fl8 yk ~ 5 ~ΰ,?4e'&d$0+ l okae yk ~ 3 '~ΰ,?4e'&d$0+%
El raif`x`r Fl8 yk ~ΰ,?4e'&d$0+ ≮ ~ ≮ ~ΰ,?4e'&d$0+
Nlena8 οk yle cly ilamkikae~ay na rabraykùe kenk}knq`cay%ok 7 ay~kg`nlray na οk
yok 7 arrlray ay~àen`r
ΰ 7 ek}ac na ykbekmki`eik` naya`nl
e 7 eøgarl na loyar}`ikleay
d 7 eøgarl na }`rk`ocay kenazaenkae~ay
~ 7 mqeikùe ay~`nçy~ki` na ~ na Ay~qnk`e~a
Ahagzcly `zcki`enl c` rabraykùe { ilrrac`ikùe gøc~kzca
Ahagzcl )=% Ae c` `nylrikùe na ~karr` { yankgae~l# c` g`bek~qn na c` `iqgqc`ikùe
ae mlrg` ilenaey`n` na cly zrlnqi~ly pqçgkily ae c` yqzarmkika ay qe`
i`r`i~arçy~ki` kgzlr~`e~a pqa kemcq{a ae c` amkikaeik` na keyai~kikn`y { }`rkly l~rly
zrlnqi~ly pqçgkily% Ac `r~çiqcl –@nylrz~kle lm Zflyzf`~a# @ryae`~a#
Ga~f`ea`ryle`~a `en I`iln{c`~a o{ C`da `en Y~ra`g Yankgae~y8 Ilgz`rkyle k~f
Ylkcy‗ &H% lm Ae}krle% Pq`c%# 0;6># zz% >;;'1:>+ zrayae~` cly ykbqkae~ay n`~ly ae c`~`oc` na `o`hl% @pqç ya ~lg` [ ilgl c` }`rk`oca nazaenkae~a# c` iq`c nael~` ac
çenkia na `nylrikùe na mlym`~l# V 0 ay qe` na c`y }`rk`ocay kenazaenkae~ay
nael~`enl c` i`e~kn`n na fkarrl av~r`çoca {# V ? ay l~r` na c`y }`rk`ocay
kenazaenkae~ay nael~`enl c` i`e~kn`n na `cqgkekl av~r`çoca% &Na}lra# ?:::+
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 586/846
Nr% Fëi~lr Pqa}anl Qrç`y
6'>?
^@OC@ 6%00% ^`oc` gly~r`enl cly n`~ly nac ahagzcl% \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
Loyar}`ikùe V 0 &Fkarrl av~r`çoca+ V ? &@cqgkekl av~r`çoca+ [ &Çenkia na `nylrikùe+ \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ 0 =0 09 >? 021 ?0 069 000 ?> 0>> 0?> ?9 061 09: => ?== 029 96 ?=2 0=; 99 ?06 0=; =0 9:
; 0=: 9; ?60: ?>> 20 9=00 ?12 00? =10? 999 66 =?09 0;; 1> >:
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
&Mqae~a8 Na}lra# ?:::+
F`iar cly iàciqcly zar~keae~ay%
Ylcqikùe8
Qy`enl qe z`pqa~a na ilgzq~`nlr` n`8 ol 7 '2%910# nay}k`ikùe ay~àen`r 7 9%>61#
o0 7 :%00?29# nay}k`ikùe ay~àen`r 7 :%:?;=;# o? 7 :%9>;::# y 7 :%:2090
C` aiq`ikùe na c` cçea` na rabraykùe ckea`c gøc~kzca ay8
[ 7 '2%910 $ &:%00?29+& V 0+ $ &:%9>;::+& V ?+
Aeyabqkn`# z`r` }ar# pqë ~`e ilemk`oca ay ac glnacl na rabraykùe i`ciqc`nl#
zrkgarl zrlianagly ` amai~q`r ac `eàckyky yqoha~k}l# ay naikr# ac `eàckyky na c`y
bràmki`y na cly rayknqly%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 587/846
Nr% Fëi~lr Pqa}anl Qrç`y
6'>9
Mkbqr` 6%0: Mkbqr` gly~r`enl c`y bràmki`y na cly rayknqly ay~`en`rkx`nly }aryqy
}`clray ayzar`nly na x &0+4 bràmki` gly~r`enl ac rayknql ay~`en`rkx`nl }aryqy c`}`rk`oca kenazaenkae~a V 0 &?+4 bràmki` gly~r`enl ac rayknql ay~`en`rkx`nl }aryqy c`
}`rk`oca kenazaenkae~a V ? &9+4 bràmki` gly~r`enl ac rayknql ay~`en`rkx`nl }aryqy ac
}`clr na [ zrlely~ki`nl &>+ {# mke`cgae~a# bràmki` na [ zrlely~ki`n` }aryqy
`nylrikùe &1+% &Ac`olr`ikùe zrlzk`+
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 588/846
Nr% Fëi~lr Pqa}anl Qrç`y
6'>>
Mkbqr` 6%00 Ay~` bràmki` gqay~r` qe aemlpqa qe zlil nkmarae~a `c na c` mkbqr`
`e~arklr# ay naikr qy`enl cly rayknqly el ay~`en`rkx`nly ae ile~r`y~a ile c` mkbqr`
6%0: pqa qy` cly rayknqly ay~`en`rkx`nly% Bràmki` gly~r`enl c` zrqao` na
elrg`ckn`n &0+% Bràmki` gly~r`enl c` zrqao` na kenazaenaeik` na rayknqly }aryqyraebcleay &?+% Bràmki` gly~r`enl cly rayknqly }aryqy }`clray zrlely~ki`nly &9+%
Bràmki` gly~r`enl cly rayknqly }aryqy }`rk`oca kenazaenkae~a na fkarrl &>+% Bràmki`
gly~r`enl cly rayknqly }aryqy }`rk`oca kenazaenkae~a `cqgkekl &1+% &Ac`olr`ikùe
zrlzk`+
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 589/846
Nr% Fëi~lr Pqa}anl Qrç`y
6'>1
Ac }`clr nac ilamkikae~a na na~argke`ikùe gøc~kzca ay8 _ ? 7 :%;>6:
Ac ilamkikae~a na na~argke`ikùe `hqy~`nl ay8 _ ?`hqy~`n` 7 :%;96:
Ac ilamkikae~a na ilrrac`ikùe gøc~kzca ay8 _ 7 :%;29=Cly ilamkikae~ay z`rik`cay ya zqanae ay~kg`r yk ya naya` y`oar c` rac`ikùe
ae~ra ac çenkia na `nylrikùe { ac `cqgkekl av~r`çoca# zlekaenl c` }`rk`oca
kenazaenkae~a# fkarrl iley~`e~a% ^`gokëe# yk ya naya`r` y`oar c` rac`ikùe ae~ra ac
çenkia na `nylrikùe { ac fkarrl av~r`çoca# ya zlenrç` c` }`rk`oca `cqgkekl iley~`e~a%
Ykgkc`rgae~a# yk ya naya`r` y`oar c` rac`ikùe ae~ra c`y }`rk`ocay `cqgkekl { c`
}`rk`oca nac fkarrl# ya zlenrç` c` }`rk`oca çenkia na `nylrikùe mkh`%
^@OC@ 6%0?% ^`oc` gly~r`enl cly ilamkikae~ay na rabraykùe# }`clray na ~ na
Ay~qnk`e~a# ek}acay na z { naikykleay ~lg`n`y ae Fl8 &Ac`olr`ikùe zrlzk`+
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ ]`rk`oca Ilamkikae~a ]`clr na ~ Ek}ac Naikykùekenazaenkae~a na rabraykùe na z &1!+
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ Ke~ariaz~l '2%91:== '?%0:;> :%:=00 @iaz~`r Fkarrl :%00?29 9%2;=; :%::91 _aif`x`r @cqgkekl :%9>;:: >%6;>> :%:::= _aif`x`r
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
^@OC@ 6%09% ^`oc` na `eàckyky na }`rk`ex`% &Ac`olr`ikùe zrlzk`+ \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ Mqae~a na b%c% Yqg` na cly Iq`nr`nl Mi`ci% ]`clr Zlnar na]`rk`ikùe iq`nr`nly gankl na z c` zrqao`
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ Ke~ariaz~l 0 0016:%90 0016:%90
_abraykùe ? 9?1;%;: 02=>%;1 ;?%:9 :%::: 0%::::Arrlr 0: 0;0%2; 0;%06
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ ^l~`c 0? 92?0%=; 90:%0>
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 590/846
Nr% Fëi~lr Pqa}anl Qrç`y
6'>=
^@OC@ 6%0>% ^`oc` gly~r`enl ac razlr~a na rayknqly% &Ac`olr`ikùe zrlzk`+ \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ _aebcùe ]`clr ]`clr _ayknql Arrlr ay~àen`r
`i~q`c zrlely~ki`nl \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ 0 > >%:=9: '=%9:1? 1%::22? 06 0;%2:== '0%2:== >%;1009 0> 09%1962 :%>=0? >%2:11> 06 0>%=11? 9%9>>2 >%=6=?1 ?= ?;%=>:= '9%=>:= 1%0:10= ?= ?1%>0>0 :%1616 >%1;;=2 ?0 ?9%?06? '?%?06? >%=>666 9: 9?%;;:? '?%;;:? >%==?9; ?6 ?>%?;2= 9%2:?> >%1=20
0: 9= >>%;91? '6%;91? >%2:0?00 =1 =:%2:;2 >%?;:? 1%>?1:0? =? =:%;:0> 0%:;6= 1%>0;109 >: 99%;?;? =%:2:2 >%1=>;
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
^@OC@ 6%01% ^@OC@ gly~r`enl cly ke~ar}`cly na ilemk`ex` z`r` ay~a zrlocag`%&Ac`olr`ikùe zrlzk`+
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ ]`rk`oca Cçgk~a kemarklr &;1!+ Cçgk~a yqzarklr &;1!+kenazaenkae~a
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ Ke~ariaz~l '01%00>; :%>092Fkarrl & V 0+ :%:>=2 :%026;@cqgkekl & V ?+ :%0;:0 :%1:2;
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 591/846
Nr% Fëi~lr Pqa}anl Qrç`y
6'>2
^@OC@ 6%0=% ^`oc` gly~r`enl c` ay~`nçy~ki` nayirkz~k}`% &Ac`olr`ikùe zrlzk`+ \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ ]`rk`oca Ile~al Zrlgankl Nay}k`ikùe ]`clr ]`clr
ay~àen`r gçekgl gàvkgl \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ Fkarrl & V 0+ 09 022%90 2:%0: =0 999@cqgkekl & V ?+ 09 >;%90 ?;%0; 09 00?Çenkia na &[ + 09 ?;%61 02%=0 > =1`nylrikùe
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
Ileicqykleay8 Ac glnacl na rabraykùe lo~aeknl ay }àcknl z`r` zrankiikùe {
ay~kg`ikùe% Cly n`~ly aei`h`e okae ile qe glnacl ckea`c gøc~kzca% Ay~` ile~aeikùeay~à o`y`n` ae ac `eàckyky yqoha~k}l na c`y bràmki`y na cly rayknqly% Zlr ahagzcl# ae
c` mkbqr` 6%0: { 6%00 c` zrqao` na elrg`ckn`n ay oqae`# zlrpqa ~lnly cly zqe~ly
ay~àe nae~rl na c`y o`en`y# { gq{ iari`ely ` c` cçea` na rabraykùe% @nagày# cly
zqe~ly ay~àe na `iqarnl ile c` rabc` nac =6!# ;1! { ;;!# ay naikr# ac =6! na cly
zqe~ly ay~àe nae~rl na x 7 °0# ac ;1! ay~àe nae~rl na x 7 °?# a~i% Ae c` mkbqr` 6%00
na cly rayknqly }aryqy cly raebcleay# ay~l y`~kym`ia c` yqzlykikùe na kenazaenaeik`#
zlrpqa f`{ ac gkygl eøgarl na rayknqly zlyk~k}ly { eab`~k}ly% @nagày# c`y
bràmki`y na cly rayknqly }aryqy c`y }`rk`ocay kenazaenkae~ay el }klc`e c` yqzlykikùe
na el ckea`ckn`n# zlrpqa el f`{ ~aenaeik`y namkekn`y% Mke`cgae~a# c` bràmki` na
rayknqly }aryqy }`clray zrlely~ki`nly ay~àe na `iqarnl ile c` yqzlykikùe na
}`rk`ex`y kbq`cay &flglyian`y~kikn`n+%
Ae iq`e~l ` cly `eàckyky loha~k}ky~`y# ay naikr# qy`enl zrqao`y ay~`nçy~ki`y#
eqa}`gae~a# zrayqzleae qe oqae `hqy~a nac glnacl na rabraykùe ay~kg`nl% Ay~l ya
naoa ` pqë# ac }`clr nac ilamkikae~a na na~argke`ikùe gøc~kzca _ ? ay~à gq{ iari`el
` qel% @nagày# ac }`clr na _ 7 :%;29= kenki` gq{ oqae` ilrrac`ikùe ae~ra c`
}`rk`oca nazaenkae~a { c`y }`rk`ocay kenazaenkae~ay% Ile rayzai~l ` c` ~`oc` nac
`eàckyky na }`rk`ex`# ac }`clr na M ay gqifl gaelr pqa ac }`clr irç~kil { ay~l ay~à
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 592/846
Nr% Fëi~lr Pqa}anl Qrç`y
6'>6
nagly~r`nl zlr ac }`clr na c` zrlo`okckn`n z ac iq`c ay gqifl gq{ ykbekmki`e~a%
C`y zrqao`y na ~ na ay~qnk`e~a# ~`gokëe yle gq{ `iaz~`ocay { nagqay~r`e pqa c`y
zaenkae~ay na οk el yle kbq`cay ` iarl% Cly ke~ar}`cly na ilemk`ex` n`e rayqc~`nlyykgkc`ray { yqbkarae pqa ac glnacl na rabraykùe ay oqae zrlely~ki`nlr% Ya zqanae
yabqkr f`ikaenl zrqao`y na fkzù~ayky z`r` ~lnly cly z`ràga~rly zloc`ikle`cay {# yke
cqb`r ` nqn`y# ëy~`y ~`gokëe `zl{`rç`e c` ile~aeikùe na pqa# ac glnacl na
rabraykùe# ay `zcki`oca%
Ahagzcl )2% Ileyknar`r cly n`~ly na c` ~`oc` na `o`hl% Qy`enl ac zrlbr`g` na
ilgzq~`nlr` Gkek~`o lo~aear ac glnacl na rabraykùe gày `zrlzk`nl# ay naikr# qe
glnacl gøc~kzca ckea`c &Glnacl 0+4 glnacl ile ~r`eymlrg`ikùe ae ac aha }ar~ki`c
&Glnacl ?+ { qe glnacl ile ~r`eymlrg`ikleay na cly ahay flrkxle~`cay { nac aha
}ar~ki`c &Glnacl 9+%
^@OC@ 6%02% ^`oc` gly~r`enl cly n`~ly ok}`rk`nly na rabraykùe% &Ac`olr`ikùe
zrlzk`+
V 0 s > > > = 9 = 9 ?
V ? s 9 > 9 > ? > ? ?
[ s 9 ? 2 = 1 = 2 >
Ylcqikùe8
@o`hl ya n`e cly rayqc~`nly na cly ~ray glnacly% @c hqxb`r zlr cly rayqc~`nly# ya ca
zkna `c cai~lr pqa naikn` iq`c glnacl ay ac gày `zrlzk`nl%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 593/846
Nr% Fëi~lr Pqa}anl Qrç`y
6'>;
^@OC@ 6%06% _ayqc~`nly gly~r`enl ac rayqgae na cly ~ray glnacly%&Ac`olr`ikùe zrlzk`+
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
_abrayykle @e`c{yky8 [ }aryqy V 0 # V ? &Glnacl 0+^fa rabrayykle apq`~kle ky8 [ 7 =%:: $ ?%::V0 ’ 9%::V?
Zranki~lr Ilam YA Ilam ^ ZIley~`e~ =%:::: 0%6:9 9%99 :%:?0V0 ?%:::: :%22>= ?%16 :%:>;V? '9%:::: 0%069 '?%1> :%:1?
y 7 0%>0> _'Yp 7 16%9! _'Yp&`nh+ 7 >0%2!Z_AYY 7 :%0?2> _'Yp&zran+ 7 10%=?!
@e`c{yky lm ]`rk`eia
Ylqria NM YY GY M Z_abrayykle ? 0>%::: 2%::: 9%1: :%00?_ayknq`c Arrlr 1 0:%::: ?%:::^l~`c 2 ?>%:::
_abrayykle @e`c{yky8 Clb [ }aryqy V 0 # V ? &Glnacl ?+
^fa rabrayykle apq`~kle ky8 Clb [ 7 :%60: $ 7%??1V0 ’ :%9>6V?
Zranki~lr Ilam YA ilam ^ ZIley~`e~ :%60:0 :%0=?? >%;; :%::>V0 :%??>6 :%:=;2 9%?9 :%:?9V? ':%9>2; :%0:=1 '9%?2 :%:??
y 7 :%0?2? _'Yp 7 =;%9! _'Yp&`nh+ 7 12%:!Z_AYY 7 :%0?2> _'Yp&zran+ 7 10%=?!
@e`c{yky lm ]`rk`eia
Ylqria NM YY GY M Z_abrayykle ? :%06?> :%:;0? 1%=9 :%:1?_ayknq`c Arrlr 1 :%:6:; :%:0=?^l~`c 2 :%?=9>
_abrayykle @e`c{yky8 Clb [ }y Clb V 0# Clb V ? &Glnacl 9+
^fa rabrayykle apq`~kle ky8 Clb [ 7 :%1;1 $ 0%69 Clb V0 ’ ?%0= Clb V?
Zranki~lr Ilam YA Ilam ^ ZIley~`e~ :%1;>; :%?:;1 ?%6> :%:9=Clb V0 0%69>? :%2?66 ?%1? :%:19Clb V? '?%0129 :%699? '?%1; :%:>;
y 7 :%0>69 _'Yp 7 16%?! _'Yp&`nh+ 7 >0%1!Z_AYY 7 :%9::1 _'Yp&zran+ 7 :%::!
@e`c{yky lm ]`rk`eia
Ylqria NM YY GY M Z_abrayykle 9 :%0199 :%:2=2 9%>6 :%009_ayknq`c Arrlr 1 :%00:: :%:??:^l~`c 2 :%?=9>
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 594/846
Nr% Fëi~lr Pqa}anl Qrç`y
6'1:
^@OC@ 6%0;% ^`oc` gly~r`enl cly n`~ly lrkbke`cay nac zrlocag`% &Ac`olr`ikùe zrlzk`+\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
I0 I? I9 I> I1 I=
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\[ V0 V? Clb [ Clb V 0 Clb V ? \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
0 9 > 9 :%>220?0 :%=:?:=: :%>220?0\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
? ? > > :%9:0:9: :%=:?:=: :%=:?:=:\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
9 2 > 9 :%6>1:;6 :%=:?:=: :%>220?0\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
> = = > :%226010 :%226010 :%:=:?:=\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
1 1 9 ? :%=;6;2: :%>220?0 :%9:0:9:\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
= = = > :%226010 :%226010 :%=:?:=:\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
2 2 9 ? :%6>1:;6 :%>220?0 :%9:0:9:\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
6 > ? ? :%=:?:=: :%9:0:9: :%9:0:9:\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
Ahagzcl )6% Ae ay~qnkly na pqçgki` `e`cç~ki`# ac qyl nac `eàckyky na mcqlrayiaeik`
na r`{ly V ya qy` ilgl qe` farr`gkae~` z`r` ay~kg`r cly zlriae~`hay na cly
kebrankae~ay na gqif`y gaxic`y% @ gaeqnl# c` ay~kg`ikùe na c`y ileiae~r`ikleay
nazaena ae c` f`okckn`n z`r` `hqy~`r glnacly na rabraykùe% Ae qe` ke}ay~kb`ikùe
ke~k~qc`n` –Ilrrai~kley mlr G`~rkv Ammai~y ke V'r`{y mcqlrayiae~ @e`ckyky Qykeb
Gqc~kzca _abrayykle Ga~flny‗# zqocki`nl zlr @e`c{~ki`c Ifagky~r{ &]lc% 92#
0;=1+ gaxic`y ile~kaenl > kebrankae~ay & V k+ mqarle zraz`r`n`y% C`y
ileiae~r`ikleay na cly ilgzleae~ay }`rk`rle ae c`y gaxic`y z`r` zrlnqikr ~kzly
ay~àen`ray na i`ckor`ikùe &[ k+% &R`czlca# 0;;?# z% >?0+% Cly n`~ly na ay~a zrlocag`
ya n`e `o`hl%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 595/846
Nr% Fëi~lr Pqa}anl Qrç`y
6'10
^@OC@ 6%?:% ^`oc` gly~r`enl cly n`~ly nac zrlocag` na `rrko`%
[ k V 0 V ? V 9 V >
:%110> 0%0?>: :%6;6: :%6?0; :%;;:=:%>>?= :%;?61 :%662? :%;9:6 :%;;>>:%1=90 0%0?0> :%6:9: :%2==6 0%0??0:%1=?> 0%0=91 :%62:= :%;?2? :%;69?:%>1:1 :%;>01 :%6:=> :%;:?= 0%00?2:%1?;: 0%:20? :%6>:> :%6==? 0%:69=:%>2:? :%;1=0 :%6290 :%6?:= 0%:?;::%1::0 0%:06= :%6>90 :%69>= 0%:1;0
:%>>?1 :%;:9; :%690> :%21;= 0%:;;>&Mqae~a8 R`czlca a~ `c% 0;;?+
&`+ @hqy~`r qe glnacl ckea`c na rabraykùe gøc~kzca ` cly n`~ly na c` ~`oc`%
Aeyabqkn`# ay~kg`r c`y ileiae~r`ikleay nac kebrankae~a @ z`r` qe` gaxic` iq{`
~`y` na ke~aeykn`nay na r`{ly'V ya`e# rayzai~k}`gae~a# V 0 7 0%0:# V ? 7 :%;::# V 9 7
:%6:: { V > 7 :%;;1%
Ylcqikùe8
&`+ Qy`enl qe z`pqa~a na ilgzq~`nlr` { `yqgkaenl qe glnacl na rabraykùe ckea`cgøc~kzca ya lo~kaea c` aiq`ikùe na rabraykùe%
[ 7 ':%9::> $ :%1962 V 0 $ :%022: V ? ’ :%:2:> V 9 $ :%01:= V >
Yqy~k~q{aenl c`y }`rk`ocay kenazaenkae~ay# ya lo~kaea ac }`clr na c` rayzqay~` [ # ay
naikr8
[ 7 ':%9::> $ :%196&0%0:+ $ :%022:&:%;:+ ’ :%:2:>&:%6:+ $ :%01:=&:%;;1+
7 :%1:Ahagzcl );% Gle~blgar{ { Zaid &0;;?+ nayirkoae ac qyl na qe glnacl na
rabraykùe z`r` rac`ikle`r c` i`e~kn`n na ~kagzl pqa rapqkara qe }aenanlr z`r` n`r
yar}kikl ` qe` gàpqke` avzaenanlr` na `r~çiqcly { ac eøgarl na agz`pqay
ile~aeknly ae c` gàpqke` { c` nky~`eik` nac }afçiqcl &zkay+ na yar}kikl nac yk~kl
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 596/846
Nr% Fëi~lr Pqa}anl Qrç`y
6'1?
nlena ya aeiqae~r` c` gàpqke`% Ay~a glnacl na rabraykùe gøc~kzca mqa q~kckx`nl
z`r` nkyað`r c` rq~`# cly flr`rkly { c` y`ckn` na cly }afçiqcly% C` ~`oc` na `o`hl
gqay~r` ?1 loyar}`ikleay nac ~kagzl na yqgkeky~rl# eøgarl na agz`pqay { c`nky~`eik`# nac }afçiqcl%
^@OC@ 6%?0% ^`oc` gly~r`enl cly n`~ly na yqgkeky~rl%
El% na loyar}`ikùe ^kagzl na yqgkeky~rl El% na ae}`yay Nky~`eik` nac }afçiqcl
0 ;%>1 ? 1:? ?>%>1 6 00:9 90%21 00 0?:> 91%:: 0: 11:
1 ?1%:? 6 ?;1= 0=%6= > ?::2 0>%96 ? 9216 ;%=: ? 921; ?>%91 ; 0::
0: ?2%1: 6 9::00 02%:6 > >0?0? 92%:: 00 >::09 >0%;1 0? 1::0> 00%== ? 9=:01 ?0%=1 > ?:10= 02%6; > >::02 =;%:: ?: =::06 0:%9: 0 1610; 9>%;9 0: 1>:?: >=%1; 01 ?1:?0 >>%66 01 ?;:?? 1>%0? 0= 10:?9 1=%?9 02 1;:
?> ??%09 = 0::?1 ?0%01 1 >::
&Mqae~a8 Gle~blgar{ a~ `c% 0;;?+
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 597/846
Nr% Fëi~lr Pqa}anl Qrç`y
6'19
Z`r` ay~a zrlocag` i`ciqc`r cly ykbqkae~ay aeqeik`nly8
&`+ Ac glnacl na rabraykùe ckea`c gøc~kzca zloc`ikle`c%
&o+ Ac glnacl na rabraykùe ckea`c gøc~kzca na c` gqay~r` pqa ay~kg` `c glnacl zloc`ikle`c%
&i+ Zranaikr ac ~kagzl na yqgkeky~rl z`r` z`ray na }`clray na c`y }`rk`ocay na
rabraykùe# eøgarl na agz`pqay & v0+ { nky~`eik` & v?+# iq`enl v0 7 0 agz`pqa { c`
nky~`eik` ay kbq`c ` v? 7 ?1 zkay%
&n+ A}`cq`r ac glnacl na rabraykùe lo~aeknl qy`enl ~ëieki`y loha~k}ky~`y {
yqha~k}ky~`y# ilgl c`y nayirk~`y ae ay~a i`zç~qcl%
Nkyiq~kr ac r`xle`gkae~l pqa ya ykbqa ae c` }`ckn`ikùe yqoha~k}` na cly bràmkily%
Ylcqikùe8
&`+ Ac glnacl na rabraykùe gøc~kzca# z`r` ? }`rk`ocay kenazaenkae~ay ay8
±[ s v0#v?s 7 οl $ ο0 v0 $ ο? v? $ α
&o+ Ac ilrrayzlenkae~a glnacl na rabraykùe ckea`c gøc~kzca gqay~r`c ay8
[ 7 ol $ o0 V 0 $ o? V ? $ a
Nlena8
[ 7 ~kagzl na yqgkeky~rl
V 0 7 el na ae}`yay
V ? 7 nky~`eik` nac }afçiqcl
Ac glnacl na rabraykùe na c` gqay~r` ay8
[ 7 0%2> $ ?%26 & V 0+ $ :%:09 & V ?+
&i+ Z`r` zranaikr ac ~kagzl na yqgkeky~rl &[ + ae rac`ikùe ile ac eøgarl na ae}`yay#iq`enl V 0 7 0 { ile c` nky~`eik` nac }afçiqcl# iq`enl V ? 7 ?1 zkay ya lo~kaea
yqy~k~q{aenl cly }`clray ae c` aiq`ikùe na rabraykùe# ay naikr8
[ 7 0%2> $ ?%26&0+ $ :%:09&?1+ 7 >%61
&n+ Cly rayqc~`nly loha~k}ky~`y ay~`nçy~kily yle8 _ ? 7 ;6%0!4 _ ?`hqy~`n` 7 ;2%;!4 y 7
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 598/846
Nr% Fëi~lr Pqa}anl Qrç`y
6'1>
?%9?4 Z_AYY 7 01;%6;%
^@OC@ 6%??% ^`oc` gly~r`enl cly }`clray na ^ { na Z% &Ac`olr`ikùe zrlzk`+%
Zranki~lr Ilamkikae~a YA ilamkikae~a ^ Z
Iley~`e~a 0%2>9 0%011 0%10 :%0>1
El% na ae}`yay ?%2;: :%:;? 9:%:; :%:::
Nky~`eik` nac }afçiqcl :%:09 :%::9 >%99 :%:::
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
^@OC@ 6%?9% ^`oc` na `eàckyky na }`rk`ex`% &Ac`olr`ikùe zrlzk`+
Mqae~a na b%c% YY GY M z]`rk`ikùeNaoknl ` c` ? 1;6>%6 ?;;?%> 111%? :%:::_abraykùeArrlr ?? 006%= 1%>
^l~`c ?>
Z`r` c` }`ckn`ikùe yqoha~k}` nac glnacl na rabraykùe# `e`ckx`enl c`y bràmki`y
na cly rayknqly ay~`en`rkx`nly# naoae avky~kr# `zrlvkg`n`gae~a# ac gkygl eøgarl
na rayknqly zlyk~k}ly { eab`~k}ly% @nagày# ae c` zrqao` na elrg`ckn`n# ~lnly cly
zqe~ly naoae ay~`r nae~rl na c`y o`en`y na ilemk`ex`% Ac ay~qnk`e~a naoarà f`iar
cly nk`beùy~kily yqoha~k}ly z`r` ilgzcagae~`r c` ramraen`ikùe l ilemk`okckn`n nac
glnacl na rabraykùe%
Zrliankgkae~l na rabraykùe gøc~kzca qy`enl ac zrlbr`g` Gkek~`o
Zrliankgkae~l80% Krya `8 Y~`~ ↟ _abrayykle ↟ _abrayykle
?% Ae c` }ae~`e` na –_abrayykle‗ `z`raiae c`y ae~r`n`y na c` }`rk`oca
nazaenaenkae~a &[ + { na c`y }`rk`ocay kenazaenkae~ay V 0# V ?# ae yqy ilcqge`y
rayzai~k}`y rac`ikle`n`y ile ac zrlocag`
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 599/846
Nr% Fëi~lr Pqa}anl Qrç`y
6'11
9% Ae c` }ae~`ekcc` na –_ayzleya‗ &na ay~` }ae~`e` na _abrayykle+ ae~r`r c` }`rk`oca
nazaenkae~a {# ae c` }ae~`ekcc` na –Zranki~lry#‗ ae~r`r c`y }`rk`ocay kenazaenkae~ay
&pqa ya ilzk`rle ae c`y ilcqge`y nac zrlbr`g`+%>% Nao`hl na ay~` }ae~` na _abrayykle ay~àe c`y }ae~`ekcc`y na –Br`zfy# Lz~kley#
_ayqc~y { Y~lr`ba‗% Zlr ahagzcl yk ya naya` qy`r Br`zfy ya zqanae yacaiikle`r cly
rayknq`cay rabqc`ray l cly ay~`en`rkx`nly% Ae c` }ae~`ekcc` na –Lz~kle rayknq`c
zcl~y‗# zqe~a`r c`y bràmki`y na c`y iq`~rl lzikleay# z`r` ac `eàckyky yqoha~k}ky~`%
1% Ae c` }ae~`e` na –_abrayykle'Lz~kley‗ zqe~a`r c`y mqeikleay naya`n`y# }%b%#
}`rk`eia –Kemc`~kle m`i~lry# Nqroke'R`~yle y~`~ky~kiy# Z_AYY‗# a~i%
=% Ae c` }ae~`e` na –_abrayykle'_ayqc~y‗ zqe~a`r c`y mqeikleay naya`n`y na c`y
iq`~rl aecky~`n`y# }%b%# –Ke `nnk~kle na yapqae~k`c yqg…%%‗
Ahagzcl )0:% Ay~a ay qe ahagzcl nac ckorl @zzckan Y~`~ky~kiy8 @eàckyky lm ]`rk`eia
`en _abraykle na cly `q~lray Nqee { Ic`rd% Ay~` ay qe` ke}ay~kb`ikùe rac`ikle`n`
ile c` ~agzar`~qr`# ~lg`n` ilgl c` }`rk`oca na rayzqay~`# ae mqeikùe na }`rk`ocay
rabraylr`y ilgl c` `c~k~qn# clebk~qn { c`~k~qn% C` ~`oc` na `o`hl gqay~r` cly
rayqc~`nly% Qy`enl ac zrlbr`g` Gkek~`o8
&`+ Aeile~r`r ac glnacl na rabraykùe gày `zrlzk`nl
&o+ ]`ckn`r ac glnacl qy`enl ga~lnly ay~`nçy~kily# ay naikr# ay~kg`enl ac
ilamkikae~a na na~argke`ikùe gøc~kzca _ ?# _ ? `hqy~`n`# y# Z_AYY# ~`oc` na @EL]@#
{ bràmki`y yqoha~k}ky~`y# ilgl rayknqly }aryqy ùrnaeay# rayknqly }aryqy }`clray
`hqy~`nly { zrqao`y na elrg`ckn`n%
&i+ F`iar ilgae~`rkly `iari` na cly rayqc~`nly
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 600/846
Nr% Fëi~lr Pqa}anl Qrç`y
6'1=
^@OC@ 6%?>% ^`oc` gly~r`enl cly }`clray na c` ~agzar`~qr` ae lM &[ +# @c~k~qn ae
zkay & V 0+# Clebk~qn ae br`nly & V ?+ { C`~k~qn ae br`nly & V 9+%
^agzar`~qr` &[ + @c~k~qn & V 0+ Clebk~qn & V ?+ C`~k~qn & V 9+
11%2 0:69 00? 99
92%6 >12 6= 96
1=%> 90? 006 9>
10%: 9:1 ;: 9?
9>%1 1??0 0:1 >:
9>%: ?6>? 00= >>
9=%2 6:2 ;> >099%> >?=: 00? >0
9?%= 601 69 >:
>;%0 9;?: 0:= 9?
>=%= 0:1> 6> 9>
9=%9 >9;2 0?: 9;
06%? 69: ;9 >1
9=%2 >=1 ;: 9;
09%9 00=? ;? >2
9:%0 262 6? >0
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
Ylcqikùe8
&`+ Ya `yyqga qe glnacl na rabraykùe ckea`c&o+ C` q~kckn`n nac glnacl ya n` zlr cly }`clray na _ ?# y# Z_AYY# a~i% gly~r`nly
zlr c`y Mkbqr`y 6%0? &`+# &o+ { &i+%
_abrayykle @e`c{yky8 &[ + ^agzar`~qr` }aryqy & V 0+ @c~k~qn# & V ? + Clebk~qn & V 9+
^fa rabrayykle apq`~kle ky8 &[+ ^agzar`~qr` 7 ;;%? ' :%::096 &V0+ @c~k~qn $ :%?;; &V?+Clebk~qn ' ?%?; &V9+ C`~k~qn
Zranki~lr Ilam YA Ilam ^ Z ]KMIley~`e~ ;;%?> 0:%2; ;%?: :%:::&V0+ @c~k~qn ':%::0926: :%:::1;=6 '?%90 :%:>: 0%2&V?+ Clebk~qn :%?;622 :%:229= 9%6= :%::? 0%2&V9+ C`~k~qn '?%?;:: :%022; '0?%62 :%::: 0%:
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 601/846
Nr% Fëi~lr Pqa}anl Qrç`y
6'12
y 7 9%0?0==# _'Yp 7 ;>%=!# _'Yp&`nh+ 7 ;9%?!# Z_AYY 7 ?0>%611# _'Yp&zran+ 7 ;:%:6!
@e`c{yky lm ]`rk`eia ^`oca
Ylqria NM YY GY M M irç~ki` Z
_abrayykle 9 ?:>6%1> =6?%61 2:%:2 M%:149#0? 7 9%>; 333 :%::0_ayknq`c Arrlr 0? 00=%;> ;%2>^l~`c 01 ?0=1%>6
Nqroke'R`~yle y~`~ky~ki 7 0%1996>
&`+ &o+ &i+
Loyar} ~kleLrnar
Y
~ ` e n ` r n k x a n _ a y k n q ` c
0=010>090?000:;62=1>9?0
?
0
:
'0
'?
_ayknq cy ]aryqy ~fa Lrnar lm ~fa N ~&rayzleya ky &[+ agzar ~qr +
Mk~~an] cqa
Y
~ ` e n ` r n k x a n _ a y k n q ` c
=:1:>:9:?:
?
0
:
'0
'?
_ayknq cy ]aryqy ~fa Mk~~an ] cqay&rayzleya ky &[+ agzar ~qr +
Y~ en rnkxan_ayknq c
Z a r i a e ~
9?0:'0'?'9
;;
;1
;:
6:
2:
=:
1:
>:
9:
?:
0:
1
0
Elrg c Zrlo okck~{ Zcl~ lm ~fa _ayknq cy&rayzleya ky &[+ agzar ~qr +
Mkbqr` 6%0?% C` Mkbqr` &`+ gqay~r` cly rayknq`cay }y% ùrnaeay4 c` mkbqr` &o+ gqay~r`
cly rayknq`cay }y% cly }`clray `hqy~`nly { c` mkbqr` &i+ n` c` zrqao` na elrg`ckn`n%
&i+ Ae ileicqykùe# na `iqarnl ` cly }`clray nac ilamkikae~a na na~argke`ikle _ ? 7
;1%=! f`{ qe oqae `hqy~a nac glnacl% Ac }`clr na M na c` @EL]@ raif`x` c`
fkzù~ayky na kbq`cn`n na zrlgankly &na `c~k~qn# clebk~qn { c`~k~qn+# ile qe }`clr na
z gq{ ykbekmki`~k}l% C`y zrqao`y na ^ yle ykbekmki`e~ay# cl pqa yqbkara pqa el f`{
zrlocag`y na gqc~kilckea`ckn`n% @eàclb`gae~a# cly }`clray o`hly na ]KM yqbkarae
kenki`e pqa el f`{ zrlocag`y na gqc~kilckea`ckn`n% Ay~l# `qe`nl# ` cly ykbely na
c`y }`rk`ocay rabraylr`y na c` aiq`ikùe na rabraykùe# cly iq`cay yk ay~àe na `iqarnl `
qe` cùbki` ` zly~arklrk% Ac }`clr na c` mqeikùe na Nqroke'R`~yle Y~`~ky~ki l na
ilrrac`ikùe ae yarka kbq`c ` 0%1996> kenki` pqa el f`{ zrlocag`y na
`q~lilrrac`ikùe &`qepqa `pqç# ay~l ya zqana kbelr`r zlrpqa ac zrlocag` el
ke}lcqir` yarkay na ~kagzl+% Ae iq`e~l ` c` Mkbqr` 6%0? c` bràmki` &`+ gqay~r` cly
rayknq`cay }aryqy cly ùrnaeay# ae c` iq`c f`{ `ca`~lrkan`n na cly n`~ly%
@eàclb`gae~a# c` bràmki` na rayknq`cay }y% }`clray `hqy~`nly &o+ kenki` `c`~alrkn`n
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 602/846
Nr% Fëi~lr Pqa}anl Qrç`y
6'16
l kenazaenaeik` na cly n`~ly# yke zrlocag`y na fa~arlyian`y~kikn`n &arrlray na
}`rk`ex`y el iley~`e~ay+# a~i% Mke`cgae~a# c` bràmki` na c` zrqao` na elrg`ckn`n &i+
kenki` pqa cly n`~ly ay~àe elrg`cgae~a nky~rkoqknly &zlrpqa ~lnly cly zqe~ly ay~àenae~rl na c`y o`en`y na ilemk`ex`+# `qepqa ile yaybl zlyk~k}l%
El~`8 ¼Iraa qy~an pqë# a}ae~q`cgae~a# ac i`cae~`gkae~l bclo`c# naoknl ` c`y
agkykleay na b`yay na ke}are`narl# baear`nly zlr agkykleay }afkiqc`ray a
kenqy~rk`cay }` ` glnkmki`r c`y ~agzar`~qr`y pqa }`e ae mqeikùe na c` c`~k~qn<
Ahagzcl )00% Ay~a ay qe ahagzcl fkzù~a~kil gly~r`enl c` rac`ikùe ae~ra c`y
ileiae~r`ikleay na lxlel `r~kmkik`c# ` ek}ac nac yqacl &zzg+ { c`y ~agzar`~qr`y &lM+%
Ay~a aharikikl ay~à aei`gke`nl ` i`ciqc`r# g`eq`cgae~a# cly rayknq`cay { na f`iar
qe` bràmki` gly~r`enl cly rayknq`cay irqnly% Cly n`~ly ya n`e ae c` ~`oc` na `o`hl%
^@OC@ 6%?1 gly~r`enl cly n`~ly na ay~a zrlocag`% \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ Ileiae~r`ikleay na L9 &{+s 21 6: 6= ;> ;; 0:2
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ ^agzar`~qr` &lM+ &v+ s =1 20 2; 61 ;9 0::
F`iar cly ykbqkae~ay iàciqcly8
&`+ I`ciqc`r ac glnacl na rabraykùe { gankr yq `naiq`ikùe ay~kg`enl _ ?# _ ?&`hqy~%+#
y# Z_AYY { c` ay~`nçy~ki` Nqroke'R`~yle
&o+ F`iar qe` ~`oc` gly~r`enl ac }`clr na c` nay}k`ikùe ae~ra cly n`~ly { ac `hqy~a#
ay naikr# na cly rayknq`cay rabqc`ray l irqnly ak 7 {k ’ { k%
&i+ F`iar qe` bràmki` na L9 &{+ { ~agzar`~qr`y &v+ gly~r`enl cly rayknq`cay irqnly
Ylcqikùe8&`+ Qy`enl qe z`pqa~a na ilgzq~`nlr` n` c` aiq`ikùe na rabraykùe8
Ileiae~r`ikùe na lxlel & { + 7 01%> $ :%;:; ^agzar`~qr` &v+
y 7 0%0:0# _ ? 7 ;;%9!# _ ?&`hqy~%+ 7 ;;%?!# Z_AYY 7 ;%>?692# ay~`nçy~ki` Nqroke'R`~yle 7 9%99
&o+ C` ^@OC@ 6%?= gqay~r` cly }`clray `hqy~`nly & { k+# cly rayknq`cay { YYA%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 603/846
Nr% Fëi~lr Pqa}anl Qrç`y
6'1;
^@OC@ 6%?=% ^`oc` gly~r`enl cly n`~ly nac zrlocag`% \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
_ayknq`c irqnl Yqg` na cly iq`nr`nly nac arrlr k vk {k { k 7 01%>> $ :%;:; v ak 7 {k ' { k YYA 7 &{k ' { k+
?
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ 0 =1 21 { 0 7 2>%19 21 ’ 2>%19 7 :%>6 :%?9:>
? 20 6: { ? 7 2;%;6 6: ’ 2;%;6 7 :%:? :%:::>
9 2; 6= { 9 7 62%?1 6= ’ 62%?1 7 '0%?1 0%1=?1
> 61 ;> { > 7 ;?%20 ;> ’ ;?%20 7 0%9: 0%=;::
1 ;9 ;; { 1 7 ;;%;6 ;; ’ ;;%;6 7 ':%;6 :%;=:>
= 0:: 0:2 { = 7 0:=%9> 0:2 ’ 0:=%9> 7 :%== :%>91=
Ω&{k ' { k+? 7 >%62;9
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
&i+ C` Mkbqr` 6%09 na `o`hl gqay~r` c`y ileiae~r`ikleay na lxlel ae mqeikùe na c`y~agzar`~qr`y ile cly }`clray na cly rayknq`cay ak%
6%09% Mkbqr` gly~r`enl c` gankikle na i`n` qel na cly }`clray rayknq`cay ile c`
cçea` na rabraykùe% &Ac`olr`ikùe zrlzk`+%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 604/846
Nr% Fëi~lr Pqa}anl Qrç`y
6'=:
Aharikikly I`zç~qcl 6
6%0% Cly n`~ly na `o`hl gqay~r`e c`y agkykleay na ùvknly na ek~rùbael &ELv+
zrl}aekae~ay na i`cnar`y na zc e~`y acëi~rki`y%
^`oc` gly~r`enl cly n`~ly z`r` ac zrlocag`% &Ac`olr`ikùe zrlzk`+
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ GO~q,fr'm~? & V + s0:: 0?1 0?1 01: 01: ?:: ?:: ?1: ?1: 9:: 9:: 91: >:: >::
ELv &[ + s01: 0>: 06: ?0: 0;: 9?: ?6: >:: >9: >>: 9;: =:: =0: 12:
&`+ I`ciqc`r c` aiq`ikùe na rabraykùe na c` gqay~r` pqa ay~kg` ` c` }arn`nar` aiq`ikùe
zloc`ikle`c% &[ 7 '?>%? $ 0%1; V +
&o+ I`ciqc`r ac ilamkikae~a na ilrrac`ikùe _ ? { _ pqa ay~kg`e ` π% &_ ? 7 :%;1+
&i+ ¼Iqàc ay c` ay~kg`ikùe ayzar`n` na c` agkykùe na ELv iq`enl c` ~`y` na ckoar`ikùe
ay na ??1 GO~q,fr'm~?< &999%=2+
&n+ Qy`r ac zrlbr`g` na ilgzq~`nlr` na Gkek~`o { `e`ckx`r c`y bràmki`y na cly
rayknqly z`r` c` zrqao` na elrg`ckn`n { na cly rayknqly ae mqeikùe nac }`clr na V % &Ac
cai~lr cl naoarà f`iar+
6%?% Ay~a ay qe ahagzcl nac ckorl na Ke~rlnqiikùe `c @e`ckyky na _abraykùe Ckea`c na
Gleblgar{ a~ `c% &?::0+# nlena f`oc` na qe ahagzcl rac`ikle`nl ile c`y
ileiae~r`ikleay na lxlel na naoknl `c i`clr% @yç# N`}knyle &–Qzn`~a le Lxlea
^raeay ke I`ckmlrek`‛y Ylq~f ILY@^ O`yke‗# @kr `en R`y~a# >9# ??=# 0;;9+ ay~qnkl
c`y ileiae~r`ikleay na lxlel ae c` iqaei` `ëra` na c` ily~` yqr na I`ckmlrek`# nqr`e~acly `ðly 0;2= ` 0;;0% Ya iraa pqa c` i`e~kn`n na nç`y ae pqa c`y ileiae~r`ikleay na
lxlel mqarle g`{lray pqa :%?: zzg nazaena nac çenkia ga~aralcùbkil ay~`ikle`c#
pqa ay ac zrlgankl ay~`ikle`c na c` ~agzar`~qr` ile 61: gkcko`ray% C` ykbqkae~a
kemlrg`ikùe gqay~r` cly n`~ly%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 605/846
Nr% Fëi~lr Pqa}anl Qrç`y
6'=0
^`oc` gly~r`enl cly n`~ly nac zrlocag`% \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
@ðl El% na Nç`y & {+ Çenkia ga~alrlcùbkil
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ 0;2= ;0 0=%20;22 0:1 02%00;26 0:= 06%?0;2; 0:6 06%00;6: 66 02%?0;60 ;0 06%?0;6? 16 0=%:0;69 6? 02%?0;6> 60 06%:0;61 =1 02%?0;6= =0 0=%;0;62 >6 02%00;66 =0 06%?0;6; >9 02%90;;: 99 02%10;;0 9= 0=%=
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ Mqae~a8 Gle~blgar{ a~ `c% ?::0
&`+ Ay~kg`r c` aiq`ikùe na rabraykùe
&o+ ¼Pqë ~`e~` ilemk`okckn`n ya ca zqana n`r `c glnacl yacaiikle`nl< Qy`r aemlpqay
ay~`nçy~kily { bràmkily z`r` hqy~kmki`r ay~` zrabqe~`%
6%9% Ae qe ay~qnkl `brçilc`# z`r` }ar cly amai~ly na cly i`gokly ickgà~kily bclo`cay
rac`ikle`nl# ile cly z`~rleay zcq}k`cay `c~ar`nly naoknl `c i`cae~`gkae~l bclo`c# zlr
c`y agkykleay na IL?# ya ay~qnkù c` zraikzk~`ikùe zcq}k`c `eq`c { ac raenkgkae~l na c`
ilyaif` na br`gçea`y% C` ~`oc` na `o`hl n` cly n`~ly%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 606/846
Nr% Fëi~lr Pqa}anl Qrç`y
6'=?
^`oc` gly~r`enl cly n`~ly nac zrlocag`% &Ac`olr`ikùe zrlzk`+
Zraikzk~`ikùe zcq}k`c _aenkgkae~l na c` ilyaif`
ae zqcb`n`y & V + ae ckor`y zlr `ira &[ +2%0? 0:92
=9%1> 96:
>2%96 >0=
>1%;? >?2
6%=6 =0;
1:%6= 966>>%6= 9?0
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
]ar iqàc glnacl na rabraykùe aei`h` gahlr ae cly n`~ly# `c hqxb`r zlr c`y ay~`nçy~ki`y
{ zlr cly `eàckyky bràmkily# ay naikr# qy`enl qe` `zrlvkg`ikùe ckea`c# qe` clb`rç~gki`
{ qe` `zrlvkg`ikùe na mqeikùe na zl~aeik` na c` mlrg` na Ce &[ + 7 Ce&`+ $ o&Ce V +%
&`+ Qy`r qe` `zrlvkg`ikùe ckea`c ilgl [ 7 ` $ oV {# `nagày# i`ciqc`r ac ilamkikae~a
na na~argke`ikùe _ ?% &[ 7 66:%> ’ ;%= & V+# _ ? 7 :%=;;+
&o+ Qy`r qe` `zrlvkg`ikùe clb`rç~gki` ilgl [ 7 ` $ o Ce & V+ { `nagày# i`ciqc`r ac
}`clr na _ ?% &[ 7 0990%:6 ’ 112%:9 Cb V +
&i+ Qy`enl qe` `zrlvkg`ikùe na mqeikùe na zl~aeik` na c` mlrg` na Ce &[ + 7 Ce &`+
$ o &Ce V + {# `nagày# i`ciqc`r _ ? &_ ? 7 :%6?;+
6%>% Ae qe ay~qnkl na pqçgki` `e`cç~ki`# ae c` ~`oc` na `o`hl ya n` c` rac`ikùe ae~ra c`~agzar`~qr` { c` glc`rkn`n &ae glcay zlr ck~rl+ na qe` yqy~`eik`% Z`r` ay~l f`iar cly
ykbqkae~ay8
&`+ Ay~kg`r ac glnacl na rabraykùe gày `zrlzk`nl o`y`nl ae `eàckyky ay~`nçy~kily na
_ ?# _ ?`hqy~`n`# Z_AYY# y# { Iz { ae `eàckyky bràmkily yqoha~k}ly na cly }`clray
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 607/846
Nr% Fëi~lr Pqa}anl Qrç`y
6'=9
rayknq`cay%
^`oc` gly~r`enl c` kemlrg`ikùe rapqarkn`% \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ ^agzar`~qr` lI s >%9 >%1 >%6 1%1 1%2 1%; =%> =%2 2%1 2%;
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ Glc`rkn`n s 0?%0 0?%1 0?%; 09%: 09%0 0>%: 0>%? 0>%6 01%: 01%1
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
6%1% Ac `cqgkekl ay ac ~ariar acagae~l gày `oqen`e~a pqa liqrra ae gkear`cay# rli`y
{ o`rrly% Ac `cqgkekl ya zqana `e`ckx`r ile ac gë~lnl na `oylrikùe `~ùgki`
ayzai~rlgë~rki` &gë~lnl @+# ac iq`c ay~à avae~l na ke~armaraeik`y ilgl mcqlrqrly {
mlym`~ly% Ac `cqgkekl ~`gokëe ya zqana `e`ckx`r zlr gankl nac gë~lnl na
i`clrkga~rç` na ik`eqrl na Arklirlgl _ &gë~lnl O+# ac iq`c ay gày ykgzca pqa ac
`e~arklr% C` ~`oc` na `o`hl gqay~r` cly rayqc~`nly na cly `eàckyky &ae gb,C+ na cly nly
gë~lnly qy`nly% F`iar cly ykbqkae~ay iàciqcly qy`enl ac zrlbr`g` na ilgzq~`nlr`
na Gkek~`o l Y@Y%
&`+ I`ciqc`r a ke~arzra~`r ac ilamkikae~a na na~argke`ikùe _ ? { ac ilamkikae~a na
ilrrac`ikùe _% &_ ?
7 :%;;??# _ 7 :%;;=0+^`oc` gly~r`enl cly n`~ly nac ahagzcl% &Ac`olr`ikùe zrlzk`+
Gë~lnl @ s 1 = = 6 0: 0: 00 00
Gë~lnl O s 6 ; ; 00 09 09 0> 0>
6%=% Ac oarkckl &Oa+ { yqy ilgzqay~ly yle av~rag`n`gae~a }aeaelyly { i`z`iay na
i`qy`r c` gqar~a ae ileiae~r`ikleay `c~`y% C` kef`c`ikùe nac Oa i`qy` qe` yark`
`maiikùe cc`g`n` oarkcklyky% Ac oarkckl ~`gokëe zqana i`qy`r narg`~k~ky# ilehqe~k}k~ky#
eaqgleç` `bqn` { oarkcklyky zqcgle`r irùeki`% Ay~a acagae~l pqçgkil ya qy` ae cly
ra`i~lray `~ùgkily# `}kleay# ilfa~ay { ae ilgoqy~kocay z`r` gçykcay% F`{ nly
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 608/846
Nr% Fëi~lr Pqa}anl Qrç`y
6'=>
gë~lnly z`r` ac `eàckyky &ae ±b,C+ nac oarkckl# ay naikr# ac gë~lnl ayzai~rlgë~rkil na
`oylrikùe `~ùgki` &gë~lnl 0+ { ac gë~lnl `cqgkele &gë~lnl ?+% Cly rayqc~`nly na
cly `eàckyky na cly nly gë~lnly ya n`e ae c` ~`oc` na `o`hl% F`iar cly ykbqkae~ayiàciqcly8
&`+ F`iar qe ay~qnkl ay~`nçy~kil loha~k}ky~`# ay naikr# ay~kg`enl cly }`clray na _ ?#
_ ?`hqy~`n`# Z_AYY# y { ~`oc`y na @EL]@% Ilgzcagae~`r ac ay~qnkl f`ikaenl `eàckyky
yqoha~k}ky~`y%
^`oc` gly~r`enl cly rayqc~`nly na cly gë~lnly 0 { ? z`r` c` gankikùe nac oarkckl%
&Ac`olr`ikùe zrlzk`+
Gë~lnl 0 s : 9 > 1 ; 0? 01 02 ?: ?:
Gë~lnl ? s 0 2 00 0; ?> 90 90 91 >0 >0
6%2% Ae ke}ay~kb`ikleay na ~lvkilclbç` avky~ae ay~qnkly pqa f`e nagly~r`nl pqa c`
zrlo`okckn`n na pqë# qe mqg`nlr na >: `ðly na an`n# pqkae f` yknl mqg`nlr cly
øc~kgly 0: `ðly ile~r`kb` ac iàeiar zqcgle`r ae cly zrùvkgly ?: `ðly ay `c~`
&`yqgkaenl pqa ile~keøa mqg`enl `c gkygl rk~gl+% Ay~` rac`ikùe }` ae mqeikùe nac
eøgarl zrlgankl na ikb`rrkccly pqa mqg`% @yqgkr qe glnacl na rabraykùe ckea`c% C`
~`oc` na `o`hl zrayae~` cly n`~ly na ay~` ke}ay~kb`ikùe na ~lvkilclbç`%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 609/846
Nr% Fëi~lr Pqa}anl Qrç`y
6'=1
^`oc` gly~r`enl cly n`~ly nac zrlocag`% &Ac`olr`ikùe zrlzk`+
Eøgarl na ikb`rrkccly Zrlo`okckn`n na iàeiar
mqg`nly zlr nç` zqcgle`r 1 %0::
0: %009?: %??19: %9::>: %>1:1: %1>:=: %2::6: %6=:
F`iar cly ykbqkae~ay iàciqcly8
&`+ Knae~kmki`r c` }`rk`oca nazaenkae~a { c` }`rk`oca kenazaenkae~a%
&o+ Nayirkokr c` aiq`ikùe na rabraykùe pqa gahlr aei`ha ae cly n`~ly% &[ 7 :%:;60 ’
:%::::?& V + $ :%:::9&V ?++
&i+ I`ciqc`r _ ?# _ ?`hqy~`n`# y# { Z_AYY% &_ ? 7 :%;;=# _ ?`hqy~`n` 7 :%;;1 y 7 :%:0;#
Z_AYY 7 :%::96+
&n+ @e`ckx`r a ke~arzra~`r cly ilgzleae~ay na c` ~`oc` na @EL]@ ilgl Mi`ci%# Mirç~ki` { ac }`clr na z%
&a+ Nkyiq~kr c` rac`ikùe avky~ae~a ae~ra _ ?# y# Z_AYY# Mi`ci%# { ac }`clr na z%
&m+ ]`ckn`r ac glnacl na rabraykùe yqoha~k}`gae~a# ay naikr# `e`ckx`enl cly bràmkily
na cly rayknqly ay~`en`rkx`nly%
6%6% Ya ra`ckxù qe ay~qnkl na pqçgki` `gokae~`c { ya rabky~r`rle c`y i`e~kn`nay na
iclrqrl na ylnkl &E`Ic+# ac iq`c# iq`enl ya nkylc}kù ae 0:: br`gly na `bq` nay~kc`n`#
` nkmarae~ay ~agzar`~qr`y &lI+ nkl cly ykbqkae~ay rayqc~`nly8
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 610/846
Nr% Fëi~lr Pqa}anl Qrç`y
6'==
^`oc` gly~r`enl cly n`~ly nac zrlocag`% &Ac`olr`ikùe zrlzk`+ \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ ^agzar`~qr` & V + E`Ic nkyqac~l ae br`gly na `bq` &[ +
: 6 = 601 0? 0: 0>9: ?1 ?0 ?>>1 90 99 ?6=: >> 9; >?21 >6 10 >>
I`ciqc`r cly ykbqkae~ay aeqeik`nly8
&`+ Br`mki`r cly n`~ly%
&o+ Aeile~r`r c` cçea` na rabraykùe { zlearc` ae c` bràmki`%
&i+ Ay~kg`r c` i`e~kn`n na E`Ic pqa ya nkylc}arà ` qe` ~agzar`~qr` na 9:: D%
&n+ @ y`okaen`y na pqa# ` gankn` pqa `qgae~` c` ~agzar`~qr`# c` nkylcqikùe na c`y
yqy~`eik`y# ilgl c`y y`cay na ylnkl# `qgae~` zrlzlrikle`c `c keiragae~l na c`
~agzar`~qr`# ae~leiay# ykaenl `yç# }arkmki`r na pqa f`{ qe` ilrrac`ikùe i`yk zarmai~`
ae~ra `go`y }`rk`ocay%&a+ F`iar qe` zrqao` na fkzù~ayky z`r` ac ilamkikae~a na ilrrac`ikùe gqay~r`c _# z`r`
}arkmki`r pqa yk avky~a qe` `ylik`ikùe ckea`c ykbekmki`e~a ae~ra c`y nly }`rk`ocay%
Yqbaraeik`8 qy`r c` ay~`nçy~ki` na ~ na Ay~qnk`e~a n`n` `o`hl8
?0?
∟∟7
e
_~
_ile η 7 e ' ? br`nly na ckoar~`n%
&m+ ^aùrki`gae~a# c` nkylcqikùe na gqif`y y`cay }` ae mqeikùe nkrai~` ` c`
~agzar`~qr` {# ae ~alrç`# ac }`clr nac ilamkikae~a na na~argke`ikùe# _ ? naoarç` na yar
na 0%:% Ykaenl `yç# aecky~`r ? m`i~lray &ae ac c`olr`~lrkl na pqçgki`+ pqa zqnkar`e
`mai~`r c` nkylcqikùe na c`y y`cay { na el n`r qe }`clr gaelr pqa 0%:%
6%;% Ae qe ay~qnkl na ga~alrlclbç` ae~ra c` i`e~kn`n na ccq}k` { c` raglikùe na
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 611/846
Nr% Fëi~lr Pqa}anl Qrç`y
6'=2
ile~`gke`e~ay `~glymërkily# ya nkl c` ykbqkae~a kemlrg`ikùe8
^`oc` gly~r`enl cly n`~ly% &Ac`olr`ikùe zrlzk`+
Zraikzk~`ikùe & V + _aglikùe na z`r~çiqc`y &[ +&:%:0 ig%,nç`+ &±b,g9+
>%9 0?=>%1 0?01%; 00=1%= 006=%0 00>1%? 0069%6 09?
?%0 0>02%1 0:6
&`+ I`ciqc`r c` raglikùe na ile~`gke`e~ay &[ + iq`enl ac }`clr na c` zraikzk~`ikùe
zcq}k c ay na V 7 6%:% &0:?%>>+
&o+ ]`ckn`r ac glnacl na rabraykùe loha~k}` { yqoha~k}`gae~a%
6%0:% Ae qe ay~qnkl z`r` a}`cq`r c` i`z`ikn`n na cly yky~ag`y na mcqhl mraà~kil
&a~c`eny+# qy`nly z`r` c` nabr`n`ikùe na c` g`~ark` lrbàeki` na c`y `bq`y rayknq`cay
ya qyl ac z`ràga~rl na c` nag`en` oklpqçgki` na lvçbael &NOL+ { }`rkly l~rly
ilgzleae~ay pqçgkily% Ay~a ay~qnkl nkl ilgl rayqc~`nl cly ykbqkae~ay n`~ly% Ay~ly
rayqc~`nly ay~àe rac`ikle`nly ile c` i`rb` na g`y` na NOL &ae Db%,fai~àra`,nç`+# c`
iq`c ya qyù ilgl c` }`rk`oca kenazaenkae~a & V + {# c` nabr`n`ikùe na c` ileiae~r`ikùe
na g`y` i`rolely` na NOL1 &ae Db%,f`,nç`+# c` iq`c ya qyl ilgl c` }`rk`oca
nazaenkae~a &[ +% &Mqae~a na kemlrg`ikùe ay –Yqrm`ia Mcllr Ra~c`eny8 @ Zarmlrg`eiaA}`cq`~kle‗% R`~ar Ae}krle% _ay%# 0;;1# zz%?>>'?>2+%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 612/846
Nr% Fëi~lr Pqa}anl Qrç`y
6'=6
^`oc` gly~r`enl cly n`~ly nac zrlocag`%
& V + s 9 6 0: 00 09 0= ?2 9: 91 92 96 >> 0:9 0>?
&[ + s > 2 6 6 0: 00 0= ?= ?0 ; 90 9: 21 ;:
&Mqae~a8 R`~ar Ae}krle% _ay%# 0;;1+
I`ciqc`r cly ykbqkae~ay aeqeik`nly8
&`+ Br`mki`r cly n`~ly
&o+ Ay~`ocaiar ac glnacl na rabraykùe gày `zrlzk`nl z`r` ay~a zrlocag`% F`iar cly
gkygly iàciqcly pqa ac zrlocag` `e~arklr%
&i+ ]`ckn`r ac glnacl na rabraykùe yacaiikle`nl# loha~k}ky~`gae~a# qy`enl cly
ykbqkae~ay irk~arkly l nk`beùy~kily8
&0+ Iàciqcl nac ilamkikae~a na na~argke`ikùe _ ?
&?+ Iàciqcl nac ilamkikae~a na na~argke`ikùe `hqy~`nl# _ ?`hqy~`nl
&9+ Ac ilamkikae~a na ilrrac`ikùe _ &>+ C` ay~`nçy~ki` Z_AYY
&1+ Ac arrlr ay~àen`r na cl ay~kg`nl# y &Q~kc z`r` gankr c` q~kckn`n nac glnacl% Ya
yacaiikle` ac glnacl pqa ~aeb` ac }`clr na y gày zapqaðl+
&n+ A}`cq`r ac glnacl i`enkn`~l ` ~r`}ëy na cly ykbqkae~ay irk~arkly bràmkily8
&0+ Zrqao` na elrg`ckn`n
&?+ _ayknq`cay ae mqeikùe na cly lrnaeay
&9+ _ayknq`cay ae mqeikùe na cly }`clray `hqy~`nly
&a+ Qe` }ax pqa ya f`{` yacaiikle`nl ac glnacl gày `zrlzk`nl# i`ciqc`r c`
raglikùe nac NOL nayzqëy na pqa ac `bq` rayknq`c ya nabr`nù ae ac a~c`en iq`enl
c` i`rb` mqa na 1: Db%,f`,nç`%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 613/846
Nr% Fëi~lr Pqa}anl Qrç`y
6'=;
El~`8 Cly yky~ag`y na mcqhl mraà~kil &àra`y z`e~`ely`y+ ya qy`e ilgl yky~ag y na
~r`~`gkae~l e`~qr`c# zlrpqa ~kaeae c` i`z`ikn`n na nabr`n`r c`y ileiae~r`ikleay
i`rolely`y na NOL `i~q`enl ilgl ayzaika na c`bqe`y na lvkn`ikùe% Ae Gke`~k~càe{ Il`~x`il`cily# ]ar`irqx# ya qy`e ay~ly ~kzly na ~r`~`gkae~ly e`~qr`cay%
Ylcqikùe8
Na `iqarnl ` c` ~`oc` na `rrko` ya ca zkna `c cai~lr naiknkr# iqàc glnacl ay yqzarklr%
6%00% Ac ~av~l na N`ekac# R%R%# H`gay I% ^arracc Oqykeayy Y~`~ky~kiy &0;6;+# z% ?12
gaeikle`e ac zrlocag` nac ae}aeae`gkae~l nac b`e`nl }`iqel# zlr cly keyai~kikn`y
ac iq`c ay qe zrlocag` gq{ yarkl# zlrpqa cly zay~kikn`y ~kaeae c` m`iqc~`n na
`iqgqc`rya ae cly ~ahknly na cly `ekg`cay {# na `fç ya z`y`e ` `pqacc`y zaryle`y pqa
cly ileyqgae% @yç# ae `ņly raikae~ay# cly `gokae~`cky~`y ya f`e zraliqz`nl gqifl
zlr cly amai~ly# ae ac gankl `gokae~a# naoknl `c qyl kenkyirkgke`nl na keyai~kikn`y%
Ay }arn`n pqa cly keyai~kikn`y g`~`e cly keyai~ly# zarl ~`gokëe g`~`e ~lnl cl nagày%
Na ay~` g`ear`# cly keyai~kikn`y ile~`gke`e cly mrq~ly na c`y zc`e~`y# ` cly `ekg`cay {
~`gokëe ` c` bae~a% Cly ke}ay~kb`nlray Glqe~ { Lafga~ ay~qnk`rle cly amai~ly na cly
keyai~kikn`y ae c`y l}ah`y rac`ikle`n` ile c` `i~k}kn`n aexkgà~ki` ae ac iaraorl%
@nagày# na l~rly `eàckyky ay~`nçy~kily# ay~ly ikae~çmkily nark}`rle qe` cçea` na
rabraykùe pqa nayirkoa c`y rac`ikùe ae~ra c` `i~k}kn`n aexkgà~ki` ae ac iaraorl na c`y
l}ah`y &[ + { ac ~kagzl# ae flr`y# nayzqëy na pqa c`y l}ah`y f`y yknl avzqay~`y ` cly
keyai~kikn`y & V +% C` mqeikùe na c` cçea` na rabraykùe ay~kg`n` zlr ay~ly ikae~çmkily ya
n` `o`hl%
[ 7 ?2%9? $ 0%9= V O`y`enl ac irk~arkl ae ay~` aiq`ikùe# ay~kg`r cl ykbqkae~a8
&`+ Yk nayzqëy na pqa f`e z`y`nl 9: flr`y# iq`enl c`y l}ah`y f`e yknl avzqay~`y `
cly keyai~kikn`y# ¼Iqàc yarç` ac }`clr na c` `i~k}kn`n aexkgà~ki`< &=6%0?+
&o+ Yk ac ilamkikae~a na ilrrac`ikùe gqay~r`c ya n` ilgl _ 7 :%6= {# ac ilamkikae~a na
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 614/846
Nr% Fëi~lr Pqa}anl Qrç`y
6'2:
na~argke`ikùe ay _ ? 7 :%2> &ac pqa gkna c` mqarx` na c` rac`ikùe ckea`c ae~ra V a [ # ay
naikr# ac ! na `ylik`ikùe ae~ra c`y nly }`rk`ocay+# ae~leiay# f`iar qe` zrqao` na
fkzù~ayky ile Fl8π 7 :# ile~r` F08π ≪ : &pqa ay cl gkygl pqa naikr pqa el f`{`ylik`ikùe ckea`c ae~ra V a [ +% @yqgkr pqa ac ~`g`ðl na c` gqay~r` ay na e 7 0= { ac
ek}ac ykbekmki`e~a na ΰ 7 :%:1%
Z`r` ay~l# yabqkr c`y ykbqkae~ay yqbaraeik`y8
Qy`r c` nky~rkoqikùe na ~ ile η 7 e ’ ? br`nly na ckoar~`n { qy`r c`y rabkleay irç~ki`y
n`n`y zlr ~T0'ΰ,?4 ηW%
6%0?% Ae ay~qnkly na pqçgki`# c` zraykùe na qe b`y pqa ilrrayzlena ` }`rkly
}lcøgaeay &na `iqarnl ` c` ca{ na cly b`yay na Ol{ca+ ya n` ae c` ~`oc` na `o`hl%
@yqgkr pqa ac }lcqgae nac b`y ay & V + { c` zraykùe ay &[ +% F`iar cly ykbqkae~ay
iàciqcly8
&`+ F`iar qe` bràmki` ile cly n`~ly%
&o+ Ay~kg`r c` cçea` na rabraykùe na c` gqay~r`%
&i+ Ay~kg`r ac ilamkikae~a na na~argke`ikùe _ ? { ac ilamkikae~a na ilrrac`ikùe _%
Ke~arzra~`r cly rayqc~`nly%
&n+ Zranaikr c` zraykùe nac b`y# iq`enl ac }lcqgae ay %::0 g9
&a+ Zranaikr c` zraykùe nac b`y# ae ckor`y zlr zqcb`n` iq`nr`n` &coy,ke?+ {# ae
`~gùymar`y &`~g+# iq`enl ac }lcqgae nac b`y ay na :%:1?6 iq`r~ly &%:1 C+%
&m+ Ae ~alrç`# naoknl ` pqa c` rac`ikùe ae~ra ac }lcqgae nac b`y { c` zraykùe ay
ke}ary`gae~a zrlzlrikle`c# ac ilamkikae~a na ilrrac`ikùe naoarç` yar na _ 7 '0%:% Yke
ago`rbl# yk _ nkmkrkar` nac }`clr na '0%:# aecky~`r 9 m`i~lray pqa zqnkar`e ke~ar}aekr z`r` avzcki`r ay~` yk~q`ikùe%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 615/846
Nr% Fëi~lr Pqa}anl Qrç`y
6'20
^`oc` gly~r`enl cly }lcøgaeay { c`y zraykleay nac b`y% &Ac`olr`ikùe zrlzk`+
]lcqgae ae ig9 s 1:%: =:%: 2:%: ;:%: 0::%:
Zraykùe ae Db%,ig? s =>%2 10%9 >:%1 ?1%; 2%6
Yqbaraeik`y8 Ya n`e cly ykbqkae~ay m`i~lray na ile}arykùe8 0 `~g 7 0>%2 coy,ke? 7 2=:
~lrr 7 0%:==6 db,ig?4 0 ig? 7 :%0= ke?4 0%:1=2 iq`r~ly 7 0 C4 0 zqcb`n` iq`nr`n` 7
=%?1 ig?4 0 g9 7 0::: C% 7 0:= ig9%
6%09% Ya ilcaiikleù qe` gqay~r` na 99 i`yly na qe` nayi`rb` na `bq`y rayknq`cay
gqekikz`cay% Ay~` gqay~r` ya `e`ckxù z`r` c` nag`en` oklpqçgki` na lvçbael na 1
nç`y &NOL1+# ae ckor`y zlr nç`# { c` nag`en` pqçgki` na lvçbael# NPL &ae ckor`y zlr
nç`+% C` ~`oc` na o`hl gqay~r` c kemlrg`ikùe rapqarkn`%
^`oc` gly~r`enl c`y gankikleay na NOL1 { NPL% &Ac`olr`ikùe zrlzk`+Nag`en` pqçgki` na lvçbael Nag`en` oklpqçgki` na lvçbael
&coy,nç`+ &coy,nç`+
>;> >6= ?0= ?:?>>> 11= ?:: ?>:1?6 =:: ?96 ?6:9;= >?6 0=> 06>19? >>: ?9: 0;>9:6 ?;0 00= 09>91: >;: 01: ?01>1= 1>1 0;: ?>=>>: 16? 0;: ?;?1>> 9=6 ?>6 022
9:; 96= 0?: 0;9196 >:: ??= 0=1>6: 9>2 ?:: 0=:1:: ?26 ??? 0?19;= 9:> 02= 092
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 616/846
Nr% Fëi~lr Pqa}anl Qrç`y
6'2?
F`iar cl ykbqkae~a8
&`+ ]ar yq avky~a qe` ilrrac`ikùe ykbekmki`e~a qy`enl cly }`clray _ nac NOL1 { ac
NPL% &_ 7 :%;=22# _ ?
7 :%;9=:+&o+ Ke~arzra~`r ac }`clr nac ilamkikae~a na ilrrac`ikùe _ { ac ilamkikae~a na
na~argke`ikùe _ ?% Qy`r ac zrlbr`g` Gkek~`o l AVIAC z`r` f`iar ac iàciqcl zanknl%
El~`% C` nag`en` oklpqçgki` na lvçbael na 1 nç`y &NOL1+ gkna c` ileiae~r`ikùe#
ae gb,C l ae ckor`y zlr nç` na c` g`~ark` i`rolely` nac `bq` rayknq`c% Na faifl ac
NOL gkna c` mr`iikùe oklnabr`n`oca nac nrae`ha# l nac `bq` rayknq`c kenqy~rk`c l
nlgëy~ki`# ae ~ërgkely nac i`rolel% Qyq`cgae~a# c`y qekn`nay yle ae gb,C% Yke
ago`rbl# ay~l ya naoa ` pqa# `e~arklrgae~a# ya qy`o` kenkyirkgke`n`gae~a c`y
qekn`nay na zzg { gb,C% Nayzqëy# ya }kl pqa# ile cly rayknqly ~ùvkily# c` br`}an`n
ayzaiçmki` ar` nkmarae~a ` c` na cly rayknqly i`rolelyly% Zlr ay~` r`xùe ay gahlr qy`r
c`y qekn`nay na gb,C% Zlr l~r` z`r~a# c` zrqao` nac NOL ay na 1 nç`y# z`r` a}k~`r c`
ek~rkmki`ikùe% Ae i`gokl# c` zrqao` na c` nag`en` pqçgki` na lvçbael gkna cly
ilgzqay~ly lrbàekily oklnabr`n`ocay { cly ilgzqay~ly lrbàekily ~ùvkily% Ay~lpqkara naikr pqa# c` nag`en` pqçgki` na lvçbael &NPL+ lvkn` c` i`e~kn`n na
g`~ark`cay ~l~`cay lvkn`ocay zrayae~ay ae ac `bq` rayknq`c { }`rç` ile c` ilgzlykikùe
nac `bq`# c` ~agzar`~qr`# ac zarklnl na ile~`i~l { l~rly m`i~lray gày%
6%0>% Ileyknar`r cly n`~ly na `o`hl rac`ikle`nly ile ac zayl nac }afçiqcl { ac
raenkgkae~l na b`ylcke`% Ac zayl nac `q~l ya n` ae ~leac`n`y {# ac raenkgkae~l nacilgoqy~koca ya n` ae gkcc`y b`cùe% Cly n`~ly ya n`e `o`hl%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 617/846
Nr% Fëi~lr Pqa}anl Qrç`y
6'29
^`oc` gly~r`enl cly n`~ly nac cly glnacly }afkiqc`ray ae mqeikùe nac zayl ae~leac`n`y &V+# { na gkcc`y zlr b`cùe &[+%
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
Glnacl }afkiqc`r Zayl &~leac`n`y+ Gkcc`y,b`cùe
Oqkid Ay~`~a R`ble >%9= 0=%;Mlrn Ilqe~r{ Ypqkra R`ble >%:1 01%1Ifa}{ G`ckoq R`ble 9%=0 0;%?Ifr{ycar Ca O`rle R`ble 9%;> 06%1^l{l~` Ilrle` ?%1= ?2%1N`~yqe 10: ?%9: ?2%?Nlnba Lgek ?%?9 9:%;@qnk 1::: ?%69 ?:%9]lc}l ;; BCA 9%0> 02%:
Y``o ;; BCA ?%6: ?0%=Zaqbl~ =;> YC 9%>0 0=%?Oqkid Iae~qr{ Yzaik`c 9%96 ?:%=Gariqr{ Xazf{r 9%:2 ?:%6Nlnba @yzae 9%=? 06%=@GI Ileilrn N,C 9%>0 06%0Ifa}{ I`zrkia Ic`yyki 9%6> 02%:Mlrn C^N 9%29 02%=Gariqr{ Br`en G`rpqky 9%;= 0=%1Mlrn Gqy~`eb ?%1; ?=%1G`xn` BCI 0%;6 9>%0Nlnba Ilc~ 0%;? 91%0]R Yikrliil 0%;; 90%1Flen` @iilrn CV ?%0> ?;%1Oqkid Yd{c`rd ?%=2 ?6%>Ifa}{ Ik~`~kle ?%=: ?6%6Lcnyglokca Lgab` ?%2: ?=%6Zc{glq~f Flrkxle ?%?: 9>%?N`~yqe ?0: ?%:? 90%6]R N`yfar ?%0; 9:%1N`~yqe 60: ?%6? ??%:OGR 9?0: ?%=: ?0%1]R _`ook~ 0%;9 90%;
&Mqae~a8 Zrlo`okckn`n { Ay~`nky~ki` @zcki`n`y ` c` Kebaekark % Gle~blgar{ a~ `c%
0;;=+
F`iar cly ykbqkae~ay iàciqcly qy`enl ac zrlbr`g` Gkek~`o%
&`+ Ay~kg`r c` cçea` na rabraykùe ae~ra c`y }`rk`ocay zayl nac }afçiqcl { ac
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 618/846
Nr% Fëi~lr Pqa}anl Qrç`y
6'2>
raenkgkae~l na b`ylcke`%
&o+ Ay~kg`r ac ilamkikae~a na ilrrac`ikùe na c` gqay~r` _ &cc`g`nl ~`gokëe
ilamkikae~a na ilrrac`ikùe na zrlnqi~l'glgae~l na Za`ryle+ { ac ilamkikae~a nana~argke`ikùe gqay~r`c _ ?%
&i+ F`iar qe` bràmki` pqa }`{` ae mqeikùe na [ { V # ~r`x`rc` ae c` bràmki` { ~`gokëe
~r`x`r c` cçea` flrkxle~`c qy`enl ac }`clr nac zrlgankl na [ %
&n+ F`iar qe` ~`oc` na @EL]@%
&a+ Ay~kg`r cly ke~ar}`cly na ilemk`ex` z`r` ΰ { ο c`y zrlo`okckn`nay
ilrrayzlenkae~ay z`r` i`n` qel na ëy~ly%
&m+ ¼Pqë l~rly m`i~lray ~aenrç`e pqa ileyknar`rya# z`r` pqa ac glnacl na rabraykùe
mqar` gày ilemk`oca<
6%01% Cly ga~`cay zay`nly ilgl ac Fb# Ir# Zo# a~i%# zqanae ke~armarkr ile ac
~r`~`gkae~l oklcùbkil ae c`y zc`e~`y gqekikz`cay na `bq`y rayknq`cay nlgëy~ki`y% Ae
ay~a ay~qnkl ya fkikarle gankikleay gaeyq`cay ae qe` zc`e~` glnacl na ~r`~`gkae~l
na c`y ileiae~r`ikleay na irlgl# Ir# ae gb,C# ~`e~l ae ac amcqae~a ilgl ae c`
ae~r`n`% Cly rayqc~`nly na c`y ileiae~r`ikleay na Ir ya n`e ae c` ~`oc` na `o`hl%
^`oc` gly~r`enl cly n`~ly% &Ac`olr`ikùe zrlzk`+Ae~r`n` & V + s ?1: ?;: ?2: 0:: 9:: >0: 00: 09: 00::&±b,C+Amcqae~a &[ + s 0; 0: 02 00 2: =: 06 9: 06:&±b,C+
F`iar cly ykbqkae~ay iàciqcly8
&`+ F`iar qe nk`br`g` na nkyzarykùe ae z`zac `rk~gë~kil%&o+ F`iar qe nk`br`g` ayz`riknl ae z`zac yagkclb`rk~gl { clb`rk~gl ilgzca~l
&~r`eymlrg`ikùe na cly ahay+%
&i+ I`ciqc`r cly glnacly na rabraykùe z`r` c`y z`r~ay &`+ { &o+%
&[ 7 ';%:= $ :%02 V 4 [ 7 '0%;= $ :%;2 V +
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 619/846
Nr% Fëi~lr Pqa}anl Qrç`y
6'21
&n+ Ay~kg`r [ iq`enl V 7 91: ae keikyly &`+ { &o+%
&a+ I`ciqc`r ac ilamkikae~a na ilrrac`ikùe z`r` &`+ { &o+% &_ ? 7 :%;>?1# _ 7 :%2>=;+
&m+ Ilgae~`r ylora cl `zrlzk`nl na [ { na _ ae i`n` i`yl%6%0=% Ae qe ay~qnkl na gkirloklclbç` `gokae~`c rac`ikle`nl ile ac iqc~k}l na qe`
gqay~r` na `bq` ya n`e cly ykbqkae~ay n`~ly%
^`oc` gly~r`enl cly n`~ly% &Ac`olr`ikùe zrlzk`+^kagzl ae nç`y nayna s 9 = ; 0? 01 06c` keliqc`ikùe & V +
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ El% na o`i~ark`y &[ + s 001#::: 0>#2:: ?9#;:: 91#=:: 12#;:: 6=#>::
F`iar cly ykbqkae~ay iàciqcly8
&`+ ^r`x`r qe` iqr}` Ce [ k }aryqy V k z`r` }ar pqë ~`e okae ya zqana `hqy~`r qe` iqr}`
avzleaeik`c ` cly n`~ly%
&o+ ^r`x`r qe` iqr}` [ k }aryqy V k z`r` }ar pqa ~`gokëe ya zqana `hqy~`r qe` cçea`
rai~` ` cly n`~ly%
&i+ Zlr ke~arzlc`ikùe# qy`enl `go`y bràmki`y ay~kg`r ac eøgarl na o`i~ark`y nayzqëy
na ?: nç`y% Iq`e~kmki`r c`y nkmaraeik`y ae `goly i`yly%
6%02% Ae ac ckorl na H% C Na}lra# Zrlo`okckn`n { Ay~`nçy~ki` z`r` Kebaekarç` {
Ikaeik`y ya n` qe` ke}ay~kb`ikùe rac`ikle`n` ile c` ~agzar`~qr` &lI+ { c`
zrlmqenkn`n na c` eka}a iqgqc`n` ae ac yqacl% Z`r` ay~l ya c` ~`oc` na `o`hl8
^`oc` gly~r`enl cly n`~ly nac zrlocag`% \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ ^agzar`~qr` &lM+ s '=? '>0 '9= ?= '99 '1= '1: '==
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ Zrlmqenkn`n na c` s ?0 09 0? 9 = ?? 0> 0;i`z` na eka}a
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ &Mqae~a8 Na}lra ?::0+ %
&`+ Knae~kmki`r c` }`rk`oca nazaenkae~a &[ + { c` }`rk`oca kenazaenkae~a & V +%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 620/846
Nr% Fëi~lr Pqa}anl Qrç`y
6'2=
&o+ Ay~kg`r qe glnacl na rabraykùe ckea`c% &[ 7 1%20 ’ :%?:?& V +# _ ? 7 :%2>0# y 7
9%21;# Z_AYY 7 >:;%:?# M 7 02%6# z 7 :%::=+
&i+ Ay~kg`r qe glnacl iq`nrà~kil% &[ 7 9%9 ’ :%:;>9& V + $ :%::?;& V ?
+# y 7 :%:0;# _ ?
7 :%;;=# Z_AYY 7 :%::96# M 7 0>%;6# z 7 %::6+
&n+ Ay~kg`r qe glnacl iøokil% &[ 7 ;%;= ’ :%09;& V + $ :%:06;& V ?+ $ :%:::??& V
9+#
_ ? 7 :%;0># y 7 ?%=1=# Z_AYY 7 6::2%21# M 7 0>%0># z 7 :%0>+
&a+ Na `iqarnl ` cly rayqc~`nly ay~`nçy~kily# ¼Iqàc na cly ~ray glnacly ay yqzarklr<
6%0;% C` ~`y` na mcqhl ae g9,gke ae qe gqay~ra`nlr na `c~l }lcqgae z`r` gankr c`
i`ckn`n nac `kra# ay naikr# z`r` z`r~çiqc`y `~glymërki`y# nazaena na c` i`çn` na
zraykùe# ae zqcb`n`y na `bq`# ` ~r`}ëy nac mkc~rl nac gqay~ra`nlr% Ykaenl `yç#
yqzùeb`ya pqa ya ilcaiikleù qe` gqay~r` na 01 }`clray na i`çn` na zraykùe { c` ~`y`
na mcqhl nac `kra ` ~r`}ëy nac mkc~rl nac yaeylr% Cly n`~ly ya n`e ae c` ~`oc` na `o`hl%
^`oc` gly~r`enl cly n`~ly z`r` ay~a zrlocag`% &Ac`olr`ikùe zrlzk`+^`y` na mcqhl nac `kra ile I`çn` na zraykùe nayzqëy nac`y z`r~çiqc`y &g9,gke+ `cbøe ~kagzl &zqcb`n`y na `bq`+
?%:: 1%:
0%;; =%:0%66 2%:0%2= 2%60%=6 6%>0%12 ;%=0%>= ;%;0%>: 0:%=0%9; 00%20%?: 0>%:
0%01 01%;0%:2 0;%:0%:0 ?>%:0%:: ?6%::%;1 91%:
&`+ I`ciqc`r ac glnacl na rabraykùe gqay~r`c pqa ay~kga ` c` }arn`nar` cçea`
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 621/846
Nr% Fëi~lr Pqa}anl Qrç`y
6'22
zloc`ikle c% Z`r` ay~l# knae~kmki`r# zrkgar`gae~a# c` }`rk`oca nazaenkae~a { c`
}`rk`oca rabraylr`% &[ 7 0%;1 ’ :%:9=> & V ++
&o+ ]`ckn`r ac glnacl na rabraykùe ay~kg`nl ae &`+ qy`enl aemlpqay yqoha~k}ly# aynaikr# ` ~r`}ëy na bràmki`y ile rayknqly ay~`en`rkx`nly }aryqy }`clray na i`çn` na
zraykùe% ^`gokëe f`iar l~r` bràmki` na rayknqly ay~`en`rkx`nly }aryqy cly raebcleay%
F`iar l~r` bràmki` gày ile cly }`clray rayknq`cay }aryqy cly }`clray na x z`r` c`
zrqao` na elrg ckn`n%
&i+ Ilgzcagae~`r c` }`ckn`ikùe nac glnacl na rabraykùe qy`enl gë~lnly ay~`nçy~kily
loha~k}ky~`y% Z`r` ay~l# ay~kg`r ac ilamkikae~a na na~argke`ikùe _ ?# ac arrlr ay~àen`r
na cl ay~kg`nl &y n`nl zlr ac Gkek~`o+ { Z_AYY% Qy`r ac zrlbr`g` Gkek~`o z`r`
ay~ly iàciqcly% &_ ? 7 2=%:!# y 7 :%06=;# Z_AYY 7 :%2>:1+
6%?0% Ya f`ia qe ay~qnkl ylora c` ileiae~r`ikùe na i`ngkl `~glymërkil# ae zzg# {k {
yq rac`ikùe ile V k 7 c` `c~qr` na cly gqay~ra`nlray { V ? 7 nky~`eik` na c` mqae~a
agkylr`% C` ~`oc` na `o`hl gqay~r` cly n`~ly% F`iar cly ykbqkae~ay iàciqcly8
&`+ @hqy~`r ac glnacl na rabraykùe pqa zqan` `hqy~`r ` cly n`~ly nac zrlocag` na c`
ileiae~r`ikùe na In% &[ 7 91:%;; ’ 0%?2 V 0 ’ :%01> V ?+
&o+ ]`ckn`r ac glnacl qy`enl aemlpqay na nk`beùy~kil na ay~`nçy~ki` na kemaraeik`
&loha~k}ky~`y+ { na `eàckyky bràmkil &yqoha~k}ky~`y+%
&i+ Qy`r ac glnacl na rabraykùe ckea`c gøc~kzca z`r` zranaikr ac c` ileiae~r`ikùe na
i`ngkl# iq`enl c` `c~qr` nac gqay~ra`nlr ay na V 0 7 ?1 ga~rly { c` nky~`eik` na c`
mqae~a agkylr`# ay V ? 7 610 ga~rly% &066%? zzg na In+
C` ~`oc` na `o`hl gqay~r` cly n`~ly rapqarknly zlr ay~a zrlocag`%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 622/846
Nr% Fëi~lr Pqa}anl Qrç`y
6'26
^`oc` gly~r`enl cly n`~ly% &Ac`olr`ikùe zrlzk`+
{ &ileiae~r`ikùe na In+ s 0;9 ?9: 02? ;0 009 0?1
V 0 &@c~qr` nac gqay~ra`nlr+ s 0%= 01%1 ??%: >9%: 99%: >:%:
V ? &Nky~`eik`+ s 610 60= 0:16 0?:0 0912 0001
6%??% Ac ~av~l @zzckan Y~`~ky~kiy8 @e`c{yky na ]`rk eia `en _abrayykle na cly
ke}ay~kb`nlray Lck}a Nqee { ]krbkek` Ic`rd# nkyiq~ae qe ahagzcl z`r` zranaikr ac
raenkgkae~l na c` ilyaif` na iao`n`# ae mqeikùe na c` zraikzk~`ikùe zcq}k`c V 0 { c`
~agzar`~qr V ?% Z`r` ay~l# f`iar cly ykbqkae~ay iàciqcly8&`+ Aecky~`r ac glnacl na rabraykùe ckea`c gøc~kzca pqa gahlr `hqy~a ` cly n`~ly%
&o+ Ay~kg`r c` aiq`ikùe na cly iq`nr`nly gçekgly pqa `hqy~a ac raenkgkae~l na ~rkbl
&[ + ` c` zraikzk~`ikùe zcq}k`c & V 0+ { c` ~agzar`~qr` & V ?+%
&i+ Zrlo`r c` fkzù~ayky na Fl8 ο? 7 : ile ΰ 7 :%:1%
&n+ Ay~kg`r ac ilamkikae~a na ilrrac`ikùe z`rik`c π?{%0 { zrlo`r Fl8π ?{%0 7 :
&a+ ]`ckn`r ac glnacl na rabraykùe nark}`nl z`r` }ar# pqë ~`e~` ilemk`okckn`n ya ca
zqana `irank~ r% F`iar ay~l# ` ~r`}ëy na hqkikly loha~k}ky~`y# ilgl cly nk`beùy~kily _ ?#
_ ?`hqy~`n`# _ ? zranaikn # y# Z_AYY { Iz% Ilgzcagae~`r c` a}`cq`ikùe nac glnacl qy`enl
~ëieki`y yqoha~k}ky~`y# ilgl cly `eàckyky na cly bràmkily na rayknq`cay ay~`en`rkx`nly {
ay~qnk`e~kx`nly# zrqao` na elrg`ckn`n# a~i%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 623/846
Nr% Fëi~lr Pqa}anl Qrç`y
6'2;
^`oc` gly~r`enl cly raenkgkae~ly na iao`n` ilgl }`rk`oca nazaenkae~a na c`
zraikzk~ ikùe zcq}k c { c` ~agzar`~qr`%
_aenkgkae~l na iao`n` & {k+ Zraikzk~`ikùe & v0+ ^agzar`~qr` & v?+&m`eab`y,`ira+ &zqcb`n`y+ &lM+
?0%: >1 1>%0?:%: >2 =0%=?0%: 99 1:%6?>%: 9; 1?%0?:%: 9: 1:%?0?%1 ?6 12%00;%: >0 11%2
?9%: >> 12%=?9%: 90 1:%00;%: ?; 96%:?0%: 9> 1=%?0?%: ?2 10%1?0%: >? 1>%0?2%: 91 >=%202%1 >9 =:%6?=%: 9; 1=%;00%: 90 =:%9
?>%: >? 1>%=?=%: >9 19%106%1 >2 =>%:01%1 ?1 >1%2
&Mqae~a8 Nqee a~ c% 0;2>% @zzckan Y~`~ky~kiy8 @e`c{yky na ]`rk`eia `en _abrayykle+
6%?9% Ac ~av~l @zzckan Y~ ~ky~kiy8 @e`c{yky na ]`rk eia `en _abrayykle na Nqee a~ `c%
0;2>+ f`ia qe ay~qnkl gënkil rac`ikle`nl ile ac i`gokl na c` faglbcloke` na c`
y`ebra na lzar`ikleay na c` bcàenqc` ~krlknay# ac iq`c ay~à rac`ikle`nl ile c` nqr`ikùena c` lzar`ikùe pqkrørbki` { ac i`gokl ae ac zlriae~`ha na c` faglbcloke` na c`
y`ebra% Cly n`~ly ya n`e ae c` ~`oc` na `o`hl%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 624/846
Nr% Fëi~lr Pqa}anl Qrç`y
6'6:
^`oc` gly~r`enl cly n`~ly% \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ El% na z`ikae~a s 0 ? 9 > 1 = 2 6
Zërnkn` na y`ebra & v0+ s 0:1 6: 6= 00? 0:; 0:: ;= 0?:Nqr`ikùe ae gkeq~ly & v?+ s 1:9 >;: >20 1:1 >6? >;: 109 >=>! na i`gokl nafaglbcloke` & {0+ s '0%2 '>%= ';%6 '0%0 '>%0 '9%9 :%> '?%;
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ Mqae~a8 Nqee a~ `c% 0;2>
F`iar cly ykbqkae~ay iàciqcly8
&`+ Aeile~r`r ac glnacl na rabraykùe gøc~kzca z`r` zranaikr ac zlriae~`ha nac i`gokl
na c` faglbcloke` & {+ ae mqeikùe na c`y }`rk`ocay kenazaenkae~ay# ay naikr# nqr`ikùena c` lzar`ikùe & v0+ { na c` zërnkn` na y`ebra & v?+% & { 7 '6>%::? $ :%0?; v?
$ :%096v?+
&o+ Zranaikr ac ! nac i`gokl ae c` faglbcloke`# iq`enl c` nqr`ikùe ae gkeq~ly na c`
lzar`ikùe ay na 6: { c` zërnkn` na y`ebra ay na 91: gc% &?1%96!+
&i+ ¼Nkyiq~kr# pqë ~`e~` mknackn`n ya ca zqana l~lrb`r `c glnacl na rabraykùe gøc~kzca
lo~aeknl ae ay~a zrlocag`<
&n+ I`ciqc`r ac ilamkikae~a na na~argke`ikùe gøc~kzca% &_ ? 7 :%609+
&a+ I`ciqc`r ac ilamkikae~a z`rik`c na ilrrac`ikùe# ay naikr# ae~ra { { v0# ile v?
iley~`e~a% &:%2;9+
6%?>% Ac ckorl na H`{ C% Na}lra ke~k~qc`nl Zrlo`okckn`n { Ay~`nçy~ki z`r` Kebaekarç`
{ Ikaeik`y nkyiq~a ac nkyaðl amkikae~a na ikar~ly keikear`nlray na nayzarnkikly
gqekikz`cay# cly iq`cay rapqkarae na kemlrg`ikùe `iari` nac ile~aeknl aearbë~kil na
cly nayzarnkikly% @ilrnagae~a# cly `q~lray nac `r~çiqcl –Glnacckeb ~fa Aearb{
Ile~ae~ lm Gqekikz`c Ylckn R`y~a Qykeb Gqc~kzca _abrayykle ^aifekpqay &H% lm ~fa
@kr `en R`y~a Gbg~% @yyli%# 0;;=# zz% =1:'=1=+ zrlzlrikle`rle cly ykbqkae~ay n`~ly
`iari` na [ 7 ile~aeknl aearbë~kil &Di`c,Db%+# ae mqeikùe na rabraylray ! na
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 625/846
Nr% Fëi~lr Pqa}anl Qrç`y
6'60
zcày~kily ae zayl# ! na z`zac ae zayl# ! na o`yqr ae zayl { ! na fqgan`n na zayl%
^`oc` gly~r`enl cly n`~ly na ay~a zrlocag`% \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ Loy% ! Zcày~kil &v0+ ! Z`zac &v?+ ! O`yqr` &v9+ ! Fqgan`n Ile~aeknl aearbë~kil &{+ \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ 0 06%=; 01%=1 >1%:0 16%?0 ;>2? 0;%>9 ?9%10 9;%=; >9%=0 0>:29 0;%?> ?>%?9 >9%0= >=%=9 0>1?> ??%=> ??%?: 91%2= >1%61 01191 0=%1> ?9%1= >0%?: 11%0> ;6;= ?0%>> ?9%=1 91%1= >?%?> 00=?2 0;%19 ?>%>1 >:%06 >2%?: 0>==6 ?9%;2 0;%9; >>%00 >9%6? 0=1=; ?0%>1 ?9%6> 91%>0 10%:0 0?1>0: ?:%9> ?=%1: 9>%?0 >;%:= 099=00 02%:9 ?9%>= 9?%>1 19%?9 0:;20? ?0%:9 ?=%;; 96%0; 10%26 0?==09 ?:%>; 0;%62 >0%91 >=%=; 0>:00> ?:%>1 ?9%:0 >9%1; 19%12 0??901 06%60 ??%=? >?%?: 1?%;6 0?0=0= 06%?6 ?0%62 >0%1: >2%>> 099>02 ?0%>0 ?:%>2 >0%?: 1>%=6 001106 ?1%00 ??%1; 92%:? >6%2> 0>190; ?0%:> ?=%?2 96%== 19%?? 0?26?: 02%;; ?6%?? >>%06 19%02 0019?0 06%29 ?;%9; 9>%22 10%:= 0??1
?? 06%>; ?=%16 92%11 1:%== 0?92?9 ??%:6 ?>%66 92%:2 1:%2? 09?2?> 0>%?6 ?=%?2 91%6: >6%?> 0??;?1 02%2> ?9%=0 92%9= >;%;? 0?:1?= ?:%1> ?=%16 91%>: 19%16 0??0?2 06%?1 09%22 10%9? 10%96 0096?6 0;%:0 ?1%=? 9;%1> 1:%09 0?;1?; ?0%?1 ?:%=9 >:%2? >6%=2 09;?9: ?0%=? ??%20 9=%?? >6%0; 092? \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
Mqae~a8 H`{ C% Na}lra% Zrlo`okck~{ `en Y~`~ky~kiy mlr Aebkeaarkeb `en ~fa Yikaeiay
&?:::+
&`+ Lo~aear ac glnacl na rabraykùe { }`ckn`rcl `ilrnagae~a# ay naikr# qy`enl
nk`beùy~kily yqoha~k}ly { nayzqëy ilgzcagae~`r c` ~`ra` qy`enl nk`beùy~kily
loha~k}ly%
6%?1% ^rake~` gqay~r`y nac amcqae~a na qe` zc`e~` na ~r`~`gkae~l ya `e`ckx`rle z`r` c`
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 626/846
Nr% Fëi~lr Pqa}anl Qrç`y
6'6?
gankikùe nac NOL1 { c` nag`en` oklpqçgki` na lvçbael &NPL+% Cly n`~ly ya
gqay~r`e ae c` ~`oc` na `o`hl% F`iar cl ykbqkae~a8
&`+ I`ciqc`r ac zrlgankl# y { ac arrlr ay~àen`r nac NOL { nac NPL% & V NOL 7 >>:%=#y 7 ;9%06# arrlr ay~àen`r 7 02%:04 V NPL 7 0;>%># y 7 >1%9# arrlr ay~àen`r 7 6%?2+
&o+ Br`mki`r cly n`~ly ae z`zac na zrlo`okckn`n%
&i+ Na~argke`r ac NOL1 { ac NPL pqa ya avianarà ac 1:! na c`y }aiay% &Ac NOL1
avianarà 0;1 coy,nç` ac 1:! na c`y }aiay% Ac NPL avianarà >>: coy,nç` ac 1:! nac
~kagzl+
&n+ Na~argke`r ac NOL1 { ac NPL pqa ya avianarà ac ;:! nac ~kagzl%^`oc` gly~r`enl c`y ileiae~r`ikleay na NPL { na NOL1% &Ac`olr`ikùe zrlzk`+
NPL s >;> >;> 1?6 9;= 19? 9:6 91: >1= >>: 1>>&coy,nç`+ s 90: 196 >6: 1:: 9;= >6= 11= =:: >?6 >>:
s ?;0 >;: 1>= 16? 9=6 96= >:: 9>2 ?26 9:>NOL1 s ?0= ?:: ?96 0=> ?9: 00= 01: 0;: 0;: ?>6
&coy,nç`+ s 0?: ??= ?:: ??? 02= ?:? ?>: ?6: 06> 0;>s 09> ?01 ?>= ?;? 022 0;9 0=1 0=: 0?1 092
6%?=% Ac nkrai~lr na c` lmkike` na zaryle`c na qe` mkrg` iley~rqi~lr` naya` y`oar yk
c` nay~rax`# ae na~argke`nl ~kzl na ~r`o`hl# nae~rl na c` agzray`# zqana yar
zrlely~ki`n` qy`enl ilgl zrlely~ki`nlray c`y }`rk`ocay an`n { avzarkaeik` na cly
agzca`nly% C` ~`oc` na `o`hl n` c` kemlrg`ikùe na qe` gqay~r` `ca`~lrk` na 01
agzca`nly% &@n`z~`ikùe nac ckorl Oqykeayy Y~`~ky~kiy na N`ekac a~ `c% 0;6;# z% 122+%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 627/846
Nr% Fëi~lr Pqa}anl Qrç`y
6'69
^`oc` gly~r`enl cly n`~ly nac zrlocag`% \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ Ek}ac na & {+ Avzarkaeik` & v0+ An`n & v?+
Nay~rax`
01 : ?001 : 06?0 : ???6 0 ?>9: 0 ?191 0 ?1>: 0 ?=
91 ? 9>9: ? ?1>1 ? 961: 9 >>=: 9 10>1 > 9;=: > 1>1: 1 11
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ Mqae~a8 N`ekac a~ `c% 0;6;% Oqykeayy Y~`~ky~kiy
F`iar cly ykbqkae~ay iàciqcly8
&`+ Aeile~r`r c` aiq`ikùe na rabraykùe na cly iq`nr`nly gçekgly%
&o+ Ilgzq~`r _ ?{%0?%
&i+ Zrlo`r Fl8ο0 7 : { Fl8ο? 7 :% Nah`r pqa ΰ 7 :%:1 { i`ciqc`r ac }`clr na z z`r`
i`n` zrqao`%
&n+ Ilgzq~`r ac ;1! na ke~ar}`cl na ilemk`ex` z`r` ο?%
&a+ Nah`r pqa v0 7 ? { v? 7 ?1 { i`ciqc`r {%&m+ Aeile~r`r ac ke~ar}`cl na ;1! z`r` {%
6%?2% C` i`z`ikn`n na cly ailclbky~`y z`r` knae~kmki`r rabkleay na gàvkg` rkpqax`
na c`y zc`e~`y zlnrç` ~aear qe kgz`i~l ylora c` zrayar}`ikùe na c` nk}arykn`n
baeë~ki`% Ay~l ay qel na cly loha~k}ly na cly ailclbky~`y pqkaeay ay~àe zraliqz`nly
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 628/846
Nr% Fëi~lr Pqa}anl Qrç`y
6'6>
zlr ac gankl `gokae~a% Ac `r~çiqcl –Zranki~kle lm _`rk~kay mrlg F`ok~`~ ]`rk`ocay8
Il`y~`c Zc`ke Zc`e~y le El}` Yil~k`e C`dayflray‗ &Ailclb{# 0;;?# zz% 061?'061;+
qyù qe` gqay~r` na 92 c`bly { ya lo~q}l c` aiq`ikùe na rabraykùe na `o`hl% Ay~a zrlocag` ya y`iù nac ckorl nac ke}ay~kb`nlr H% C% Na}lra &?::0+%
{ 7 9%6; $ %:99 v0 $ %:?> v? $ %:?9 v9 ' %::6: v> ' %09 v1 ' %2?v=
Nlena8
{ 7 rkpqax` na ayzaikay na zc`e~`y
v0 7 àra` na c` iqaei`
v? 7 `eifl na c` zc`{`
v9 7 g`c nrae`nl &!+
v> 7 ilclr nac `bq`
v1 7 ! na `rae`
v= 7 `ci`ckekn`n%
Ac ay~qnkl razlr~ù qe ilamkikae~a na na~argke`ikùe gøc~kzca na _ ? 7 :%69% _a`ckx`r
qe` zrqao` na c` q~kckn`n nac glnacl na rabraykùe% Yqbaraeik`8 qy`r c` mqeikùe
ay~`nçy~ki`8 M 7 T_ ?,d W , T&0 ' _ ?+,&e ' &d $ 0++W# ile rabkùe na raif`xl z`r` qe`
zrqao` na ek}ac na M ≡ Mΰ#d#e'&d$0+# nlena d ay ac eøgarl na zrlely~ki`nlray qy`nly%
Qy`r c` ~`oc` na c` nky~rkoqikùe M% ]`clr`r c` q~kckn`n nac glnacl na `iqarnl `c
}`clr na c` zrlo`okckn`n z%
6%?6% Ay~a ay aharikikl pqa ke}lcqir` c` yacaiikùe na qe glnacl na rabraykùe ile ;
}`rk`ocay kenazaenkae~ay l zranki~lr`y# ay naikr# v0 # v? # v9 # v> # v1 # v= # v2 # v6 { v;%
O`y`enl ac irk~arkl ae cly nk`beùy~kily _ ?
# GYA { Iz &irk~arkl na G`ccl+# naikr iqàc glnacl na rabraykùe ay ac gày `zrlzk`nl% Ay~l ay# yacaiikle`enl cly gahlray
yqoilehqe~ly zlykocay% Cly n`~ly ya n`e `o`hl%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 629/846
Nr% Fëi~lr Pqa}anl Qrç`y
6'61
^`oc` gly~r`enl cly n`~ly% &Ac`olr`ikùe zrlzk`+ \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
Yqoilehqe~l na zranki~lray
0 ? 9 > 1 = 2 6 ;Irk~arkly \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
_ ?d s %91> %>19 %100 %11: %1=? %12: %12? %121 %121
GYAd s ??;1 0;>6 02>? 0=:2 01== 01>0 0191 019: 019?
Izd s 90> 029 6;%= 91%2 0;%; 00%: ;%> 6%? 0:%: \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
6%?;% Ae qe ay~qnkl na c`olr`~lrkl z`r` }ar c` rac`ikùe ae~ra cly yùcknly
yqyzaenknly { c`y ileiae~r`ikleay na NOL ya y`iù qe` gqay~r` ile cly n`~ly pqa
ya gqay~r`e ae c` ~`oc` na `o`hl%
^`oc` gly~r`enl cly n`~ly% &Ac`olr`ikùe zrlzk`+
Yùcknly yqyzaenknlys 06 2 0> 90 ?0 1 00 0= ?= ?;NOL1 s 11 02 9= 61 =? 06 99 >0 =9 62
&`+ F`iar qe` bràmki` pqa }`{` ae mqeikùe na c` }`rk`oca nazaenkae~a { na c`
}`rk`oca kenazaenkae~a%
&o+ Lo~aear ac glnacl na c` aiq`ikùe na rabraykùe { ~r`x`rc` ae c` bràmki`% &Yùcknly
yqyzaenknly [ 7 :%9? $ :%91? & V +
&i+ ]`ckn`r ac glnacl na rabraykùe loha~k}`gae~a# i`ciqc`enl ac ilamkikae~a na
na~argke`rle _ ?# y { Z_AYY% &_ ? 7 :%;=?# y 7 :%;12# y 7 0%61# Z_AYY 7 >?%96+
&n+ F`iar qe` ~`oc` na @EL]@ pqa keicq{` ac }`clr na M { z% &Ilgzca~`r c` ~`oc`
na `o`hl%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 630/846
Nr% Fëi~lr Pqa}anl Qrç`y
6'6=
^`oc` na @EL]@% &Ac`olr`ikùe zrlzk`+ \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ Mqae~a b%c% YY GY Mi`ci% M~`o% ]`clr z
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ Naoknl ` c` 0 =;>%0=rabraykùeArrlr ?2%>> 9%>9^l~`c ; 2?0%=:
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
&a+ F`iar qe nk`beùy~kil bràmkil z`r` }`ckn`r c` `q~ae~kikn`n nac glnacl na
rabraykùe yacaiikle`nl% Yqbaraeik`8 qy`r ac zrlbr`g` Gkek~`o%6%9:% ^rake~` i`yly nac amcqae~a na qe` zc`e~` na ~r`~`gkae~l ya `e`ckx`rle z`r` ac
NOL { ac NPL% Cly n`~ly ya gqay~r`e ae c` ~`oc` na `o`hl% F`iar cly ykbqkae~ay
iàciqcly8
^`oc` gly~r`enl cly n`~ly nac zrlocag`% &Ac`olr`ikùe zrlzk`+
NPL &coy,Nç`+s >;> >>> 1?6 9;= 19? 9:6 91: >1= >>: 1>> 90: 196
s >6: 1:: 9;= >6= 11= =:: >?6 >>: ?;0 >;: 1>= 16?s 9=6 96= >:: 9>2 ?26 9:>
NOL &coy,Nç` s ?0= ?:: ?96 0=> ?9: 00= 01: 0;: 0;: ?>6 0?: ??=
s ?:: ??? 02= ?:? ?>: ?6: 06> 0;> 09> ?01 ?>= ?;?
s 022 0;9 0=1 0=: 0?1 092
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
&`+ Na~argke`r _ ? { _% &_ ? 7 :%;91:# _ 7 :%;=2+
&o+ Br`mki`r cly n`~ly ae z`zac na zrlo`okckn`n { na~argke`r cl ykbqkae~a8
&0+ Na~argke`r cly }`clray na NOL { ac NPL pqa avianaràe ac 1:! { ac ;:! na
c`y }aiay% &0;1 coy,Nç` { >>: coy,Nç`+
&?+ Na~argke`r cly }`clray na NOL { nac NPL pqa ya clbr`ràe ac ;:! na c`y }aiay%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 631/846
Nr% Fëi~lr Pqa}anl Qrç`y
6'62
&9+ Na~argke`r cly }`clray na NOL { nac NPL pqa zqan`e yar avianknly ac 0:! na
c`y }aiay% &?=: coy,Nç` { 16: coy,Nç`+&>+ I`ciqc`r ac zrlgankl { c` nay}k`ikùe ay~àen`r nac NOL { nac NPL%
&i+ A}`cq`r ac glnacl na rabraykùe `zcki`enl aemlpqay yqoha~k}ky~`y# ay naikr# ile
bràmki`y na cly }`clray rayknq`cay ae mqeikùe na }`clray `hqy~`nly &z`r` c` zrqao` na
kenazaenaeik`+# zrqao`y na elrg`ckn`n# a~i%
6%90% Ay~a ay qe zrlocag` `n`z~`nl nac ckorl Ke~rlnqiikùe `c @eàckyky na _abraykùe
Ckea`c na cly `q~lray Gle~blgar{# Zaid { ]kekeb &?::0+% Ay~a zrl{ai~l ay~à
rac`ikle`nl ile qe ay~qnkl na aearbç` ylc`r ae ac ^aielcùbkil na Balrbk`# Ay~`nly
Qeknly% Ac zrl{ai~l ke}lcqir` n`~ly na zrqao`y na aearbç` ~ërgki` ile qe`
}`rk`oca nazaenkae~a &{+# pqa rac`ikle` `c mcqhl ~l~`c na i`clr &D`~~y+ { ikeil
}`rk`ocay kenazaenkae~ay pqa ay~àe rac`ikle`n`y ile c` keylc`ikùe &`~~y,g?+# c`
zlykikùe nac mlil ae nkraiikùe nac ay~a &ae zqcb`n`y+# c` zlykikùe nac mlil ae
nkraiikùe nac yqr &ae zqcb`n`y+# c` zlykikùe nac mlil ae nkraiikùe elr~a &ae
zqcb`n`y+ { c` flr` nac nç`% Z`r` ay~l# ay~kg`r cly ykbqkae~ay aeqeik`nly8
&`+ Zrlo`r ac glnacl na rabraykùe pqa gahlr `hqy~a ` cly n`~ly%
&o+ A}`cq`r ac glnacl na rabraykùe yacaiikle`nl# ay naikr# ` ~r`}ëy na irk~arkly
ay~`nçy~kily { ilgzcagae~`r c` naikykùe qy`enl bràmkily yqoha~k}ky~`y%
C` ~`oc` na `o`hl gqay~r` c` kemlrg`ikùe rapqarkn` z`r` ylcqikùe ~lnly cly
aeqeik`nly rapqarknly zlr ay~a zrlocag`%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 632/846
Nr% Fëi~lr Pqa}anl Qrç`y
6'66
^`oc` gly~r`enl cly n`~ly na c`y zrqao`y na aearbç` ylc`r ~ërgki`% \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
{ v0 v? v9 v> v1 \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
?20%6 269%91 99%19 >:%11 0=%== 09%?:
?=>%: 2>6%>1 9=%1: 9:%0; 0=%>= 0>%00
?96%6 =6>%>1 9>%== 92%90 02%== 01%=6
?9:%2 6?2%6: 99%09 9?%1? 02%1: 0:%19
?10%= 6=:%>1 91%21 99%20 0=%>: 00%::
?12%; 621%01 9>%>= 9>%0> 0=%?6 00%90
?=9%; ;:;%>1 9>%=: 9>%61 0=%:= 00%;=
?==%0 ;:1%11 91%96 91%6; 01%;9 0?%16
??;%0 21=%:: 91%61 99%19 0=%=: 0:%==
?9;%9 2=;%91 91%=6 99%2; 0=%>0 0:%61?16%: 2;9%1: 91%91 9>%2? 0=%02 00%>0
?12%= 6:0%=1 91%:> 91%?? 01%;? 00%;0
?=2%9 60;%=1 9>%:2 9=%1: 0=%:> 0?%61
?=2%: 6:6%11 9?%?: 92%=: 0=%0; 09%16
?1;%= 22>%;1 9>%9? 92%6; 0=%=? 0>%?0
?>:%> 200%61 90%:6 92%20 02%92 01%1=
??2%? =;>%61 91%29 92%:: 06%0? 01%69
0;=%: =96%0: 9>%00 9=%2= 06%19 0=%>0
?26%2 22>%11 9>%2; 9>%=? 01%1> 09%0:
?2?%9 212%;: 91%22 91%>: 01%2: 09%=9
?=2%> 219%91 9=%>> 91%;= 0=%>1 0>%10
?1>%1 2:>%2: 92%6? 9=%?= 02%=? 01%96
??>%2 ===%6: 91%:2 9=%9> 06%0? 0=%0:
060%1 1=6%11 91%?= 91%;: 0;%:1 0=%29
??2%1 =19%0: 91%1= 90%6> 0=%10 0:%16
?19%= 2:>%:1 91%29 99%0= 0=%:? 00%?6
?=9%: 2:;%=: 9=%>= 99%69 01%6; 00%;0
?=1%6 2?=%;: 9=%?= 9>%6; 01%69 0?%=1
?=9%6 =;2%01 92%?: 9=%?2 0=%20 0>%:= \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ { 7 Mcqhl ~l~`c na i`clr &d`~~y+4 v0 7 Keylc`ikùe &`~~y,g?+4 v? 7 Zlykikùe nac mlil ae nkraiikùe ay~a &zqcb`n`y+4 v9 7 Zlykikùe
nac mlil ae nkraiikùe yqr &zqcb`n`y+4 v> 7 Zlykikùe nac mlil ae nkraiikùe elr~a &zqcb`n`y+4 v1 7 Flr` nac nç`
Mqae~a8 Ke~rlnqiikùe `c @eàckyky na _abraykùe Ckea`c % Gle~blgar{ a~ `c% ?::0%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 633/846
Nr% Fëi~lr Pqa}anl Qrç`y
6'6;
6%9?% C` ke~aeikùe na ay~a aharikikl ay c` na f`iar qe` bràmki`# ile c` }`rk`oca na
rayzqay~` &[+ { ile iq`~rl }`rk`ocay rabrayk}`y &V0# V?# V9# V>+ qy`enl ac zrlbr`g`
Gkek~`o% Ykaenl `yç# na c` ilemkbqr`ikùe na cly zqe~ly ayz`riknly lo~aekn` zranaikr# pqë ~kzl na mqeikùe na rabraykùe ay~`nçy~ki` aei`h`rç` gahlr ae cly n`~ly%
@nagày# a}`cq`r ac glnacl na rabraykùe i`enkn`~l l yqzarklr qy`enl gë~lnly
ay~`nçy~kily { bràmkily% Yqbaraeik`8 z`r` f`iar c` bràmki` zankn`# qy`r ac zrlbr`g`
Gke~`o zrliankaenl na c` ykbqkae~a g`ear`8 Krya ` Br`zf ↟ Nr`m~yg`e Zcl~% Ay~l
cca}` `c raiq`nrl na –Nr`m~yg`e Zcl~‗% Aeyabqkn`# ae c` }ae~`ekcc` na –[ }`rk`oca‗
zlear c` }`rk`oca nazaenkae~a &[+ {# ae c` }ae~`ekcc` na –V }`rk`oca‗ zlear c`y
}`rk`ocay kenazaenkae~ay &V+%
^`oc` gly~r`enl cly n`~ly na ay~a zrlocag`% &Ac`olr`ikùe zrlzk`+% \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ ]`rk`oca na rayzqay~` &[+ ]`rk`oca rabrayk}` V0 ]`rk`oca rabrayk}` V? ]`rk`oca rabrayk}` V9 ]`rk`oca rabrayk}` V>
?91 ?: 0; 6= ;1
?90 ?2 02 61 ;:
?61 >: ?: 69 0:1
?2: 11 ?: 6? 69?;= =: ?: 62 ;:
90? =6 ?0 6; ;>
?;1 21 ?: 69 ;?
?;? 6: ?: 60 ;?
?=9 2: ?: 16 0:1
?20 1: 01 2; 0::
?69 >: 01 6: ;:
?1= 9: 01 2; 66
6%99% Ay~a ay qe zrlocag` `n`z~`nl nac ~av~l na H`{ C% Na}lra &?::0+# ae ac iq`c ya
n` c` kemlrg`ikùe rapqarkn` z`r` c` yacaiikùe nac glnacl na rabraykùe yqzarklr#
o`y`nl ae c` keicqykùe nac eøgarl na }`rk`ocay rabraylr`y# yacaiikle`nl ae~ra
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 634/846
Nr% Fëi~lr Pqa}anl Qrç`y
6';:
iq`~rl glnacly i`enkn`~ly% Ac ahagzcl ay~à rac`ikle`nl ile ac i`clr `iqgqc`nl nac
aenqraikgkae~l nac iagae~l ~lg`nl ilgl c` }`rk`oca nazaenkae~a# ae mqeikùe na
cly zranki~lray v0 7 ! na `cqgke`~l na ~rki`cikl# v? 7 ! na ykcki`~l na ~rki`cikl# v9 7! marr`~l na `cqgkekl { v> 7 ykcki`~l na nki`cikl% Ya n` qe ~`g`ðl na gqay~r` kbq`c
` 09 loyar}`ikleay {# nlena c` yqg` na cly iq`nr`nly nac ~l~`c ay kbq`c ` ?#201%2=%
Z`r` ay~l# ya zkna `c cai~lr ccae`r cly m`c~`e~ay na c` ~`oc` na `o`hl { naiknkr iq`c ay
ac glnacl yqzarklr pqa ~kaea ac eøgarl `naiq`nl na }`rk`ocay rabraylr`y%
^`oc` gly~r`enl c` kemlrg`ikùe% Ccae`r cly m`c~`e~ay% El% na rabraylray d _abraylr &ay+d YYAd _ ?d _ ?&`hqy~`n`+d Izd M&i`ci%+d
0 v> 66:%61 :%=2= :%=>2 096%?? v0# v? 16%:0 ?%29 v0# v?# v9 :%;6? :%62= 9%?> v0# v?# v9# v> :%;6? >%:
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 635/846
Nr% Fëi~lr Pqa}anl Qrç`y
;'0
I@ZK^QCL ;
_abraykùe zlckelgk`c
Glnacly zlckelgk`cay na yabqenl lrnae &d 7 ?+ ile qe` }`rk`oca kenazaenkae~a%'
Glnacl na zlckelgkly na ~ariar lrnae &d 7 9+# ile qe` }`rk`oca kenazaenkae~a%'
Glnacl na yabqenl lrnae &iq`nrà~kil+ ile ke~ar`iikùe%' Glnacl zlckelgk c &na
yabqenl lrnae l iq`nrà~kil+# ile ~ray }`rk`ocay kenazaenkae~ay ile ke~ar`iikùe%'
A}`cq`ikùe na cly glnacly na rabraykùe%' Zrqao` ay~`nçy~ki` z`r` ilgz`r`r c`
yqg` na cly iq`nr`nly nac arrlr &YYA+ na i`n` glnacl zrlo`nl# z`r` y`oar iq`c
glnacl ay yqzarklr%' Glnacly na rabraykùe el ckea`cay { na rabraykùe clbçy~ki`%'
Glnacly na rabraykùe avzleaeik`cay z`r`gë~rkily# ile qe` ylc` }`rk`oca
kenazaenkae~a%' Zrliankgkae~ly z`r` c knae~kmki`ikùe na }`clray `~çzkily
av~ragly% Nk`beùy~kil { gk~kb`ikùe na gqc~kilckea`ckn`n%' Gankn`y z`r` ilrrabkr
gqc~kilckea`ckn`n ya}ar`%' Ahagzcly na zrlocag`y na rabraykùe zlckelgk`c
qy`enl ac zrlbr`g` na ilgzq~`nlr` Gkek~`o%' @q~lilrrac`ikùe ae n`~ly nayarkay na ~kagzl%' Fa~arlyian`y~kikn`n { flglyian`y~kikn`n%' Zrqao` na Rfk~a
z`r` ac zrlocag` na fa~arlyian`y~kikn`n%'
C` rabraykùe zlckelgk`c ay qe i`yl ayzaik`c na c` rabraykùe ckea`c ykgzca l gøc~kzca%
F`{ glnacly zlckelgk`cay na yabqenl l ~ariar lrnae% Ile c` rabraykùe zlckelgk`c
avky~ae glnacly ile qe` }`rk`oca kenazaenkae~a# ile aiq`ikleay iq`nrà~ki`y# iøoki`y
l ile ùrnaeay gày `c~ly pqa d 7 9% ^`gokëe f`{ glnacly zlckelgk`cay ile nly l gày
}`rk`ocay kenazaenkae~ay# ile aiq`ikleay na yabqenl# ~ariar lrnae# a~i% Kbq`cgae~a#
zqana f`oar glnacly na yabqenl lrnae l ~ariar lrnae ile ke~ar iikùe% Yke ago`rbl#
cly glnacly zlckeùgkily pqa ~kaeae ~ray l gày }`rk`ocay kenazaenkae~ay# ile }`clray
na d 5 9 yle `zcki`ikleay gq{ nkmkiqc~ly`y { r`r`y%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 636/846
Nr% Fëi~lr Pqa}anl Qrç`y
;'?
Nae~rl nac ~ùzkil na rabraykùe# ~`gokëe f`{ glnacly na rabraykùe el ckea`c#
ilgl cly glnacly na rabraykùe avzleaeik`cay# ae cly iq`cay cly z`ràga~rly el yle
ckea`cay%Ac glnacl na rabraykùe zlckelgk`c z`r`gë~rkil l zloc`ikle`c ay8
{ 7 οl $ ο0 v $ ο? v? $ ……… $ οd v
d $ α
Ac ay~kg`nlr l glnacl na rabraykùe ay~`nçy~kil ay8
{ 7 ol $ o0 v $ o? v? $ …… $ od v
d $ a
Glnacly zlckelgk`cay na yabqenl lrnae &d 7 ?+ ile qe` }`rk`oca kenazaenkae~a
Ac glnacl zlckeùgkil na yabqenl lrnae &d 7 ?+# ile qe` }`rk`oca kenazaenkae~a#
cc`g`n` mqeikùe na rayzqay~` iq`nrà~ki` ay8
{ 7 οl $ ο0 v $ ο00 v? $ α &;'0+
Pqa ~`gokëe ya zqana avzray`r ile nkmarae~a `el~`ikùe ilgl8
{ 7 οl $ ο0 v $ ο? v? $ α &;'0`+
Nlena8
{ 7 }`rk`oca nazaenkae~a l mqeikùe na rayzqay~`
οl 7 ke~ariaz~l ae c` lrnae`n`% Ay~a ilamkikae~a na rabraykùe razrayae~` c` rayzqay~`
zrlgankl na {# iq`enl v 7 :
ο0 7 ilamkikae~a na amai~l ckea`c
ο00 l ο? ay ac ilamkikae~a na amai~l iq`nrà~kil
v 7 }`rk`oca kenazaenkae~a
α 7 ~ërgkel na arrlr l rayknql
C` mqeikùe na rayzqay~` z`r` ay~a glnacl na rabraykùe &Ea~ar a~ `c% 0;;=+ ay8At[w 7 οl $ ο0 v $ ο00 v
? &;'0o+
Ay~` mqeikùe ay c` mlrg` oàyki` na qe` z`ràolc` ile}av`# ay naikr# iq`enl ο? 3 :% Yke
ago`rbl# iq`enl ο? 5 :# c` z`ràolc` ay iùei`}`% Ay~`y yk~q`ikleay ya }ae ae c` Mkbqr`
;%:&`+ { ae c` Mkbqr` ;%: &o+% Ac ilamkikae~a οl razrayae~` ac ke~ariaz~l ae c` lrnae`n`%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 637/846
Nr% Fëi~lr Pqa}anl Qrç`y
;'9
Cly ilamkikae~ay ο0 { ο? ile~rlc`e c` z`ràolc`# rac`~k}l ` c` lrnae`n`% Zlr ahagzcl# yk ο0
7 :# c` z`ràolc` ay ykgë~rki` { iae~r`n` `crananlr na { 7 :% El loy~`e~a# yk ο0 { ο? ~kaeae
ac gkygl ykbel# c` z`ràolc` ya nay}ç` f`ik` c` kxpqkarn`# zarl yk ο0 { ο? ~kaeae ykbelylzqay~ly# c` z`ràolc` ya nay}ç` f`ik` c` naraif`% @nagày# ac ilamkikae~a ο? nayirkoa c`
iqr}`~qr`% Zlr l~r` z`r~a# yk ο? 7 :# el f`{ iqr}`~qr`% Ay~l ya }a ae c` Mkbqr` ;%:&i+%
Ae~ra gày br`ena ya` ac }`clr na ο?# g`{lr yarà c` ~`y` na iqr}`~qr`% Yke ago`rbl# ae~ra
gày zapqaðl ya` ac }`clr na ο?# gaelr yarà c` iqr}`~qr` &Daccar a~ `c% 0;;:+% ^ln`y ay~`y
yk~q`ikleay ya }ae ae ay~`y bràmki`y%
Glnacl na zlckelgkly na ~ariar lrnae &d 7 9+# ile qe` }`rk`oca kenazaenkae~a
{ 7 οl $ ο0 v $ ο00 v? $ ο000 v
9 $ α &;'?+
Nlena8
{ 7 }`rk`oca nazaenkae~a
ο0 7 ilamkikae~a na amai~l ckea`c
ο00 7 ilamkikae~a na amai~l iq`nrà~kil
ο000 7 ilamkikae~a na amai~l iøokil%
C`y Mkbqr`y ;%: &n+ { &a+ na `o`hl gqay~r`e ay~a ~kzl na aiq`ikùe% Ilgl ya }a# iq`enl
ο9 3 :# ylora ac r`ebl na v# ac }`clr na { nkygkeq{a# zarl iq`enl ο9 5 :# ac }`clr na {
`qgae~`% Yke ago`rbl# c`y `zcki`ikleay nac glnacl iøokil yle gq{ zli`y%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 638/846
Nr% Fëi~lr Pqa}anl Qrç`y
;'>
Mkbqr` ;%:% Mkbqr`y gly~r`enl c`y bràmki`y nac glnacl iq`nrà~kil { iøokil% Zlr
ahagzcl# bràmki` &`+ gqay~r` ac glnacl na yabqenl lrnae# ile ο? 3 :4 c` bràmki` &o+
gqay~r` ac glnacl ile ο? 5 : { ile }`rkly }`clray na ο?% C` bràmki` &i+ gqay~r` cly
glnacly na ~ariar lrnae ile ο9 3 : {# &n+# ile ο9 5 :% &Mqae~a8 Daccar a~ `c% 0;;:+
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 639/846
Nr% Fëi~lr Pqa}anl Qrç`y
;'1
Cly glnacly zlckelgk`cay na zlnaray gày `c~ly pqa d 7 9 naoae na qy`rya ile
zrai`qikùe% Ay~l ya naoa ` pqa# c` ke~arzra~`ikùe na cly ilamkikae~ay ay nkmçikc# { c`y
ke~arzlc`ikleay zqanae yar zackbrly`y% @nagày# iq`enl f`oc`gly na glnacly ile}`clray na d 7 ># l d 7 1# ac ilgzlr~`gkae~l na yagah`e~ay glnacly ay av~r`ðl { na
`zcki`ikleay r`r`y {# zlr cl ~`e~l# el ya nkyiq~kràe pqç%
Glnacl na yabqenl lrnae ile gày na nly }`rk`ocay kenazaenkae~ay ile
ke~ar`iikùe
@e~ay na nkyiq~kr ay~ly glnacly na rabraykùe f`{ pqa namkekr ac ~ërgkel ke~ar`iikùe%
Ke~ar`iikùe ykbekmki` pqa# ac amai~l na v0 ylora {# ay kemcqaeik`nl zlr ac }`clr na v?# pqa
~`gokëe ykbekmki` pqa# ac amai~l na v? ylora {# ay kemcqaeik`nl zlr v0%
Z`r` }ar ac amai~l na ke~ar`iikùe# yqzùeb`ya pqa c` aiq`ikùe na c` cçea` na
rabraykùe gqay~r`c ay { 7 = $ >v0 $ 1v? ’ 9v0v?% Z`r` avzcki`r ay~a amai~l yqzùeb`ya pqa
ca nagly }`clray ` v? na 0# ?# { 9% @c yqy~k~qkr cly }`clray na v? 7 0# ?# { 9# ae c` aiq`ikùe
gqay~r`c na `rrko`# ya zrlnqiae c`y ykbqkae~ay aiq`ikleay8 { 7 1 $ v0# ile v? 7 04 { 7
0: ’ ?v0 ile v? 7 ? {# nagày# { 7 01 ’ 1v0# ile v? 7 9% @e`ckx`enl ay~`y ~ray aiq`ikleay
glnkmki`n y }agly pqa ac ke~ariaz~l { cly ilamkikae~ay na v0 ~`gokëe }`rç`e% @pqç ya }a
pqa ac amai~l na v0 ylora { ay kemcqaeik`nl zlr ac }`clr na v?% @c br`mki`r ay~`y ~ray
aiq`ikleay }agly pqa c`y ~ray cçea`y rai~`y ya irqx`e ae~ra yk% Ay~l ya }a ae c` Mkbqr`
;%0 &o+% Ae ay~` bràmki`# ic`r`gae~a# ya }a pqa f`{ ke~ar`iikùe# ay naikr# iq`enl c`y
cçea`y rai~`y ya irqx`e ae~ra yk%
Glnacl na yabqenl lrnae &iq`nrà~kil+ ile ke~ar`iikùe
Yk qe ke}ay~kb`nlr iraa pqa ae yqy n`~ly avky~a qe` rac`ikùe iq`nrà~ki` ae~ra c` }`rk`ocanazaenkae~a & {+ { i`n` qe` na c`y }`rk`ocay kenazaenkae~ay v0 { v?# ay naikr# iq`enl c`y
}`rk`ocay kenazaenkae~ay ke~ar`iikle`e ae~ra yk &naikykùe pqa ya clbrù nayzqëy na
`e`ckx`r c`y bràmki`y ile ~ray iqr}`y ke~ar`iikle`enl ae~ra yk+# ae~leiay# ya
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 640/846
Nr% Fëi~lr Pqa}anl Qrç`y
;'=
naoa na keicke`r zlr ac glnacl na yabqenl lrnae ile ke~ar`iikùe%
Ac glnacl zlckelgk`c ile nly }`rk`ocay kenazaenkae~ay ile ke~ar`iikùe ya n`
ilgl8 { 7 οl $ ο0 v0 $ ο? v? $ ο9 v
?0 $ ο> v
?? $ ο1 v0 v? $ α &;'9+
Ay~a glnacl# ~`gokëe ya zqana avzray`r ile nkmarae~ay `el~`ikleay# ilgl c`y
yað`c`n`y `o`hl8
{ 7 οl $ ο0 v0 $ ο? v? $ ο0? v?0 $ ο?? v
?? $ ο0? v0 v? $ α &;'9`+
Nlena8
ο0? 7 ilamkikae~a na amai~l na ke~ar`iikùe# nlena v0 { v? razrayae~`e c` ke~ar`iikùe ae~ra
cly zrlely~ki`nlray l }`rk`ocay kenazaenkae~ay v0 { v?%
@pqç# eù~aya pqa# c` nkmaraeik` ae~ra c` aiq`ikleay &;'?+ { &;'9+# ay ac øc~kgl ~ërgkel na
c` naraif`# ac iq`c nael~` ac amai~l na c` ke~ar`iikùe%
Glnacl zlckelgk`c &na yabqenl lrnae l iq`nrà~kil+ ile ~ray }`rk`ocay
kenazaenkae~ay yke ke~ar`iikùe
Ac glnacl na yabqenl lrnae ile ~ray }`rk`ocay kenazaenkae~ay# iq`enl ay~`y }`rk`ocay
el ke~ar`iikle`e ae~ra yk# ay8
{ 7 οl $ ο0 v0 $ ο? v? $ ο9 v9 $ ο00 v?0 $ ο?? v
?? $ ο99 v
?9 $ α &;'>+
Glnacl zlckelgk`c &na yabqenl lrnae l iq`nrà~kil+# ile ~ray }`rk`ocay
kenazaenkae~ay ile ke~ar`iikùe
Ac glnacl na yabqenl lrnae ile ~ray }`rk`ocay kenazaenkae~ay# ile ke~ar`iikùe
&Ea~ar a~ `c% 0;;=+ ay8
{ 7 οl $ ο0v0 $ ο?v? $ ο9v9 $ ο00 v?
0 $ ο??v?
? $ ο99 v?
9 $ ο0?v0v? $ ο09v0v9 $ ο?9v?v9 $ α &;'1+ Nlena8
{ 7 }`rk`oca nazaenkae~a l mqeikùe na rayzqay~`
οl 7 ke~ariaz~l ae c` lrnae`n`
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 641/846
Nr% Fëi~lr Pqa}anl Qrç`y
;'2
ο0?# ο09# ο?9 7 cly ilamkikae~ay nac amai~l na ke~ar`iikùe ae~ra cly z`ray na }`rk`ocay
na zrankiikùe v0 v? # v0 v9 { v? v9 v0 v? # v0 v9 # v? v9 razrayae~`e c` ke~ar`iikùe ae~ra c`y
}`rk`ocay kenazaenkae~ay v0 # v?# v9# v0 # v? # v9 7 }`rk`ocay kenazaenkae~ayAe c` ylcqikùe na zrlocag`y rac`ikle`nly ile glnacly na rabraykùe ckea`c#
gøc~kzca l na rabraykùe zlckelgk`c# ile qe` l gày }`rk`ocay kenazaenkae~ay ay ykagzra
ile}aekae~a br`mki`r cly n`~ly { av`gke`r ac nk`br`g` ayz`riknl% Ay~l ya f`ia ile ac
loha~l na `e`ckx`r# }kyq`cgae~a# ac nk`br`g` ayz`riknl { }ar ac ~kzl na iqr}` gly~r`nl
{# zlr ileykbqkae~a# ac glnacl na rabraykùe l mqeikùe pqa zqan` aei`h`r gahlr ae cly
n`~ly%
Mkbqr` ;%0% Bràmki`y gly~r`enl glnacly zlckelgk cay na zrkgarl { yabqenl lrnae# ile
nly }`rk`ocay kenazaenkae~ay% C` bràmki` &`+ gqay~r` c` aiq`ikùe { 7 = $ > v0 $ 1 v?%
Iq`enl v? 7 0# ? { 9# c`y aiq`ikleay glnkmki`n`y ya }ae ae c` bràmki` ae i`n` qel na yqy
i`yly%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 642/846
Nr% Fëi~lr Pqa}anl Qrç`y
;'6
Ae ay~`y mkbqr`y ya }a pqa el f`{ ke~ar`iikùe &c`y cçea`y el ya irqx`e# zlrpqa ay qe
glnacl `nk~k}l+% C` bràmki` &o+ gqay~r` c` aiq`ikùe { 7 = $ >v0 $ 1v? ’ 9v0v?% Iq`enl
v? 7 0# ? { 9# c` bràmki` gqay~r` c`y aiq`ikleay glnkmki`n`y% @pqç ya }a pqa# c` aiq`ikùe zlckelgk c na zrkgar lrnae ~kaea ke~ar`iikùe% Mke cgae~a# c`y bràmki`y &i+ { &n+
gqay~r`e cly glnacly na rabraykùe zlckelgk`c na yabqenl lrnae# yke ke~ar`iikùe {# ile
amai~l na ke~ar`iikùe na kemaraeik`# rayzai~k}`gae~a% Ay~l ya }arç` nayzqëy na pqa ya
yqy~k~q{ar`e cly }`clray na v? 7 0# ? { 9 ` qe` aiq`ikùe gqay~r`c pqa agqc`rà `c glnacl
&;'=+% &Daccar a~ `c% 0;;:+%
A}`cq`ikùe na cly glnacly na rabraykùe
C` rabraykùe zlckelgk`c ay qe i`yl ayzaik`c na cly glnacly na rabraykùe ckea`c ykgzca
{ gøc~kzca% C` }`ckn`ikùe na ay~ly glnacly ay `eàclb` ` c` na cly glnacly na rabraykùe
ckea`c% Yke ago`rbl# `e~ay na ay~`r ~l~`cgae~a yabqrly `iari` na c` q~kckn`n nac glnacl
na rabraykùe yacaiikle`nl# z`r` mkeay na zrankiikùe { ay~kg`ikùe# f`{ pqa }ar pqa ac
glnacl razrayae~a `naiq`n`gae~a c` rac`ikùe ae~ra c`y }`rk`ocay% Ay~l ya zqana f`iar
` ~r`}ëy na ay~`nçy~ki` na kemaraeik` { na `eàckyky na bràmkily%
Z`r` c` a}`cq`ikùe na cly glnacly ya zqana zrlianar# haràrpqki`gae~a# `hqy~`enl
glnacly na yabqenl { ~ariar lrnae# ile ke~ar`iikùe { yke ke~ar`iikùe {# cqabl ya
avzclr` c` zlykokckn`n na `hqy~`r qe glnacl na lrnae gày o`hl ilgl glnacly na
rabraykùe ckea`c gøc~kzca# zarl# eqa}`gae~a# ile ke~ar`iikùe { yke ke~ar`iikùe%
Na iq`cpqkar g`ear`# ilgl ya nkhl `e~ay# z`r` a}`cq`r cly glnacly na rabraykùe
ya zrliana avzclr`enl cly irk~arkly ay~`nçy~kily# ilgl ac ilamkikae~a na na~argke`ikùe
gøc~kzca &_ ?
+# ac arrlr ay~àen`r na cl ay~kg`nl &yα+# ac ilamkikae~a na na~argke`ikùegøc~kzca &_ ?+# ac irk~arkl Iz na G`ccl# Z_AYY &c` ykbc` na yqg` na iq`nr`nly na arrlr
na zrankiikùe+ l# cly }`clray na ~# a~i% @nagày# ya ra}ky`e cly }`clray na ]KM &m`i~lray
na }`rk`ex` kemc`n`4 ae nlena }`clray br`enay na ]KMy kenki`e br`enay nkmaraeik`y
ae~ra cly ilamkikae~ay na rabraykùe ay~kg`nly { cly ay~`en`rkx`nly+# z`r` }ar zlykocay
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 643/846
Nr% Fëi~lr Pqa}anl Qrç`y
;';
zrlocag y na ilckea ckn`n% ^`gokëe# ya zqana qy`r c` ay~`nçy~ki` na Nqroke'R ~yle
z`r` ra}ky`r zrlocag`y na `q~lilrrac`ikùe na cly rayknqly ae yarkay na ~kagzl% @pqç#
z`r` rabraykùe gøc~kzca# na `iqarnl ` c` cùbki` nac zrlbr`g` EIYY# ëy~` nkia pqa# ykay~` mqeikùe ay~à iari`e` ` ?# el f`{ `q~lilrrac`ikùe# zarl yk ay gq{ nkmarae~a na ?#
ae~leiay# yk c` f`{% @eàclb`gae~a# ya zqanae qy`r l~rly gë~lnly ilgl –_abraykùe zlr
Z`yly‗ l –^ln`y c`y _abraykleay Zlykocay‗# pqa yacaiikle`e cly glnacly ùz~kgly
o`yàenlya ae cly irk~arkly `rrko` ik~`nly# ay naikr# `brab`enl {,l ackgke`enl c`y
}`rk`ocay kenazaenkae~ay l na rayzqay~`%
Mke`cgae~a# ~lnl ay~l ya zqana ilgzcagae~` qy`enl qe `eàckyky yqoha~k}l# ay
naikr# `e`ckx`enl cly bràmkily na cly rayknqly ay~`en`rkx`nly l el ay~`en`rkx`nly# ay~l
ay# av`gke`enl c` zrqao` na elrg`ckn`n# rayknqly }aryqy }`clray `hqy~`nly# na cly
ùrnaeay# a~i%
@% @eàckyky na ay~`nçy~ki` na kemaraeik` &loha~k}l+ z`r` ilgzcagae~`r c` }`ckn`ikùe nac
glnacl
0% Iàciqcl nac ilamkikae~a na na~argke`ikùe _ ?% Ay~a irk~arkl kenki`# pqë zrlzlrikùe na
c` }`rk`ikùe ~l~`c ae c` rayzqay~` [ ya avzcki` ile ac glnacl `hqy~`nl% Ae ~ërgkely
ykgzcay# ay~l nkia pqa _ ? kenki` c` zrlzlrikùe na }`rk`ikùe avzcki`n` zlr c`y }`rk`ocay
kenazaenkae~ay v0# v?# v9# …%# vd % Ay~a ilamkikae~a na na~argke`ikùe _ ? {` ya nayirkokù
`e~arklrgae~a# ay naikr8
_ ? 7 &Ω v{+? , Ω v?Ω {? &;'=+
Nlena8 Ω v{ 7 ΩV[ ’ ΩVΩ[,e# Ωv? 7 ΩV ? ’ &ΩV+?,e { Ω {? 7 Ω[ ? ’ &Ω[+?,e# c`y iq`cay
ya namkeae zlr c`y aiq`ikleay &6'6+# &6';+ { &6'0:+ n`n`y ae ac i`zç~qcl 6%?% Ac arrlr ay~àen`r na cl ay~kg`nl8
yα 7d e
YYA
∟∟0 7 Ω& { ’ { z+
?,&e ’ ?+ &;'2+
Nlena# YYA 7 Ωa?k ya ramkara c` yqg` na cly iq`nr`nly nac arrlr l rayknql# { z ay cçea`
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 644/846
Nr% Fëi~lr Pqa}anl Qrç`y
;'0:
na rabraykùe# e ay ac ~`g`ðl na gqay~r` {# d# ay ac eøgarl na ilamkikae~ay οk zrlo`nly
ae ac glnacl na rabraykùe zlckelgk`c% Qe }`clr zapqaðl na YYA# kenki` qe oqae `hqy~a
nac glnacl% C` mqeikùe YYA ay qe nk`beùy~kil gq{ kgzlr~`e~a%9% Irk~arkl Iz% Ay~a nk`beùy~kil ay~à rac`ikle`nl ile ac arrlr iq`nrà~kil gankl na qe
}`clr `hqy~`nl% Ae baear`c# ya zramkarae }`clray zapqaðly na Iz% Ac glnacl ùz~kgl
~kaea qe }`clr na Iz iari`el ` &z $ 0+# nlena# z ay ac eøgarl na }`rk`ocay
kenazaenkae~ay% Qe Iz g`{lr pqa &z $ 0+ kenki` pqa ac glnacl na rabraykùe ile~kaea
}`rk`ocay keeaiay`rk`y pqa zqan`e n`r zrlocag`y na ilckek`ckn`n# zarl yk ac Iz ay
gaelr pqa &z $ 0+# ay~l kenki` pqa ya f`e lgk~knl }`rk`ocay kgzlr~`e~ay%
>% @eàckyky na @EL]@Y { zrqao`y na ~ na Ay~qnk`e~a z`r` }ar iq`c glnacl na
rabraykùe hqy~` gahlr cly n`~ly%
O% @eàckyky bràmkil &yqoha~k}l+% Z`r` f`iar c` a}`cq`ikùe# yqoha~k}`gae~a# na c` olen`n
na `hqy~a na cly glnacly qy`nly ya `e`ckx`e cly ykbqkae~ay bràmkily8
0% Zrqao` na elrg`ckn`n% Z`r` pqa avky~` elrg`ckn`n# cly rayknqly naoaràe mlrg`r qe`
cçea` rai~` l ay~`r nae~rl na c`y o`en`y na ilemk`ex`% Yk el ay `yç# c` yqzlykikùe na
elrg`ckn`n ay ke}àckn`%
?% Fky~lbr`g`y na rayknqly% Ay~` bràmki` naoarà yagah`rya qe` nky~rkoqikùe elrg`c%
9% Bràmki` na rayknqly }aryqy }`clray `hqy~`nly na [ z`r` c` zrqao` na kenazaenaeik`%
@pqç# naoa f`oar `ca`~lrkan`n na cly rayknqly% El naoa f`oar ~aenaeik`y iraikae~ay l
nairaikae~ay% @nagày# naoa f`oar ac gkygl eøgarl na rayknqly zlyk~k}ly { eab`~k}ly%
Na el yar `yç# ya }klc`e c`y yqzlykikleay nac glnacl%
>% @q~lilrrac`ikùe &}`clray na α mkhly+% Z`r` nk`bely~ki`r c` q~lilrrac`ikùe ae yarkay na~kagzl# bràmki`r rayknqly }y% ~kagzl% Qy`r zrqao` na Nqroke'R`~yle z`r` }ar yk avky~a
`q~lilrrac`ikùe na zrkgar lrnae% Ya gk~kb` f`ikaenl ~r`eymlrg`ikleay nac aha [¾ %
1% @eàckyky na bràmkily z`r` nk`bely~ki`r ilckek`ckn`n &ilrrac`ikùe l nazaenaeik` i`yk
ckea`c ae~ra c`y }`rk`ocay na rabraykùe+% Z`r` gk~kb`r ay~l f`iar ~r`eymlrg`ikleay ilgl
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 645/846
Nr% Fëi~lr Pqa}anl Qrç`y
;'00
[¾ 7 Clb [ # [ 7 [ ?# [ ¾7 ∘ [ a [ ¾7 0,[%
=% Zrqao` na fa~arlyian`y~kikn`n &fa~arl' 7 naykbq`c4 'yian`y~kikn`n 7 ayz`riknl+ l na
rayknq`cay el qekmlrgay &kgzcki` arrlr na }`rk`ex` na ω?α el iley~`e~a ae ~lnly cly
i`yly# ae ile~r`y~a ile flglyian`y~kikn`n# c` iq`c kgzcki` arrlr na }`rk`ex` ω?α
iley~`e~a+% Z`r` nk`bely~ki`r ac zrlocag` na fa~arlyian`y~kikn`n br`mki`r cly rayknq`cay
}aryqy }`clray zranaiknly# [ % @eàclb`gae~a# z`r` nk`bely~ki`r ay~a zrlocag` na
fa~arlyian`y~kikn`n ya zqana f`iar `zcki`enl c`y zrqao`y na Rfk~a { na
Oraqyif'Z`b`e% Z`r` gk~kb`r ac zrlocag` na c` m`c~` na flglyian`y~kikn`n# ay~l ya
zqana f`iar zlr gankl na ~r`eymlrg`ikleay# ilgl ae ac keikl &1+% ^`gokëe ya zqana
f`iar zrlo`enl l~rly glnacly pqa hqy~ae gahlr cly n`~ly%
_ayqgae na cly glnacly na rabraykùe qy`nly
@% Glnacl na rabraykùe ckea`c ykgzca &na zrkgar lrnae+# ile qe` }`rk`oca
kenazaenkae~a
{ 7 οl $ ο0 v0 $ α
O% Glnacl na rabraykùe ckea`c gøc~kzca# ile nly }`rk`ocay kenazaenkae~ay# yke
ke~ar`iikùe
{ 7 οl $ ο0 v0 $ ο? v? $ α
I% Glnacl na rabraykùe ckea`c gøc~kzca# ile nly }`rk`ocay kenazaenkae~ay# ile
ke~ar`iikùe
{ 7 οl $ ο0 v0 $ ο? v? $ ο0? v0 v? $ α
N% Glnacl iq`nrà~kil# ile qe` }`rk`oca kenazaenkae~a
{ 7 οl $ ο0 v0 $ ο? v?
? $ α
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 646/846
Nr% Fëi~lr Pqa}anl Qrç`y
;'0?
M% Glnacl iøokil# ile qe` }`rk`oca kenazaenkae~a
{ 7 οl $ ο0 v0 $ ο? v?? $ ο9 v
99 $ α
M% Glnacl na yabqenl lrnae &iq`nrà~kil+# ile ? }`rk`ocay kenazaenkae~ay# ykeke~ar`iikùe
{ 7 οl $ ο0 v0 $ ο? v? $ ο00 v?0 $ ο?? v
?? $α
B% Glnacl iq`nrà~kil ile nly }`rk`ocay kenazaenkae~ay ile ke~ar`iikùe
{ 7 οl $ ο0 v0 $ ο? v? $ ο00 v?0 $ ο?? v
?? $ ο0? v0 v? $ α
F% Glnacl na yabqenl lrnae ile 9 }`rk`ocay kenazaenkae~ay# yke ke~ar`iikùe
{ 7 οl $ ο0 v0 $ ο? v? $ ο9 v9 $ ο00 v?0 $ ο?? v
?? $ ο99 v
?9 $ α
K% Glnacl iq`nrà~kil ile 9 }`rk`ocay kenazaenkae~ay ile ke~ar`iikùe
{ 7 οl $ ο0 v0 $ ο? v? $ ο9 v9 $ ο00 v?0 $ ο?? v
?? $ ο99 v
?9 $ ο0? v0 v? $ ο09 v0 v9 $ ο?9 v? v9 $ α
Ahagzcl )0% Ae qe `r~çiqcl nac H% @brkiqc~qr`c Aeb% _aya`rif# 0;21 &z% 919'9=0+ ya
razlr~`e cly n`~ly ile ac eøgarl na nç`y nayzqëy na c` mclr`ikùe &v+# ac raenkgkae~l na
c` ilyaif`# ae Db%,f` &{+% &Na}lra# ?::0+% C` ~`oc` na `o`hl gqay~r` cly n`~ly%
^@OC@ ;%0% ^`oc` gly~r`enl cly n`~ly nac zrlocag`%
v s 0= 06 ?: ?? ?> ?= ?6 9: 9? 9> 9= 96 >: >? >> >=
{ s ?1:6 ?106 99:> 9>?9 91:2 90;: 91:: 9669 96?9 9=>= 92:6 9999 9102 9?>0 90:9 ?22=
&Mqae~a8 Na}lra# ?::0+
F`iar cly ykbqkae~ay iàciqcly8
&`+ F`iar qe` bràmki` pqa }`{` ae mqeikùe na & {+ { na cly nç`y na mclr`ikùe & v+%
&o+ @hqy~`r ac glnacl na rabraykùe gày zrlzk`nl%
&i+ F`iar qe` rac`ikùe na cly iàciqcly na cly ilamkikae~ay na c` nay}k`ikùe ay~àen`r { naqe` ~`oc` na `eàckyky na }`rk`ex`%
&n+ Ay~kg`r ac }`clr nac ilamkikae~a na na~argke`ikùe gøc~kzca _ ?%
&a+ F`iar qe` zrqao` na fkzù~ayky Fl8ο? 7 : }aryqy F08ο? ≪ :% F`iar l~r` zrqao` gày
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 647/846
Nr% Fëi~lr Pqa}anl Qrç`y
;'09
ile Fl8ο0 7 : }aryqy F08ο0 ≪ :%
Ylcqikùe8
&`+ C` bràmki` na `o`hl kenki` pqa# qe` mqeikùe zlckelgk`c iq`nrà~ki`# ile ο 3 :# yarç`c` gày `zrlzk`n`%
Mkbqr` ;%?% Nk`br`g` ayz`riknl na cly n`~ly nac raenkgkae~l na c` ilyaif` &{+ { ac
eøgarl na nç`y na mclr`ikùe &v+% &Mqae~a8 Na}lra# ?::0+%
&o+ C` ~`oc` na `o`hl gqay~r` cly rayqc~`nly% &Ac`olr`ikùe zrlzk`+ \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ Zrlely~ki`nlr Ilamkikae~a Nay}k`ikùe ay~àen`r ]`clr na ~ ]`clr na z
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ Iley~`e~a '0:2:%> =02%9 '0%29 :%0:2ο0 ?;9%>6 >?%06 =%;= :%:::ο? '>%1916 :%=2>> '=%29 :%:::
YYA 7 ?:9%; _ ? 7 :%2;> _ ?`hqy~`n` 7 :%2=? \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
Cly ek}acay irç~kily z`r` qe` zrqao` okc`~ar`c# ile qe ek}ac ykbekmki`e~a na ΰ 7 :%:1 yle8
'?%0= ≮ ~T%:?1409W ≮ ?%0=:
&i+ Cly rayqc~`nly lo~aeknly na ay~a keikyl ya gqay~r`e ae c` ~`oc` na `o`hl%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 648/846
Nr% Fëi~lr Pqa}anl Qrç`y
;'0>
^@OC@ ;%:% ^`oc` na `eàckyky na }`rk`ex`% &Ac`olr`ikùe zrlzk`+ \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ Mqae~a na b%c% YY GY Mi`ci% ]`clr na z
]`rk`ikùe \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ Naoknl ` c` ? ?#:6>#22; 0#:>?#96; ?1%:6 :%::::rabraykùeArrlr 09 1>:#966 >0#1=6
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ ^l~`c 01 ?#=?1#0=2
Ileicqykùe8 Naoknl ` pqa Mi`ci% 7 ?1%:6 555 Mirç~ki` 7 9%60# ya raif`x` c` fkzù~ayky eqc`
Fl8ο? 7 : {# zlr cl ~`e~l# ya keicke` zlr c` fkzù~ayky `c~are`~k}` na F08ο? ≪ :%
&n+ C` ay~kg`ikùe nac ilamkikae~a na na~argke`ikùe ay8
_ ? 7 0 ’ YYarrlr ,YY~l~`c
7 0 ’ 1>:#966,?#=?1#0=2
7 :%2;>
&a+ Z`r` c` zrqao` na fkzù~ayky eqc` Fl8ο? 7 : { c` fkzù~ayky `c~are`~k}` F08ο? ≪ :#
qy`gly cly n`~ly na `rrko`% Zlr ahagzcl# ο? 7 '>%1916 { c` nay}k`ikùe ay~àen`r ay na yο?
7 :%=2>>% C` zrqao` na Fl8ο? 7 : ay cl gkygl pqa naikr pqa ac glnacl zlckelgk`c
iq`nrà~kil el `zcki` ` cly n`~ly {# F08ο? ≪ : nkia pqa yk `zcki`% C` mqeikùe na ~ qy`n`
ay8
~ 7 ο? , yο? &;'6+
Yqy~k~q{aenl cly }`clray ilrrayzlenkae~ay ely n`8~ 7 '>%1916 , :%=2>>
7 '=%29
C` zrqao` ay~à o`y`n` ae e ' &d $ 0+ br`nly na ckoar~`n & η+# ay naikr# ile e 7 0= { d 7 ?%
Zlr cl ~`e~l# η 7 09% C`y rabkleay irç~ki`y yle8 '?%0=: ≮ ~%:?1409 ≮ ?%0=:%
Ae ileicqykùe# naoknl ` pqa c` ~i`ci% 7 '=%29 3 ~irç~ki` 7 '?%0=:# ya raif`x` c` fkzù~ayky eqc`
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 649/846
Nr% Fëi~lr Pqa}anl Qrç`y
;'01
na Fl8ο? 7 : { ya keicke` zlr c` zrqao` na fkzù~ayky `c~are`~k}` na F08ο? ≪ :% El
loy~`e~a# yk ya raif`x` c` fkzù~ayky eqc`# ya nkia pqa ac glnacl iq`nrà~kil yk aei`h` okae
ae cly n`~ly4 na l~r` g`ear` ya `iaz~` c` fkzù~ayky eqc`% Z`r` f`iar c` zrqao` nafkzù~ayky eqc` na Fl8ο0 7 : }aryqy F08ο0 ≪ :# ya zrliana ae mlrg` ykgkc`r%
&m+ Z`r` c`y bràmki`y na cly rayknqly ay~`en`rkx`nly# ae mqeikùe na c`y loyar}`ikleay#
ay~`y bràmki`y ya zqanae mlrgqc`r qy`enl c` kemlrg`ikùe n`n` ae c` ~`oc` na `o`hl%
Zrqao` ay~`nçy~ki` z`r` ilgz`r`r c` yqg` na cly iq`nr`nly nac arrlr &YYA+ na
i`n` glnacl zrlo`nl# z`r` y`oar iq`c glnacl ay yqzarklr
Cly q~lray Daccar a~ c% &0;;:+ nac ckorl Y~`~ky~kiy mlr G`e`bagae~ `en Ailelgkiy n`e
qe` zrqao` ay~`nçy~ki` pqa gkna c`y nkmaraeik`y na c` yqg` na cly iq`nr`nly nac arrlr
&YYA+# z`r` zrlo`r c` yqzarklrkn`n na i`n` glnacl zrlo`nl% Ay~l ya naoa ` pqa YYA
gkna# pqë ~`e okae aei`h`e cly n`~ly ae ac glnacl% Ay~` zrqao` ya f`ia ilgz`r`enl c`
yqg` na cly iq`nr`nly nac arrlr &YYA0+ nac glnacl ykgzca l `ora}k`nl {# c` yqg` na cly
iq`nr`nly nac arrlr &YYA?+ nac glnacl ilgzca~l l ilgzcahl% Ay~l ya f`ia# zlrpqa
ykagzra ay ile}aekae~a qy`r glnacly ykgzcay &ac qyl na glnacly ilgzcahly el
eaiay`rk`gae~a cly f`ia yqzarklray+% C` zrqao` ay~`nçy~ki` z`r` gankr c` rac`ikùe ae~ra
YYA0 { YYA? ay8
&YYA0 ’ YYA?+,&d ? ’ d 0+M 7 „„„„„„„„„„„ &;';+
YYA? , &e ’ d ? ’ 0+Nlena8
M 7 nky~rkoqikùe na Mkyfar# ile η0 7 d ? ’ d 0 { η? 7 e ’ d ? ’ 0 br`nly na ckoar~`n%
Nlena8 e ’ d ? ’ 0 7 eøgarl na br`nly na ckoar~`n `ylik`nly ile ac glnacl ilgzca~l%Nlena8 d ? 7 eøgarl na ilamkikae~ay &οk+ zrlo`nly nac glnacl ilgzca~l
d 0 7 eøgarl na ilamkikae~ay &οk+ zrlo`nly nac glnacl ykgzca%
e 7 ~`g`ðl na c` gqay~r`
YYA0 7 yqg` na cly iq`nr`nly nac arrlr nac glnacl ykgzca zrlo`nl
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 650/846
Nr% Fëi~lr Pqa}anl Qrç`y
;'0=
YYA? 7 yqg` na cly iq`nr`nly nac glnacl ilgzca~l zrlo`nl
El~ 8 Yk ac `hqy~a nac glnacl ilgzca~l el ay ykbekmki`e~agae~a gahlr pqa ac glnacl
ykgzca l `ora}k`nl# ac }`clr na YY0 yarà zapqaðl% Zlr aena# c` rac`ikùe YY0 ’ YY? yarà zapqað` {# zlr cl ~`e~l# ac } clr na M yarà zapqaðl { el ya zlnrà raif`x`r c fkzù~ayky
eqc`% Yke ago`rbl# yk ac `hqy~a nac glnacl ilgzca~l ay oqael# ac }`clr na YY? yarà
zapqaðl { c` rac`ikùe YY0 ’ YY? yarà br`ena {# zlr ileykbqkae~a# ac }`clr na M yarà
br`ena { ya raif`x`rà c` fkzù~ayky eqc`%
C` rabkùe na raif`xl z`r` c` aiq`ikùe na `rrko` &;';+ ay n`n` zlr c` ykbqkae~a mqeikùe
ay~`nçy~ki`8
M 5 MTΰ4d?'d0#e'd?'0W &;';`+
Nlena8
M 7 ac }`clr na c` ay~`nçy~ki` M i`ciqc`n`
ΰ 7 ek}ac ykbekmki`e~a na :%:1 l :%:0 na c` nky~rkoqikùe na M
d ? 7 eøgarl na ilamkikae~ay οk nac glnacl yqzarklr
d 0 7 eøgarl na ilamkikae~ay οk nac glnacl `ora}k`nl
e 7 ~`g`ðl na c` gqay~r`
Ahagzcl )?% Ac ckorl Y~`~ky~kiy mlr G`e`bagae~ `en Ailelgkiy na Daccar a~ `c% &0;;:+
n` qe ahagzcl# z`r` na~argke`r ac glnacl na rabraykùe gày `zrlzk`nl% Z`r` ay~l# ya
y`i` qe` gqay~r` na ?1 àra`y &i`yly+% I`n` àra` ileyky~a ae# `zrlvkg`n`gae~a# 1#:::
}k}kaen`y% Ya rabky~r` c` b`e`eik` `eq`c ~l~`c na c`y }ae~`y# ac kebrayl zrlgankl `eq`c
na c`y }k}kaen`y { c` an`n zrlgankl na cly ekðly na ay~a zrlocag`%
F`iar cly ykbqkae~ay iàciqcly8&`+ Zrlo`r qe glnacl na rabraykùe iq`nrà~kil# ile ke~ar`iikùe% Ae ay~a i`yl# cl
cc`g`ragly glnacl yqzarklr l glnacl ilgzca~l%
&o+ Nayzqëy# zrlo`r qe glnacl na rabraykùe iq`nrà~kil# yke ke~ar`iikùe% Ac ay~a i`yl# cl
cc`g`ragly glnacl `ora}k`nl%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 651/846
Nr% Fëi~lr Pqa}anl Qrç`y
;'02
&i+ Mke`cgae~a# zrlo`r qe glnacl na rabraykùe ckea`c gøc~kzca# yke ke~ar`iikùe% Ay~a
glnacl# cl cc`g`ragly glnacl ckea`c ykgzca%
&n+ Nayirkokr c`y aiq`ikleay na cly glnacly na rabraykùe zloc`ikle`cay na cly keikyly&`+# &o+ { &i+%
&a+ Z`r` naiknkr iq`c glnacl ay gahlr# f`iar qe` ~`oc` ile cly rayqc~`nly na cly ~ray
glnacly# o`yàenlya ae cly nk`beùy~kily loha~k}ky~`y ilgl c`y ay~`nçy~ki`y _ ?# _ ?`hqy~`n`#
y# Z_AYY# @EL]@# a~i%
&m+ F`iar qe` zrqao` na fkzù~ayky z`r` }ar yk ac amai~l na ke~ar`iikùe ay }k`oca% @nagày#
qy`r c` aiq`ikùe ;';# z`r` yacaiikle`r ac glnacl na rabraykùe gày `zrlzk`nl%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 652/846
Nr% Fëi~lr Pqa}anl Qrç`y
;'06
^@OC@ ;%?% ^`oc` gly~r`enl cly n`~ly z`r` ay~a zrlocag`%
Kebrayl `eq`c Kebrayl zrlgankl An`n zrlgankl
na }ae~`y `eq`c na cly ekðly \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ Àra` & {+ & v0+ & v?+0 0#0?6 ?9%1 0:%1? 0#::1 02%= 2%?9 0#?0? ?=%9 2%=> 6;9 0=%1 1%;1 0#:29 ??%9 =%== 0#02; ?=%0 =%92 0#0:; ?>%9 0?%0
6 0#:0; ?:%; 0>%;; 0#??6 ?2%0 6%;0: 60? 01%= 9%>00 0#0;9 ?1%2 0:%=0? ;69 9:%1 =%:09 0#?60 ?=%1 6%=0> 0#01= ?1%2 00%=01 0#:9? ?0%6 09%20= 61= 99%= 1%6
02 ;26 02%; 0:%906 0#:02 06%9 1%90; 0#:;0 9:%0 =%9?: 0#:>6 ?;%6 1%9?0 0#0;? ?6%1 0:%>?? 0#?1= ?2%1 6%2?9 0#?01 ?=%6 ;%1?> 0#?99 ?>%9 6%9?1 ;1: 02%6 =%0
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
&Mqae~a8 Y~`~ky~kiy mlr G`e`bagae~ `en Ailelgkiy na Daccar a~ `c%# 0;;:+
Ylcqikùe8
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 653/846
Nr% Fëi~lr Pqa}anl Qrç`y
;'0;
Mkbqr` ;%9% Mkbqr` gly~r`enl cly rayqc~`nly qy`enl ac zrlbr`g` Gkek~`o na cly ~ray
glnacly zrlo`nly% &Ac`olr`ikùe zrlzk`+
&`+ Zrkgarl# ya zrqao` ac glnacl na rabraykùe zlckelgk`c iq`nrà~kil# ile ke~ar`iikùe#ay naikr# ac glnacl ilgzca~l%
^fa _abrayykle Apq`~kle ky8
&{+ 7 '0091 $ 029&V0+ $ ?9%=&V?+ ’ 9%29&V
?0+ ’ 9%6&V
??+ $ 0%;2&V0V?+
Zranki~lr Ilam YA Ilam ^ z
Iley~`e~ '009>%2 90;%6 '9%11 :%::?
&V0+ 029%?> ?6%0; =%01 :%:::
&V?+ ?9%=? 9?%?0 :%29 :%>2?
&V0YP_+ '9%2?2: :%1>?: '=%66 :%:::
&V?YP_+ '9%62?: 0%02;: '9%?6 :%::>
&V0V?+ 0%;=20 :%;>?> ?%:; :%:10
y 7 >>%=6 _'Yp 7 ;:%2! _'Yp&`nh+ 7 66%?!
Z_AYY 7 2?96:%= _'Yp&zran+ 7 6?%06!
@e`c{yky lm ]`rk`eia ^`oca
Ylqria NM YY GY M z
Nqa ~l rabrayykle 1 9=60=? 29=9? 9=%66 :%:::
_ayknq`c Arrlr 0; 92;9> 0:;2
^l~`c ?> >:=:;=
El~ 8 @pqç# c` rabkùe irç~ki` na M# ile ΰ 7 :%:1 { ile 1 { 0; br`nly na ckoar~`n# ay ?%2>%&o+ Aeyabqkn` ya zrqao` ac glnacl na rabraykùe iq`nrà~kil# yke ke~ar`iikùe# ay naikr# ac
glnacl `ora}k`nl% Ac zrlbr`g` Gkek~`o `rrlh` cly ykbqkae~ay rayqc~`nly8
^fa _abrayykle Apq`~kle ky8
&{ + 7 '0116 $ 0;6&V0+ $ 2:%6&V?+ ’ 9%;6&V
?0+ ’ >%0?&V
??+
Zranki~lr Ilam YA Ilam ^ z
Iley~`e~ '0116%9: ?=2%0 '1%69 :%:::
&V0+ 0;6%:2 ?2%=? 2%02 :%:::
&V?+ 2:%2= ?>%69 ?%61 :%:0:
&V0YP_+ '9%;;2 :%12:; '=%;2 :%:::&V?YP_+ '>%002 0%?=6 '9%?1 :%::>
y 7 >6%?; _'Yp 7 66%1! _'Yp&`nh+ 7 6=%?!
Z_AYY 7 26:1> _'Yp&zran+ 7 6:%26!
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 654/846
Nr% Fëi~lr Pqa}anl Qrç`y
;'?:
@e`c{yky lm ]`rk`eia ^`oca
Ylqria NM YY GY M z
Nqa ~l _abrayykle > 91;>=9 6;6== 96%1> :%:::
_ayknq`c Arrlr ?: >==99 ?69?^l~`c ?> >:=:;=
&i+ Mke`cgae~a# ya zrqao` ac glnacl na rabraykùe ckea`c yke ke~ar`iikùe# ay naikr# ac
glnacl ykgzca% Ac zrlbr`g` Gkek~`o rrlh` cly ykbqkae~ay rayqc~`nly8
^fa _abrayykle Apq`~kle ky8
&{ + 7 ==6 $ 00%>&V0+ $ 0=%6&V?+
Zranki~lr Ilam YA Ilam ^ z
Iley~`e~ ==2%6 09?%? 1%:1 :%:::
&V0+ 00%>?1 >%=2= ?%>> :%:?9
&V?+ 0=%6?; 2%;66 ?%00 :%:>2
y 7 000%= _'Yp 7 9?%=! _'Yp&`nh+ 7 ?=%>!
Z_AYY 7 9;?=2> _'Yp&zran+ 7 9%90!
@e`c{yky lm ]`rk`eia
Ylqria NM YY GY M z
Nqa ~l rabrayykle ? 09??19 ==0?= 1%90 :%:09
_ayknq`c Arrlr ?? ?296>> 0?>>2
^l~`c ?> >:=:;=
El~ 8 Z r` zrlo`r pqa cly ilamkikae~ay yle kbq cay# ae iq`e~l c `eàckyky na } rk`ex`# c`mqeikùe na @EL]@ zrqao` c` clebk~qn ~l~`c na c` q~kckn`n nac glnacl%
&n+ C` nayirkzikùe na cly ~ray glnacly zloc`ikle`cay# ` ay~kg`rya# zlr cly glnacly na
rabraykùe ay~`nçy~ki` yle8
0% Ac glnacl iq`nrà~kil ile ke~ar`iikùe l ilgzca~l ay8
{ 7 οl $ ο0 v0 $ ο? v? $ ο9 v?0 $ ο> v
?? $ ο1 v0 v? $ α
?% Ac glnacl iq`nrà~kil yke ke~ar`iikùe l `ora}k`nl ay8
{ 7 οl $ ο0 v0 $ ο? v? $ ο9 v?0 $ ο> v
?? $ α
9% Ac glnacl na rabraykùe ckea`c gøc~kzca ay8
{ 7 οl $ ο0 v0 $ ο? v? $ α
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 655/846
Nr% Fëi~lr Pqa}anl Qrç`y
;'?0
&a+ Ac rayqgae na cly rayqc~`nly na cly ~ray glnacly ya n` ae c` ~`oc` na `o`hl%
^@OC@ ;%9% ^`oc` gly~r`enl cly rayqc~`nly na c`y ay~`nçy~ki`y na cly ~ray glnacly na
rabraykùe zrlo`nly% &Ac`olr`ikùe zrlzk`+^kzl na glnacl _ ? y Z_AYY _ ?&`hqy~`n`+
Glnacl ilgzca~l ;:%2! >>%2 2?#96:%= 66%?!
Glnacl `ora}k`nl 66%1! >6%9 26#:1>%: 6=%?!
Glnacl ckea`c ykgzca 9?%=! 000%= 9;?#=2>%: ?=%>!
@c hqxb`r zlr cly rayqc~`nly# ~`c z`raia pqa cly gahlray glnacly i`enkn`~ly yle acglnacl ilgzca~l { ac `ora}k`nl% &Z`r` f`iar qe` naikykle mke`c qy`r c` mqeikùe &;';+%
&m+ @flr` ya }` ` kepqkrkr yk avky~a yqmkikae~a a}knaeik`# z`r` ileicqkr pqa# ac glnacl
iq`nrà~kil# ile ke~ar`iikùe# ay ac glnacl ùz~kgl% Ay~l ya naoa ` pqë# yk ` cly glnacly
na rabraykùe ya cay `brab`e }`rk`ocay keeaiay`rk`y# pqa zqnkar`e n`r qe gahlr`gkae~l
zapqaðl# el ay railgaen`oca% O`hl ay~`y ilenkikleay# c` `nkikùe na }`rk ocay
keeaiay`rk`y ilecca}` ` zrlocag`y na ilckea`ckn`n &ilrrac`ikùe ae~ra c`y }`rk`ocay
rabraylr`y+% Zlr ay~` r`xùe# ay ile}aekae~a nah`r ac glnacl na rabraykùe# cl gày ykgzca
zlykoca# ` gaely na pqa avky~`e oqae`y r`xleay ay~`nçy~ki`y z`r` `brab`rca }`rk`ocay
`nkikle`cay%
Qe` g`ear` na ra}ky`r# yk ac amai~l na ke~ar`iikùe ay eaiay`rkl# ya zrqao` ο1 &ac
ilamkikae~a na ke~ar`iikùe+ f`ikaenl qe` zrqao` na fkzù~ayky# ilgl ykbqa8
C` zrqao` na fkzù~ayky eqc` ay8 Fl8ο1 7 :
C` zrqao` na fkzù~ayky `c~are`~k}` ay8 F08ο1 ≪ :
Ile qe ek}ac ykbekmki`e~a na ΰ 7 :%:1# c` rabkùe na raif`xl ay8
s~s 5 ~ΰ,?4e'd'04 5 ~:%:1,?4?1'='04 5 ~%:?140;4 5 ?%:;9
Na c` Mkbqr` ;%9# ae c` ilcqge` na c`y zrqao`y na ~# ya }a pqa# z`r` ac amai~l na
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 656/846
Nr% Fëi~lr Pqa}anl Qrç`y
;'??
ke~ar`iikùe na & v0 v?+ ac }`clr na ^ ay kbq`c ` ?%:; ile z 7 %:10% Zlr ileykbqkae~a# naoknl
` pqa ^ 7 ?%:; ay gaelr pqa c` ^ irç~ki` na ?%:;9# ay~l kenki` pqa# c` keicqykùe nac
~ërgkel na ke~ar`iikùe ο1 v0 v? el gahlr` `c glnacl ilgzca~l# ay naikr# `c keicqkr acm`i~lr na ke~ar`iikùe% Ay~` naikykùe ely cca}` `c glnacl iq`nrà~kil# yke ke~ar`iikùe#
ilgl ac gahlr glnacl z`r` ay~a zrlocag`%
L~r` mlrg` na ilgzrlo`r cl gkygl pqa `rrko`# ya zqana f`iar qy`enl c`
aiq`ikùe &;';+% Ay~` ay~`nçy~ki` n`n` zlr Daccar a~ `c% &0;;:+ ay~à rac`ikle`n` ile c`
yqg` na cly iq`nr`nly YY# c` iq`c gkna# pqë ~`e okae aei`h`e cly n`~ly ae ac glnacl%
Ilgl ya nkhl `e~ay# ay~a zrliankgkae~l ileyky~a ae ilgz`r`r c` yqg` na cly iq`nr`nly
YY0 nac glnacl `ora}k`nl { YY? nac glnacl ilgzca~l% Zlr ahagzcl# yk YY? ay
ykbekmki`~k}`gae~a gày zapqað` pqa YY0# ya ileicq{a pqa ac glnacl ilgzca~l ay
yqzarklr `c glnacl `ora}k`nl4 na l~r` g`ear`# ya ileicqkrç` pqa# ac glnacl ilgzca~l el
yarç`# ra`cgae~a# yqzarklr% Z`r` ~`cay mkeay ya qy` c` ay~`nçy~ki` &;';+ { ya zrliana `
yqy~k~qkr cly ykbqkae~ay }`clray y`i`nly na c` Mkbqr` ;%9 ay naikr# YY0 7 >==99# YY? 7
92;9># d ? 7 1# d 0 7 ># e 7 ?1%
C` zrqao` na fkzù~ayky eqc` ay8 Fl8ο9 7 ο> 7 ο1 7 :% C` zrqao` na fkzù~ayky
`c~are`~k}` ay pqa cly ilamkikae~ay na rabraykùe el yle kbq`c ` : l# iq`enl gaely# qel
na cly ilamkikae~ay ο9# ο> { ο1 el ay kbq`c ` :% Yk ac glnacl ilgzca~l &ile ke~ar`iikùe ae
ay~a i`yl+ ay gahlr pqa ac `ora}k`nl &yke ke~ar`iikùe ae ay~` key~`eik`+# ac }`clr na YYA ?
yarà gày zapqaðl pqa YYA0# ac }`clr na M yarà br`ena# { ya raif`x`rà Fl8# { ya ileicqkrà
pqa yk f`{ a}knaeik` z`r` `mkrg`r pqa ac glnacl ilgzca~l# ile ke~ar`iikùe# ay gahlr
pqa ac glnacl yke ke~ar`iikùe% Yke ago`rbl# yk ac glnacl ilgzca~l el ayykbekmki`e~agae~a gahlr pqa ac glnacl `ora}k`nl# ae~leiay# c` rac`ikùe YYA0 ’ YYA?#
yarà# `zrlvkg`n`gae~a# kbq`c ` iarl% Zlr ileykbqkae~a# ac }`clr na M yarà zapqaðl { el
ya raif`x`rà c` fkzù~ayky eqc` Fl8 O`hl ay~`y ilenkikleay ya ileicqkrà pqa# ac glnacl
`ora}k`nl &yke ke~ar`iikùe+# ay gahlr%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 657/846
Nr% Fëi~lr Pqa}anl Qrç`y
;'?9
C` rabkùe irç~ki`# ile ΰ 7 :%:1# ay qy`enl c` nky~rkoqikùe M%
M 5 MTΰ&d?'d0+#&e'd?'0+W
5 MT:%:1&0+#&0;+W 5 >%96
@flr` qy`enl c` aiq`ikùe &;';+ { yqy~k~q{aenl cly }`clray n`8
&>=#=99 ’ 92#;9>+,&1 ’ >+M 7 „„„„„„„„„„„ 7 >%91
92#;9>,&?1 ’ 1 ’ 0+
Ae ileicqykùe# naoknl ` pqa c` Mi`ci% 7 >%91 3 Mirç~ki` 7 >%96# ya nkia pqa el f`{ a}knaeik`
z`r` `mkrg`r pqa ac glnacl na rabraykùe ile ke~ar`iikùe ay yqzarklr `c glnacl
`ora}k`nl%
Ahagzcl )9% Ac nay`rrlccl na gkirllrb`ekygly ykbqa ` qe iraikgkae~l avzleaeik`c
g`~agà~kil% Z`r` ay~l naiknkù qy`r qe glnacl iøokil# nlena [ ay ac ile~al na
gkirllrb`ekygly { V ay ac eøgarl na flr`y pqa f`e z`y`nl% Qy`r ac zrlbr`g` Gkek~`o
z`r` ~`cay zrlzùyk~ly%
Ylcqikùe8
C` aiq`ikùe ay8 [ 7 '6%0: $ 0?%2 V ’ :%;:1& V ?+ $ ?%0>& V 9+
y 7 >0%6>1 _ ? 7 :%;;6 _ ?&`hqy~`n`+ 7 ;;%6!
^@OC@ ;%>% ^`oc` na `eàckyky na }`rk`ex`% &Ac`olr`ikùe zrlzk`+
Mqae~a na }`rk`ikùe b%c% YY GY Mi`ci%
Naoknl ` c` rabraykùe 9 0?#990#606 >#00:#=:= 092:?:?
_ayknql &arrlr+ 09 ??#2=: 0#210
^l~`c 0= 0?#91>#126
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 658/846
Nr% Fëi~lr Pqa}anl Qrç`y
;'?>
Glnacly na rabraykùe el ckea`cay { na rabraykùe clbçy~ki`
Nae~rl na ay~` i`~ablrç`# f`{ glnacly na rabraykùe avzleaeik`c { glnacly na
rabraykùe clbçy~ki`% Ac glnacl na rabraykùe avzleaeik`c ya qy` ae ay~qnkly rac`ikle`nlyile ac iraikgkae~l na `cbøe zrliayl# nlena c` ~`y` na iraikgkae~l# ` qe ~kagzl V n`nl#
ay zrlzlrikle`c ` c` i`e~kn`n na iraikgkae~l pqa pqan`# ` gankn` pqa ac ~kagzl ya
keiragae~`% L~rl qyl ay ac ay~qnkl na c` rac`ikùe ae~ra c` ileiae~r`ikùe na qe` yqy~`eik`
& {+# ae mqeikùe nac ~kagzl ~r`eyiqrrknl & V +% Ae mlrg` `eàclb`# cly glnacly na rabraykùe
clbçy~ki` ya qy`e ae ay~qnkly zloc`ikle`cay# z`r` rac`ikle`r ac eøgarl na ayzaikay &[ +
ae mqeikùe nac ~kagzl & V +% Ay~ly glnacly ~`gokëe ya zqanae qy`r iq`enl c` }`rk`oca
nazaenkae~a ay iq`ck~`~k}`4 yke ago`rbl# ay~ly glnacly el ya nkyiq~kràe ae ay~a ~av~l#
ilgl ae ac i`yl na cly glnacly avzleaeik`cay%
Glnacly na rabraykùe avzleaeik`cay z`r`gë~rkily# ile qe` ylc` }`rk`oca
kenazaenkae~a &Ea~ar a~ c% 0;;=+
[ k 7 ιl avz&ι0 V k+ $ αk &;'0:+
Qe glnacl gày baear`ckx`nl na rabraykùe avzleaeik`c el ckea`c ay8
[ k 7 ιl $ ι0 avz&ι? V k+ $ αk &;'00+
Nlena8
[ k ay c` mqeikùe na rayzqay~` l }`rk`oca nazaenkae~a
ιl# ι0# ι? 7 cly z`ràga~rly ` ay~kg`rya zlr `# o { i
V k 7 }`rk`ocay iley~`e~ay
αk 7 arrlr l rayknql elrg`cgae~a nky~rkoqknl { ile }`rk`ex` iley~`e~a
Ac ilrrayzlenkae~a ay~kg`nlr ay~`nçy~kil ay8 { 7 ` $ o avz&'iv+ $ ak &;'0?+
Nlena8
`# o# i 7 ay~kg`nlray ay~`nçy~kily na ιl#ι0 { ι?# rayzai~k}`gae~a
ak 7 ac arrlr l rayknq`c ay~`nçy~kil%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 659/846
Nr% Fëi~lr Pqa}anl Qrç`y
;'?1
Cly glnacly avzleaeik`cay ya qy`e ae ahagzcly na iraikgkae~l pqa }` ae
mqeikùe nac ~kagzl &ilgl ac iraikgkae~l na o`i~ark`y ae qe zc`~l na `b`r# l z`r`
nayirkokr ac iraikgkae~l avzleaeik`c na cly b`yay na ke}are`narl ilgl ac IL?# aemqeikùe nac ~kagzl# pqa ay~àe li`ykle`enl ac i`cae~`gkae~l bclo`c { c` ilrrqzikùe nac
ickg` nac zc`ea~`+%
Ile rac`ikùe ` cly glnacly na rabraykùe clbçy~kily# c` mqeikùe zloc`ikle`c pqa
nayirkoa ay~ly glnacly ya n` ilgl &Ea~ar a~ `c% 0;;=+8
ιl [ k 7 „„„„„„„„ $ α &;'09+
0 $ ι0 avz&ι? V k+Nlena8
[ k 7 mqeikùe na rayzqay~`
ιl 7 z`ràga~rl ` ay~kg`rya zlr yqy ay~`nçy~ki` ilrrayzlenkae~a
ι0 7 z`ràga~rl ` ay~kg`rya zlr yq ay~`nçy~ki`
ι? 7 z`ràga~rl ay~kg`rya zlr yqy ay~`nçy~ki`
C` a}`cq`ikùe { ay~kg`ikùe na cly z`ràga~rly na rabraykùe el ckea`c ya f`iae
kbq`c pqa ile c` rabraykùe ckea`c% Zlr ahagzcl# z`r` ac nk`beùy~kil yqoha~k}l# ya `e`ckx`e
c`y bràmki`y z`r` c` zrqao` na elrg`ckn`n# cly bràmkily na cly rayknqly ae mqeikùe nac
~kagzl# { ~`gokëe ae mqeikùe na cly }`clray `hqy~`nly# a~i% Yke ago`rbl# ae c`
ke~arzra~`ikùe na c`y bràmki`y na cly rayknq`cay na c` rabraykùe el ckea`c f`{ pqa railrn`r
pqa# cly rayknq`cay# el eaiay`rk`gae~a# yqg`e ` iarl% ^`gokëe# ya zqanae f`iar
~r`eymlrg`ikleay na c`y }`rk`ocay z`r` f`iar qe gahlr `hqy~a nac glnacl yqzarklr% Ae
iq`e~l ` c`y kemaraeik`y ay~`nçy~ki`y# ile c` rabraykùe el ckea`c# ya o`y`e ae c` ~alrç` nagqay~ral br`ena# ay~l ay# ile ~`g`ðly na gqay~r`y br`enay% C`y mqeikleay na glnacly
na rabraykùe avzleaeik`c# ~`gokëe ya zqanae ~r`~`r nae~rl nac
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 660/846
Nr% Fëi~lr Pqa}anl Qrç`y
;'?=
~ùzkil na yarkay na ~kagzl%
Ahagzcl )>% Ay~a ay qe zrlocag` y`i`nl nac ckorl Y~`~ky~kiy mlr G`e`bagae~ `en
Ailelgkiy na Daccar &0;;:+# R`rr`id { O`r~ac rac`ikle`nl ile ac `hqy~`gkae~l {`eàckyky na glnacly na rabraykùe el ckea`cay rac`ikle`nl ile n`~ly ay~`nçy~kily nac
YKN@ ae cly Ay~`nly Qeknly na Elr~a`gërki`# ae mqeikùe nac ~kagzl%
El~ 8 Kenazaenkae~agae~a na cly n`~ly n`nly zlr ac q~lr na ay~a zrlocag # ` hqkikl nac
nac `q~lr na ay~a ckorl na ay~`nçy~ki`# ac elgora gënkil ile}aeikle`c nac `irùekgl
YKN@ &@KNY ae c`y ykbc`y nac kebcëy# ay naikr# `ipqkran kegqelnamkikaei{ yçenrlga+
ay~à nkikaenl pqa ac cc`g`nl YKN@ ay qe` aemargan`n l qe` namkikaeik`# ae z`r~kiqc`r#
nac yky~ag` kegqea nac iqarzl% Ay~l el ay zlykoca# zlrpqa ac yky~ag` kegqelcùbkil nac
iqarzl ay qe` z`r~a nazaenkaea na ~lnl ac lrb`ekygl# ilgl qekn`n kenazaenkae~a% Yk
ay~a ~ërgkel –YKN@‗ mqar` ilrrai~l# ae~leiay# ya ~aenrç` pqa naikr pqa ac yky~ag`
kegqea nac iqarzl ay qe` z`r~a kenazaenkae~a nac ray~l nac lrb`ekygl# { el qe` z`r~a
nazaenkae~a na ~lnl ac iqarzl ilgl qekn`n kenazaenkae~a% Ae ~ërgkely gày ykgzcay#
ay~l ykbekmki` pqë# ac lrb`ekygl fqg`el ay~à ilgzqay~l zlr ùrb`ely# z`r~ay l yky~ag`y
ile~kebae~ay# iq{` mqeikùe# ae ~qrel# nazaena na ~lnl ac iqarzl ae~arl# ilgl qekn`n
kenazaenkae~a# ay naikr# iq`enl ac lrb`ekygl ay~à ae qe ay~`nl na y`cqn zarmai~l% Na
`iqarnl ` ay~a r`xle`gkae~l# ac cc`g`nl YKN@ ay qe yçe~lg` na aemargan`n &zarl el
na qe` aemargan`n ae z`r~kiqc`r+# pqa `iqy` pqa ~lnl ac iqarzl ay~à aemargl &~ln` c`
qekn`n lrbàeki` nky~lrykle`n` zlr }kn` `e~ke`~qr`c+# el øeki`gae~a# ac yky~ag`
kegqelcùbkil# ilgl ilgøegae~a ya iraa% Na g`ear` pqë# z`r` iqr`r cly yçe~lg`y na
ay~a g`c# ay eaiay`rkl iqr`r ~lnl ac ilgzcahl lrbàekil# ` ~r`}ëy na `r~ay gënki`ye`~qr`cay% C` cùbki` ykagzra ileyah`rà pqë# z`r` iqr`r qe amai~l &yçe~lg`y nac YKN@+#
zrkgarl f`{ pqa ~aenar ac lrkbae i`qy c gày raiùenk~l# pqa el ay l~r` ily` gày pqa c`
}kn` el e`~qr`c% @c zrlianar na l~r` g`ear`# ykagzra f`orà ilgzcki`ikleay pqa
`br`}`ràe ac zrlocag` nac aemargl%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 661/846
Nr% Fëi~lr Pqa}anl Qrç`y
;'?2
^@OC@ ;%1% ^`oc` gly~r`enl cly n`~ly na ay~a zrlocag`%
@ðly Zarklnl na ~kagzl ~ Eøgarl na i`yly na YKN@
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ 0;60 0 0#:::0;6? ? =#:::0;69 9 0:#:::0;6> > 0>#:::0;61 1 ?1#:::0;6= = >6#:::0;62 2 =9#:::0;66 6 0:6#:::0;6; ; 0=0#:::
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ &Mqae~a8 Daccar a~ `c%# 0;;:+
F`iar cly ykbqkae~ay iàciqcly8
&`+ F`iar qe` bràmki` ile cly n`~ly { lo~aear c` aiq`ikùe na rabraykùe nac glnacl
`zrlzk`nl% Zlear c` aiq`ikùe ylora c` bràmki`%
&o+ Zranaikr ac eøgarl na i`yly na YKN@ z`r` ac `ðl ?::: &~ 7 ?:+%
Ylcqikùe8
&`+ Qy`enl ac zrlbr`g` Gkek~`o ile qe` mqeikùe na rabraykùe ay~`nçy~ki` na yarkay
na ~kagzl { `eàckyky na ~aenaeik` &~raen `e`c{yky+# n` c` bràmki` { c` aiq`ikùe
yað`c`n` o`hl%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 662/846
Nr% Fëi~lr Pqa}anl Qrç`y
;'?6
Mkbqr` ;%>% Bràmki` gly~r`enl cly i`yly na YKN@# ae mqeikùe nac ~kagzl na ay~a
zrlocag`% &Ac`olr`ikùe zrlzk`+
&o+ Iq`enl ~ 7 ?: &`ðl ?:::+# ac eøgarl na i`yly na YKN@ yarç`8
{ 7 &0?;:%6>+&0%21;2>+?:
7 0:>#=2>#6;>%;
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 663/846
Nr% Fëi~lr Pqa}anl Qrç`y
;'?;
@hqy~`gkae~l na iqr}`y
Ae ac `hqy~`gkae~l na iqr}`y# z`r` yacaiikle`r ac glnacl pqa gahlr aei`h` ae cly n`~ly
ya n`e `o`hl }`rkly ~kzly na iqr}`y% Ay~`y mqeikleay `{qn`e ` yacaiikle`r c` mlrg` gày`zrlzk`n` z`r` cly n`~ly% Ay~ly ~kzly na iqr}`y yle yqbarknly zlr ac zrlbr`g` na
ilgzq~`nlr` EIYY%
Mkbqr` ;%1% Bràmki`y gly~r`enl cly nkmarae~ay ~kzly na mqeikleay qy`nly ae cly`hqy~ay na iqr}`y# z`r` yacaiikle`r ac gahlr glnacl na rabraykùe pqa zqan` aei`h`r ae cly n`~ly%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 664/846
Nr% Fëi~lr Pqa}anl Qrç`y
;'9:
Mkbqr` ;%=% Bràmki`y gly~r`enl cly nkmarae~ay ~kzly na iqr}`y qy`nly ae ac `hqy~`gkae~lna glnacly na rabraykùe gày `zrlzk`nly%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 665/846
Nr% Fëi~lr Pqa}anl Qrç`y
;'90
Zrliankgkae~ly z`r` c` Knae~kmki`ikùe na }`clray `~çzkily av~ragly% Nk`beùy~kil
{ gk~kb`ikùe na gqc~kilckea`ckn`n
Cly zrliankgkae~ly z`r` ramke`r ac glnacl na rabraykùe yle c` knae~kmki`ikùe {ackgke`ikùe na }`clray keqyq`cay av~ragly% Ae `cbqe`y li`ykleay# ay~ly }`clray
av~ragly ya aeiqae~r`e ae c` baear`ikùe na n`~ly gqay~r`cay% Ay~ly }`clray av~ragly ya
ramkarae ` n`~ly qek}`rk`nly pqa yle keileyky~ae~ay ile ac ray~l na c` kemlrg`ikùe% Cly
}`clray av~ragly liqrrae ` gaeqnl naoknl ` arrlray na gankikùe# {` ya` zlr g`c
mqeikle`gkae~l nac apqkzl l zlr eabckbaeik` nac zaryle`c# m`c~` na g`e~aekgkae~l na
cly key~rqgae~ly# a~i% Ae rabraykùe gøc~kzca# cly }`clray av~ragly zqanae liqrrkr ile
c`y }`rk`ocay kenazaenkae~ay { ile c` }`rk`oca nazaenkae~a% Ay~ly }`clray# qe` }ax
`e`ckx`nly ya zqanae ackgke`r l ra~aear# yk ya y`oa pqa yle# ae ra`ckn`n# }`clray
av~ragly% Ykaenl `yç# ay eaiay`rkl ackgke`rcly# zlrpqa zqanae nky~lrykle`r ac glnacl
na rabraykùe hqy~`nl l i`qy`r yarkly arrlray ae cly iàciqcly na rabraykùe%
C` knae~kmki`ikùe na }`clray av~ragly ya zqana f`iar na c`y ykbqkae~ay g`ear`y8
0% Qy`enl bràmki`y na ~`ccl { flh`%
?% Qy`enl bràmki`y na i`h`%
9% Qy`enl bràmkily na zrlo`okckn`n elrg`c%
>% Qy`enl c` ay~`nçy~ki` NMK^Y pqa knae~kmki` }`clray av~ragly zl~aeik`cay# iq`enl
NMK^Y 5 ? e z , # nlena z ay ac eøgarl na }`rk`ocay kenazaenkae~ay {# e# ay ac ~`g`ðl
na c` gqay~r`%
1% Qy`enl bràmkily na rayknqly yagkay~qnae~kx`nly# cly iq`cay knae~kmki`e cly }`clray
av~ragly# iq`enl cly }`clray `oylcq~ly na cly rayknq`cay yagkay~qnae~kx`nly yle ≡ >%=% Qy`enl NMOA^@Y iq`enl ay~ly }`clray yle ≡ °?, e %
2% Qy`enl cly bràmkily na _y~qnae~ }y% F`~ Nk`ble`c%
6% Qy`enl rabraykùe rloqy~` &rloqy~ rabrayykle+ railgaen`nly zlr c` cùbki` nac
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 666/846
Nr% Fëi~lr Pqa}anl Qrç`y
;'9?
zrlbr`g` EIYY%
;% ^`gokëe ya zqana f`iar qy`enl ac }`clr irç~kil na Olemarrlek# pqa knae~kmki` cly
}`clray `oylcq~ly na cly rayknq`cay ay~qnae~kx`nly% Ay~` zrqao` ik~`n` zlr Ea~ar a~ `c%&0;;=+ ya n` ilgl ~&0 ’ ΰ,?e4e ’ z ’ 0+%
0:% ^`gokëe ya f`ia ile c` ay~`nçy~ki` Illd¾y Nky~`eia &cùbki` nac zrlbr`g` EIYY+# c`
iq`c nkia pqa# yk ëy~` ay g`{lr pqa M&%1:#z#e'z+# nlena M ay qe }`clr na c` nky~rkoqikùe M#
ae~leiay# ay~l yqbkara qe }`clr av~ragl%
00% Cly }`clray av~ragly ~`gokëe ya zqanae knae~kmki`r ile cly bràmkily na cly rayknqly
pqa }`e ae mqeikùe na V l na [ %
Nk`beùy~kil na gqc~kilckea`ckn`n
Ae rabraykùe gøc~kzca f`{ cl pqa ya cc`g` ilckea`ckn`n# gqc~kilckea`ckn`n l
ke~arilrrac`ikùe% Ay~` yk~q`ikùe avky~a iq`enl c`y }`rk`ocay kenazaenkae~ay ay~àe
ilrrac`ikle`n`y ae~ra yk% Cl kna`c ae rabraykùe gøc~kzca ay na pqa c`y }`rk`ocay
kenazaenkae~ay v0 # v? #…# vde el ay~ëe ilrrac`ikle`n`y# na ~`c g`ear` pqa# i`n` qe`
avzckpqa qe zlriae~`ha yaz`r`nl na c` }`rk`ikùe ae c` }`rk`oca nazaenkae~a%
Ac g`c amai~l na gqc~kilckea`ckn`n ay pqa c`y nay}k`ikleay ay~àen`r na cly
ilamkikae~ay nac glnacl na rabraykùe ay~àe yloraay~kg`n`y% Ilgl rayqc~`nl na ay~l#
iq`enl ya f`iae c`y zrqao`y na fkzù~ayky# c` ay~`nçy~ki` ~ ay gày zapqað` na cl pqa
naoarç` yar% @nagày# `cbqe`y }`rk`ocay kenazaenkae~ay l avùbae`y `z`raiae ilgl yk el
ay~q}kar`e rac`ikle`n`y ckea`cgae~a ile c` }`rk`oca [ # iq`enl ae ra`ckn`n yk cl ay~àe%
Avky~ae nly gë~lnly z`r` nayiqorkr c` gqc~kilckea`ckn`n# ay naikr# gë~lnly
kemlrg`cay { gë~lnly mlrg`cay% Cly gë~lnly kemlrg`cay z`r` na~ai~`r ilckea`ckn`nya}ar` yle8
0% Ay~qnkly na cly ykbely `cbaor`kily na cly ilamkikae~ay nac glnacl na rabraykùe% Yk
f`{ ilckea`ckn`n# cly ykbely `cbaor`kily na cly ilamkikae~ay yle lzqay~ly# ` cl pqa ya
naoarç` ayzar`r na ileyknar`ikleay ~aùrki`y l na avzarkaeik` ` zly~arklrk%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 667/846
Nr% Fëi~lr Pqa}anl Qrç`y
;'99
?% L~r` yk~q`ikùe pqa zqnkar` kenki`r gqc~kilckea`ckn`n ay ac faifl na pqa liqrrae
br`enay i`gokly ae cly ilamkikae~ay ay~kg`nly na rabraykùe# iq`enl qe` }`rk`oca
avzcki`~k}` l kenazaenkae~a ya `brab` l ya ackgke`%9% Iq`enl ya f`iae zrqao`y na fkzù~ayky na Fl8ο¾y 7 :# c`y zrqao`y na ~ el yle
ykbekmki`e~ay% Ay~` ilenkikùe ~`gokëe zqnkar` kenki`r ilckea`ckn`n%
>% Iq`enl f`{ br`enay ilrrac`ikleay ae~ra z`ray na }`rk`ocay kenazaenkae~ay# ay~`
yk~q`ikùe ~`gokëe kenki` gqc~kilckea`ckn`n%
1% Ile c` cùbki` nac glnacl na ilgzq~`nlr` EIYY# ile eøgarly na cly Akbae}`cqay
g`{lray pqa 0:::# ay~` ilenkikùe kenki` ilckea`ckn`n ya}ar`% Yke ago`rbl# ile }`clray
na Akbae}`cqay ae~ra 0:: { 0:::# ay~` ilenkikùe kgzcki` ilckea`ckn`n glnar`n` `
mqar~a%
=% Eqa}`gae~a# ile c` cùbki` nac zrlbr`g` EIYY# ae c` yaiikùe na ilrrac`ikùe na
g`~rkiay# br`enay ilrrac`ikleay ae~ra c`y }`rk`ocay avzcki`~k}`y ilecca}`e nk`beùy~kily
na ilckea`ckn`n%
2% Cly }`clray av~ragly# ~`gokëe zqanae i`qy`r zrlocag`y na ilckea`ckn`n%
Zlr l~r` z`r~a# cly gë~lnly mlrg`cay z`r` na~ai~`r gqc~kilckea`ckn`n yle cly
m`i~lray na kemc`ikùe na }`rk`ex` &]`rk`eia Kemc`~kle M`i~lry# ]KM+% Ae ay~a ile~av~l# ac
zrlocag` na gqc~kilckea`ckn`n ya ileyknar` ya}arl# iq`enl ac gàvkgl } clr na ]KZ ay
g`{lr pqa 0: l okae# iq`enl ac zrlgankl na cly ]KM¾y ay ileyknar`ocagae~a 5 0%
&Zm`mmaeoarbar# 0;62+%
Ae iq`e~l ` yk~q`ikleay rac`ikle`n`y ile c` gqc~kilckea`ckn`n ya aecky~`e cly
ykbqkae~ay zly~qc`nly80% Yk ac glnacl ya }` ` qy`r# øeki`gae~a# z`r` ay~kg`r rayzqay~`y zrlgankl l z`r` f`iar
zrankiikleay na cly }`clray na c` }`rk`oca nazaenkae~a [ # { c`y zrankiikleay yle
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 668/846
Nr% Fëi~lr Pqa}anl Qrç`y
;'9>
faif`y# ylc`gae~a# ylora c`y rabkùe na cly }`clray na c`y }`rk`ocay kenazaenkae~ay {# cly
ilamkikae~ay na rabraykùe ay~kg`nly el ya qy`ràe z`r` zrlzùyk~ly na ke~arzra~`ikùe#
ileiarekaenl c`y rac`ikleay na c`y }`rk`ocay avzcki`~k}`y & V¾ y+ { na c` }`rk`oca narayzqay~` &[ +# ae~leiay# c` gqc~kilckea`ckn`n# `qe iq`enl ya` ya}ar`# el yarà qe
zrlocag` &Zm`mmaeoarbar# 0;62+% @pqç# yke ago`rbl# c` na~argke`ikùe na c` rabkùe
gqay~ra`n` ay nkmçikc% Zlr ahagzcl# yk f`{ qe` }`rk`oca kenazaenkae~a# ae~leiay# c`
rabkùe ay qe ke~ar}`cl ylora c` cçea` ra`c ae~ra ac }`clr gçekgl na v { ac }`clr gàvkgl
na v ae c` gqay~r`% @nagày# ile iq`~rl }`rk`ocay kenazaenkae~ay# c` rabkùe gqay~ra`n`
ay ae ac ayz`ikl na iq`~rl nkgaeykleay na c`y v¾y { yqy ckenarly el yle lo}kly% Zlr cl
~`e~l# o`hl ay~`y ilenkikleay# f`{ pqa ahariar zrai`qikùe# na ~`c g`ear` pqa# c`
zrankiikùe el razrayae~a qe` av~r zlc ikùe gày ccà na c` rabkùe gqay~ra`n` na c`y v¾y#
iq`enl avky~a gqc~kilckea`ckn`n ya}ar`% Zlr l~r` z`r~a# yk ya naya` f`iar
ke~arzra~`ikleay na cly ilamkikae~ay na ilrrac`ikùe &ok+# ae~leiay c` gqc~kilckea`ckn`n el
ya zqana ~lcar`r%
?% Ac faifl na pqa `cbqely l ~ln`y c`y }`rk`ocay kenazaenkae~ay ay~ëe ilrrac`ikle`n`y
ae~ra yk# ae baear`c# el loy~rq{a c` f`okckn`n z`r` lo~aear qe oqae `hqy~a na cly n`~ly%
Ay~` yk~q`ikùe ~`gzlil ke~armkara ae c`y kemaraeik`y `iari` na c`y rayzqay~`y zrlgankl
na zrankiikleay na eqa}`y loyar}`ikleay# ykagzra { iq`enl# ay~`y kemaraeik`y ya`e
faif`y nae~rl na c` rabkùe na c`y loyar}`ikleay%
9% Iq`enl c`y }`rk`ocay kenazaenkae~ay ay~àe `c~`gae~a ilrrac`ikle`n`y# cly
ilamkikae~ay na rabraykùe ay~kg`nly ~kaenae ` ~aear qe` br`e }`rk`ikùe na gqay~ral% Zlr
cl ~`e~l# o`hl ay~`y ilenkikleay# cly ilamkikae~ay na rabraykùe ~kaenae ` }`rk`r `gzck`gae~a na qe` gqay~r` ` l~r`% Ilgl rayqc~`nl na ay~l# ylc`gae~a# ya lo~kaea
kemlrg`ikùe kgzraiky` `iari` na cly ilamkikae~ay kenk}knq`cay%
>% Iq`enl f`{ gqc~kilckea`ckn`n# c` ke~arzra~`ikùe na qe ilamkikae~a na rabraykùe#
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 669/846
Nr% Fëi~lr Pqa}anl Qrç`y
;'91
ilgl gankn` na qe i`gokl ae ac }`clr ayzar`nl ae [ # iq`enl qe` }`rk`oca
kenazaenkae~a# nkb`gly V 0 ya keiragae~` zlr qe` qekn`n# g`e~aekaenl iley~`e~ay c`y
nagày }`rk`ocay# el ay ~l~`cgae~a `zcki`oca%1% L~rly amai~ly i`qy`nly zlr c` gqc~kilckea`ckn`n ay~àe rac`ikle`nly ile c` yqg` na
cly iq`nr`nly# cly amai~ly ae cly ilamkikae~ay na na~argke`ikùe z`rik`c# amai~ly ae ac
arrlr ay~àen`r na cl ay~kg`nl y# amai~ly ylora cly }`clray `hqy~`nly# amai~ly ae c`y
zrqao`y ykgqc~àea`y na cly ilamkikae~ay ο¾y# a~i% &Ea~ar a~ `c% 0;;=+%
Gankn`y z`r` ilrrabkr gqc~kilckea`ckn`n ya}ar`
0% Ac gë~lnl gày lo}kl z`r` ragank`r c` gqc~kilckea`ckn`n ay ac na el keicqkr ae ac
glnacl c`y }`rk`ocay kenazaenkae~ay pqa ay~àe `c~`gae~a ilrrac`ikle`n`y% Ay~l ya f`ia
z`r` ranqikr cly arrlray ay~àen`r na cly ilamkikae~ay na rabraykùe ay~kg`nly na c`y
}`rk`ocay kenazaenkae~ay pqa pqanae ae ac glnacl% Yke ago`rbl# ay~a ragankl ~kaea
nly ckgk~`ikleay zlrpqa# na ay~` g`ear`# {` el f`orà kemlrg`ikùe nkrai~` na c` }`rk`oca
kenazaenkae~a avicqkn`% Ae yabqenl cqb`r# c`y g`bek~qnay na cly ilamkikae~ay na
rabraykùe# z`r` cly ilamkikae~ay ray~`e~ay yle `mai~`n`y zlr c`y }`rk`ocay kenazaenkae~ay
ilrrac`ikle`n`y# pqa el ya keicq{`e ae ac glnacl%
?% L~rl gë~lnl z`r` ilrrabkr c` gqc~kilckea`ckn`n ya ramkara ilgl rabraykùe na ikg`
&rknba rabrayykle+% Ykaenl `yç# iq`enl f`{ gqc~kilckea`ckn`n# cly ay~kg`nly na cly
iq`nr`nly gçekgly yle kgz`rik`cay# zarl yqy }`rk`ex`y yle br`enay# na ~`c g`ear` pqa
zqan`e ay~ r `cah nly nac }`clr }arn`narl% @brab`enl qe br`nl na z`rik`ckn`n ` cly
ay~kg`nly na c` rabraykùe# c` rabraykùe na ikg` &l rknba rabrayykle+ ranqia cly arrlray
ay~àen`ray# na ~`c g`ear` pqa ac amai~l ea~l n`rà ilamkikae~ay ay~kg`nlray gàyilemk`ocay &Ea~ar a~ `c% 0;;=+%
9% L~rl gë~lnl z`r` ranqikr c` gqc~kilckea`ckn`n ya}ar` ay c` rabraykùe zlr z`yly% C`
rabraykùe zlr z`yly keicq{a# ylc`gae~a# c`y }`rk`ocay kenazaenkae~ay pqa ay~àe
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 670/846
Nr% Fëi~lr Pqa}anl Qrç`y
;'9=
ykbekmki`e~agae~a rac`ikle`n`y ckea`cgae~a# ile c` }`rk`oca nazaenkae~a% Ay~l ~kaena `
ranqikr c` ilckea`ckn`n zlrpqa# yk f`{ nly }`rk`ocay kenazaenkae~ay# `c~`gae~a
ilrrac`ikle`n`y ae~ra yk# `c keicqkr qe`# qyq`cgae~a ya ackgke` c` yabqen`% Ae acgai`ekygl# ae c` rabraykùe zlr z`yly# qe` }`rk`oca kenazaenkae~a# ` qe ~kagzl# ay
keicqkn` ae c` aiq`ikùe% Ae ac z`yl 0# c` }`rk`oca kenazaenkae~a# gày mqar~agae~a
rac`ikle`n` ile c` }`rk`oca nazaenkae~a# ay keicqkn` ae ac glnacl% Ae ac z`yl ?# c`
ykbqkae~a }`rk`oca kenazaenkae~a &ae~ra c`y }`rk`ocay kenazaenkae~ay ray~`e~ay+ gày
mqar~agae~a rac`ikle`n` ile c` }`rk`oca nazaenkae~a# ya keicq{ae ae ac glnacl% Ay~`
yk~q`ikùe ile~keø` f`y~` pqë# ylc`gae~a# c`y }`rk`ocay kenazaenkae~ay# pqa el ay~àe
rac`ikle`n`y ile c` }`rk`oca nazaenkae~a &n`nl pqa c`y l~r`y }`rk`ocay {` ay~àe ae ac
glnacl+ zarg`eaiae mqar` na c` aiq`ikùe% Na iq`cpqkar g`ear`# z`r` a}k~`r zrlocag`y#
c` rabraykùe zlr z`yly naoa qy`rya ae ilehqeikùe ile qe zrlmqenl r`xle`gkae~l
ay~`nçy~kil%
C` zrabqe~` na iq`e~`y }`rk`ocay kenazaenkae~ay &keicq{aenl c`y }`rk`ocay
~r`eymlrg`n`y+ naoae na keicqkrya ae ac glnacl na rabraykùe ay ac ~ag` ` ~r`~`r# iq`enl
ya f`oc` na cly zrliankgkae~ly qy`nly ae ac zrlbr`g` Gkek~`o# ilgl –^ln`y c`y
_abraykleay Zlykocay‗ &@cc Zlyykoca _abrayykley+# –_abraykùe zlr Z`yly‗ &Y~azkya
_abrayykle+ { –_abraykùe na cly Gahlray Ilehqe~ly‗ &Oay~ Yqoya~ _abrayykle+%
Z`r` aeile~r`r ac eøgarl kna`c na }`rk`ocay kenazaenkae~ay# ay~l ke}lcqir` nly
loha~k}ly lzqay~ly% Zrkgarl ya naya` pqa ac glnacl na rabraykùe ya` cl gày ilgzca~l
{ ra`cky~` zlykoca% Ay~l nkia pqa ya naoa keicqkr i`n` }`rk`oca kenazaenkae~a# `qepqa
z`raxi` ragl~`gae~a rac`ikle`n` ile c } rk`oca nazaenkae~a% Ae yabqenl ~ërgkel# yanaoa na keicqkr cl gaely zlykoca na }`rk`ocay kenazaenkae~ay% Ay~l ya naoa ` pqa# i`n`
}`rk`oca kenazaenkae~a# pqa el ya` raca}`e~a `c glnacl# nkygkeq{a c` zraikykùe
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 671/846
Nr% Fëi~lr Pqa}anl Qrç`y
;'92
na cly ilamkikae~ay i`ciqc`nly { na cly }`clray zrlely~ki`nly% Na ay~` g`ear`# c`
mke`ckn`n na c` yacaiikùe na c`y }`rk`ocay ay z`rykglekly`# ay~l pqkara naikr pqa naoa
f`oar qe o`c`eia ae~ra cl ykgzca &cl gaely zlykoca na }`rk`ocay+ { ac `hqy~a &c` keicqykùena ~ln`y c`y }`rk`ocay pqa ya`e zar~keae~ay+%
F`{ nkmarae~ay ay~r`~abk`y z`r` c` yacaiikùe na c`y }`rk`ocay gày `zrlzk`n`y z`r`
ac glnacl na rabraykùe% Zlr ahagzcl ac glnacl EIYY railgkaen` pqa# yk el f`{ gày
na pqkeia i`enkn`~ly na }`rk`ocay kenazaenkae~ay &yke keicqkr ac ke~ariaz~l+# ae~leiay#
ya naoa qy`r ac zrliankgkae~l na –^ln`y c`y _abraykleay Zlykocay‗ &@cc Zlyykoca
_abrayykley+% Ay~l ya naoa ` pqa ay~a zrliankgkae~l n`rà glnacly ~`e oqaely l
gahlray pqa ac zrliankgkae~l na –_abraykùe zlr Z`yly‗% Yke ago`rbl# yk f`{ gày na
pqkeia i`enkn`~ly na }`rk`ocay# ae~leiay# ya railgkaen` ac zrliankgkae~l na
–_abraykùe zlr Z`yly‗ &Y~azkya _abrayykle+% L~r` mqeikùe n`n` zlr ac zrlbr`g`
Gkek~`o ay~à rac`ikle`n` ile c` –_abraykùe na Gahlray Ilehqe~ly‗ &Oay~ Yqoya~y
_abrayykle+%
Nayzqëy na pqa ya f`{` mlrg`nl qe ilehqe~l na i`enkn`~ly na }`rk`ocay
kenazaenkae~ay &qe` }ax pqa ya ackgke`rle c`y loyar}`ikleay av~rag`y { ya gk~kbù c`
gqc~kilckea`ckn`n+# c` ykbqkae~a ~`ra` ay c` na ay~`ocaiar qe` o`ya z`r` ilgz`r`r nly
glnacly mke`cky~`y% ¼Iùgl ya zqana naikr# yk ac glnacl @ ay gahlr pqa ac glnacl O<
Z`r` f`iar ay~` naikykùe irç~ki` ac ileyaeyl na ke}ay~kb`nlray na ay~`nçy~ki` ay~à o`y`nl
ae c`y mqeikleay ay~`nçy~ki`y ik~`n`y `e~arklrgae~a# ilgl _ ?# y# Z_AYY# a~i% Ilgl {`
ya avzckiù `e~arklrgae~a# ay~`y mqeikleay yle8
&`+ Ac ilamkikae~a na na~argke`ikùe _ ?
&o+ Ac arrlr ay~àen`r ay~kg`nl y
&i+ Ac irk~arkl Iz na G`ccl
&n+ Z_AYY
L~rly irk~arkly yle cly }`clray na ~# ~`oc`y na @EL]@# `eàckyky na bràmkily# a~i%# zarl
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 672/846
Nr% Fëi~lr Pqa}anl Qrç`y
;'96
cly zaeøc~kgly iq`~rl nk`beùy~kily yle cly gày zlzqc`ray%
Ahagzcly na zrlocag`y na rabraykùe zlckelgk`c qy`enl ac zrlbr`g` na
ilgzq~`nlr` Gkek~`o
Ahagzcl )01% Ay~a zrlocag` ay~à rac`ikle`nl ile qe avzarkgae~l nac ileyqgl na
b`ylcke` qy`enl c` }aclikn`n o`h` &l}arnrk}a+ na qe` i`gklea~` ck}k`e`% @pqç c`
}`rk`oca kenazaenkae~a ay c` }aclikn`n iley~`e~a n`n`# ae gkcc`y zlr flr` & V +% @nagày#
c` }`rk`oca nazaenkae~a &[ +# ay~à rac`ikle`n` ile c`y gkcc`y zlr b`cùe lo~aekn`y o`hl
ay~`y ilenkikleay na g`eahl% F`iar cly ykbqkae~ay iàciqcly8
&`+ Br`mki`r cly n`~ly%
&o+ @hqy~`r qe glnacl iq`nrà~kil%
&i+ @hqy~`r qe glnacl iøokil%
&n+ Ilgzcagae~`r ac nk`beùy~kil nac keikyl &n+ ile cly `eàckyky na cly bràmkily
yqoha~k}ly z`r` ac glnacl yqzarklr%
&m+ Na `iqarnl ` cly `eàckyky na cly irk~arkly loha~k}k~`y { yqoha~k}ky~`y# naiknkr iq`c na
cly nly glnacly ay yqzarklr%
^@OC@ ;%=% ^`oc` gly~r`enl cly }`clray lrkbke`cay { cly }`clray nac iq`nr`nl { nac
iqol na cly }`clray na V % &Ac`olr`ikùe zrlzk`+
El~ 8 z`r f`iar ay~ ~`oc` iq`nr r { iqoki`r cly } clray na V `e~ay na zlearcly ae c`y
ilcqge`y% Nayzqëy na ay~l# ya ilrra ac zrlbr`g` ilgl yk mqar` qe` rabraykùe ckea`c%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 673/846
Nr% Fëi~lr Pqa}anl Qrç`y
;'9;
Ylcqikùe8
&`+ C` mkbqr` na `o`hl gqay~r` c` bràmki` na cly n`~ly%
=:1:>:
>:
9:
?:
]aclikn`n iley~`e~a &v+
G k c c ` y z l r b ` c l e & { +
Br`mki` gly ~r`enl c` rac`ikle ae~ra ac ~kzl na g`eahl { ac ileyqgl na b`y lcke`
Mkbqr` ;%2% Mkbqr` gly~r`enl ac raenkgkae~l na b`ylcke` ae mqeikùe nac ~kzl na g`eahl%
&Ac`olr`ikùe zrlzk`+
&o+ Cly rayqc~`nly yqgkaenl qe glnacl iq`nrà~kil yle8
{ 7 '069 $ 6%;6& V + ’ :%:;00& V ?+
Ile y 7 0%2?2# _ ? 7 :%;>2# Z_AYY 7 >;%?=
^@OC@ ;%2% ^`oc` na @EL]@ z`r` ac `hqy~a na qe glnacl iq`nrà~kil% &Ac`olr`ikùe
zrlzk`+
Mqae~a na }`rk`ikùe b%c% YY GY Mi`ci% ]`clr z
Naoknl ` c` rabraykùe ? >69%02 ?>0%16 60%: :%:::
Arrlr l rayknq`c ; ?=%69 ?%;6
^l~`c 00 10:%::
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 674/846
Nr% Fëi~lr Pqa}anl Qrç`y
;'>:
^@OC@ ;%6% ^`oc` gly~r`enl cly ilamkikae~ay# cly arrlray ay~àen`ray na clyilamkikae~ay# cly }`clray na ~ { na z z`r` ac glnacl iq`nrà~kil% &Ac`olr`ikùe zrlzk`+
Zranki~lray Ilamkikae~ay Arrlr ay~àen`r ]`clr ~ ]`clr zna ilamkikae~ay
Iley~`e~a '06?%16 02%=6 '0:%99 :%:::V0 6%;6 :%2= 00%6: :%:::VYP_ ':%:; :%::6 '00%9; :%:::
&i+ C` aiq`ikùe na qe glnacl na rabraykùe iøokil `hqy~`nl ay8
{ 7 '2> $ 0%61& V + $ 0%61 $ :%:=?& V ?+ ’ :%::0& V 9+
Cly nk`beùy~kily ay~`nçy~kily yle8
_ ? 7 :%;1?# y 7 0%21# Z_AYY 7 1;%??
^@OC@ ;%;% ^`oc` na z`r` ac glnacl iøokil% &Ac`olr`ikùe zrlzk`+
Mqae~a na }`rk`ikùe b%c% YY GY Mi`ci% ]`clr na z
Naoknl ` c` rabraykùe 9 >61%1: 0=0%69 1?%61 :%:::
Arrlr l rayknq`c 6 ?>%1: 9%:=
^l~`c 00 10:%::
^@OC@ ;%0:% ^`oc` gly~r`enl cly ilamkikae~ay# cly arrlray ay~àen`ray na clyilamkikae~ay# cly }`clray na ~ { na z z`r` ac glnacl iøokil% &Ac`olr`ikùe zrlzk`+Zranki~lray Ilamkikae~ay Arrlr ay~àen`r ]`clr ~ ]`clr z
na ilamkikae~ayIley~`e~a '29%; 0?1%2 ':%1; :%12V0 0%61 6%? :%?9 :%69VYP_ :%:= 02%1 :%91 :%29VIQOA ':%::0 :%::0 ':%62 :%>0
C` mkbqr` na `o`hl gqay~r` cly rayknqly ay~`en`rkx`nly ae mqeikùe nac lrnae na c`loyar}`ikùe z`r` ac glnacl na rabraykùe iq`nrà~kil
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 675/846
Nr% Fëi~lr Pqa}anl Qrç`y
;'>0
? > = 6 0: 0?
'?
'0
:
0
Loyar}`~kle Lrnar
Y ~ ` e n ` r n k x a n _ a y k n q ` c
_ayknq`cy ]aryqy ~fa Lrnar lm ~fa N`~`
&rayzleya ky Gkcc`y z+
Mkbqr` ;%6% _ayknqly ay~`en`rkx`nly ae mqeikùe nac lrnae na c` loyar}`ikùe z`r` ac
glnacl na rabraykùe iq`nrà~kil% @pqç# eù~aya pqa avky~ae `zrlvkg`n`gae~a# ac gkygl
eøgarl na rayknqly zlyk~k}ly { eab`~k}ly% Ae ile~r`y~a# c` bràmki` nac glnacl iøokil
&pqa el ya gqay~r` `pqç+# el gqay~r` ac gkygl eøgarl na rayknqly zlyk~k}ly {
eab`~k}ly% &Ac`olr`ikùe zrlzk`+
>:9:?:
0
:
'0
'?
Mk~~an ]`cqa
Y ~ ` e n ` r n k x a n _ a y k n q ` c
_ayknq`cy ]aryqy ~fa Mk~~an ]`cqay
&rayzleya ky Gkcc`y z+
Mkbqr` ;%;% Bràmki` na cly rayknqly ay~`en`rkx`nly }aryqy cly }`clray `hqy~`nly na [ z`r` ac glnacl iøokil% Eù~aya pqa# ae ay~` bràmki` f`{ ac gkygl eøgarl na }`clray
zlyk~k}ly { eab`~k}ly% Ae ile~r`y~a# ac glnacl iøokil `hqy~`nl &el gly~r`nl `pqç+ el
gqay~r` ac gkygl eøgarl na rayknqly zlyk~k}ly { eab`~k}ly% &Ac`olr`ikùe zrlzk`+%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 676/846
Nr% Fëi~lr Pqa}anl Qrç`y
;'>?
'? '0 : 0
'?
'0
:
0
?
E l r g ` c Y i l r a
Y~ en`rnkxan _ayknq c
Elrg`c Zrlo`okck~{ Zc l~ lm ~fa _ayknq`c y
&rayzleya ky Gkcc`y z+
Mkbqr` ;%0:% Bràmki` gly~r`enl c` zrqao` na elrg`ckn`n z`r` ac glnacl iq`nrà~kil%
&Ac`olr`ikùe zrlzk`+%
&n+ Na `iqarnl ` cly n`~ly ~`oqc`nly na `o`hl# { na cly nk`beùy~kily bràmkily# ~`c z`raia
pqa ac gahlr glnacl ay ac glnacl iq`nrà~kil% Ay~l ya naoa ` pqa# ` zay`r na pqa cly
}`clray na _ ? { y na cly nly glnacly yle z`raiknly# cly }`clray na Z_AYY nkmkarae qel
nac l~rl% @nagày# cly }`clray na ~ nac glnacl iq`nrà~kil yle gq{ ykbekmki`e~ay ae
ilgz`r`ikùe ile cly nac glnacl iøokil &^@OC@Y ;%6 { ;%0:+% ^`gokëe c` Mkbqr` ;%2
na [ }aryqy V yqbkara ` qe` mqeikùe iq`nrà~ki`4 el iøoki`% Mke`cgae~a# cly `eàckyky na
cly bràmkily na cly rayknq`cay z`r` c` mqeikùe iq`nrà~ki` yle gày ile}keiae~ay pqa cly
nac glnacl iøokil%
^@OC@ ;%00% ^`oc` gly~r`enl cly n`~ly nac zrlocag`% \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
Nk`beùy~kily ay~`nçy~kily
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ Ic`ya na Glnacl _ ? y Z_AYY \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ Glnacl iq`nrà~kil :%;>2 0%2?2 >;%?=
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ Glnacl iøokil :%;1? 0%21: 1;%??
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 677/846
Nr% Fëi~lr Pqa}anl Qrç`y
;'>9
Ahagzcl )0=% Ya lo~kaea qe` gqay~r` `ca`~lrk` na ?1 gankikleay na z`r~çiqc`y
`~glymërki`y &ae gkir`y+% Ya naya` y`oar yk f`{ }`clray keqyq`cay av~ragly l
glnar`nly% Qy`r qe nk`br`g` na i`h`%
^`oc` ;%0>% ^`oc` gly~r`enl cly n`~ly &Ac`olr`ikùe zrlzk`+
1 6 0> 2> 61 66 ;: ;? ;? ;9 ;> ;> ;1 ;1 ;= ;= ;= ;2 ;2 ;6 ;; 0:0 0:> 0:= 00>
Ylcqikùe8
@e~ay na ilgaex`r# }`gly ` ~lg`r ae iqae~` c` namkekikùe pqa nkia ilgl i`ciqc`r ac
iq`r~l kemarklr { ac iq`r~l yqzarklr na qe nk`br`g` na i`h`% Ay~` namkekikùe nkia pqa#qe` }ax pqa ya lrnae`e cly n`~ly ae mlrg` `yiaenae~a# ac iq`r~l kemarklr { ac iq`r~l
yqzarklr ya namkeae ilgl8
Iq`r~l 7 Gank`e` na cly gçekgly e,? i`yly# iq`enl e ay z`r
kemarklr Gank`e` na cly gçekgly &e $ 0+,? i`yly# iq`enl e ay kgz`r
Iq`r~l 7 Gank`e` na cly gàvkgly e,? i`yly# iq`enl e ay z`r
yqzarklr Gank`e` na cly gàvkgly &e $ 0+,? i`yly# iq`enl e ay kgz`r
Ac ke}ay~kb`nlr Na}lra &?::0+ aecky~` cly }`clray `~çzkily qy`enl qe nk`br`g`
na i`h`% Ay~ly n`~ly yle8 Ac }`clr gçekgl { ac }`clr gàvkgl# ac iq`r~l kemarklr { ac
iq`r~l yqzarklr# c` gank`e`# c` iq`r~` nkyzarykùe m y &c` iq`c ay c` nkmaraeik` ae~ra ac
iq`r~l yqzarklr { ac iq`r~l kemarklr+%
@nagày# z`r` knae~kmki`r c` zrayaeik` na }`clray keqyq`cay glnar`nly {
av~ragly ya nkia pqa# ~ln` loyar}`ikùe g`{lr pqa 0%1m y# nac iq`r~l gày iari`el# ay qe
}`clr keqyq`c% @eàclb`gae~a# yk 9m y ay g`{lr pqa ac iq`r~l gày iari`el# ae~leiay# ac
}`clr keqyq`c ay av~ragl%
Cly iàciqcly z`r` ay~a zrlocag` yle8
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 678/846
Nr% Fëi~lr Pqa}anl Qrç`y
;'>>
V 7 ;1%:# e 7 ?1# }`clr gçekgl 7 1%:# }`clr gàvkgl 7 00>%:# V 7 6>%;?# y 7 ?;%11#
arrlr ay~àen`r nac zrlgankl 7 1%;0# P0 7 6;%:# P9 7 ;2%1
Iq`r~l kemarklr z`r` loyar}`ikleay kgz`ray 7 gank`e` na cly gçekgly &?1 $ 0+,? 7 09Iq`r~` nkyzarykùe m y 7 iq`r~l yqzarklr ’ iq`r~l kemarklr
7 ;2 ' ;: 7 2
@nagày# 0%1m y 7 &0%1+&2+ 7 0:%1 { 9m y 7 &9+&2+ 7 ?0
Z`r` ay~kg`r cly }`clray `~çzkily keqyq`cay# ac irk~arkl ay8 iq`cpqkar loyar}`ikùe
gaelr pqa ac iq`r~l kemarklr# gaely 0%1m y l g`{lr pqa ac iq`r~l yqzarklr gày 0%1m y ay qe
}`clr `~çzkil keqyq`c% Ay~l ay8 ;: ’ 0:%1 7 2;%1 { ;2 $ 0:%1 7 0:2%1
@e`ckx`enl cly n`~ly na c` ^@OC@ ;%0># ya }a pqa f`{ qe }`clr `~çzkil &00>+
g`{lr ae ac av~ragl yqzarklr na c` gqay~r`% @nagày f`{ iq`~rl }`clray# na ay~a ~kzl &1#
6# 0># 2>+# ae ac av~ragl kemarklr% Z`r` knae~kmki`r cly }`clray av~ragly ya i`ciqc` c`
nkmaraeik` ae~ra ac iq`r~l kemarklr { 9m y# ay naikr# ;: ’ ?0 7 =;% _amkrkëenlya c` @OC@
;%0> { c` Mkbqr` ;%00# }agly pqa c`y ~ray loyar}`ikleay 1# 6 { 0> yle }`clray av~ragly
&pqa ya ackgke`ràe+ { cly }`clray 61 { 00> yle }`clray `~çzkily glnar`nly%
0::1::
I0
Olvzcl~ lm I0
Mkbqr` ;%00% Nk`br`g` na i`h` ile cly 9 }`clray `~çzkily av~ragly &1# 6# 0>+ { cly
}`clray `~çzkily glnar`nly &61# 00>+% &Ac`olr`ikùe zrlzk`+
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 679/846
Nr% Fëi~lr Pqa}anl Qrç`y
;'>1
Ahagzcl )02% Ay~a ahagzcl ay~à aei`gke`nl ` `e`ckx`r ac amai~l pqa zqan` liqrrkr ae ac
glnacl na rabraykùe ay~kg`nl# iq`enl ya ackgke`e }`clray av~ragly% Z`r` cly n`~ly na
c` ~`oc` na `o`hl# `yqgkr qe glnacl zlckelgk`c iøokil% Ae c` zrkgar` key~`eik`# ay~kg`r ac glnacl iøokil keicq{aenl ~ln`y c`y }`rk`ocay% Aeyabqkn`# `hqy~`r qe glnacl na
rabraykùe zlckelgk`c# ilgl ac `e~arklr# zarl ay~` }ax avicq{aenl cly }`clray av~ragly
&1# 6 { 0>+ ay~kg`nly ae ac ahagzcl `e~arklr% @e`ckx`r ae i`n` i`yl# cly }`clray na _ ?#
_ ?`hqy~`n`# ac arrlr ay~àen`r na cl ay~kg`nl y# Z_AYY &c` ykbc` na yqg` na iq`nr`nly na
zrankiikùe+# @EL]@# a~i% ]ar yk f`{ nkmaraeik`y ykbekmki`e~ay ae i`n` qel na cly nly
i`yly% F`iar qe` ~`oc` ile cly nly glnacly na rabraykùe pqa keicq{` c`y ay~`nçy~ki`y
`e~arklray# ilrrayzlenkae~ay ` i`n` qel na cly nly glnacly zrlo`nly# o`hl c`y nly
ilenkikleay%
^@OC@ ;%01% ^`oc` gly~r`enl cly n`~ly na gankikleay &gkir`y+ na z`r~çiqc`y
`~glymërki`y na c` }`rk`oca nazaenkae~a# ae mqeikùe na yqy rayzai~k}ly i`yly & V +%
&Ac`olr`ikùe zrlzk`+
= 6 0> 61 66 ;: ;? ;? ;9 ;> ;> ;1 ;1 ;= ;= ;= ;2 ;2 ;6 ;; 0:0 0:> 0:= 00>
0 ? 9 > 1 = 2 6 ; 0: 00 0? 09 0> 01 0= 02 06 0; ?: ?0 ?? ?9 ?> ?1
Ac aypqag` gly~r`enl cly rayqc~`nly nac Gkek~`o# ae ac `hqy~`gkae~l na qe glnacl na
rabraykùe zlckelgk`c iøokil# ac iq`c keicq{a ~lnly cly n`~ly {# l~rl `hqy~`gkae~l gày#
na qe glnacl na rabraykùe zlckelgk`c iøokil# ac iq`c avicq{a cly }`clray av~ragly ya
n` ae c` ^@OC@ ;%0=% Ilgl ya }a ae ay~` ~`oc`# zrkgar`gae~a# ya `hqy~` qe glnacl narabraykùe zlckelgk`c iøokil8 &[ + }aryqy & V +# & VYP_+# & VIQOA +% Ay~a glnacl keicq{a
cly }`clray av~ragly% Nayzqëy ya keicq{a l~rl glnacl na rabraykùe zlckelgk`c pqa el
keicq{a cly }`clray keqyq`cay av~ragly% Cly rayqc~`nly lo~aeknly qy`enl ac zrlbr`g`
Gkek~`o ya n`e ae c` ^@OC@ ;%0= na `o`hl%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 680/846
Nr% Fëi~lr Pqa}anl Qrç`y
;'>=
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 681/846
Nr% Fëi~lr Pqa}anl Qrç`y
;'>2
^@OC@ ;%0=% ^`oc` gly~r`enl cly rayqc~`nly nac Gkek~`o%^fa rabrayykle apq`~kle ky8
&[ + 7 '?=%> $ ?2%0&V+ ’ 0%6= &V?+ $ :%:>:9 &V
9+
Zranki~lr Ilam YAIlam ^ z ]KM
Iley~`e~ '?=%>9> 6%6;0 '?%;2 :%::2
&V+ ?2%:2; ?%;:> ;%99 :%::: 0?0%2
&VYP_+ '0%61;1 :%?1=6 '2%?> :%::: =6?%;
&VIQOA+ :%:>:9 :%::=1 =%?: :%::: ?1;%9
y 7 ;%>;0 _'Yp 7 ;0%:! _'Yp&`nh+ 7 6;%2!
Z_AYY 7 ?2>;%;6 _'Yp&zran+ 7 6=%66!
@e`c{yky lm ]`rk`eia ^`oca
Ylqria lm }`rk`~kle NM YY GY M z
Nqa ~l _abrayykle 9 0;:2?%0 =912%> 2:%12 :%:::
_ayknq`c Arrlr ?0 06;0%2 ;:%0
^l~`c ?> ?:;=9%6
Nqroke'R`~yle y~`~ky~ki 7 0%>: &ga`yqray `q~lilrrac`~kle mlr ~kga yarkay+
@hqy~`enl qe Glnacl na _abraykùe Zlckelgk`c Iøokil8 &[ + }aryqy & V +# &VYP_+#
&VIQOA+% Ay~a glnacl el keicq{a cly }`clray keqyq`cay av~ragly%
^fa rabrayykle apq`~kle ky8
&[ + 7 20%6 $ =%?0 &V+ ’ :%1>: &V?+ $ :%:011 &V
9+
Zranki~lr Ilam YA Ilam ^ z ]KMIley~`e~ 20%60; 0%10> >2%>? :%:::
&V+ =%?:;? :%112= 00%0> :%::: 0?1%9
&VYP_+ ':%1>:: :%:112 ';%2: :%::: 2::%;
&VIQOA+ :%:011 :%::0= ;%21 :%::: ?=1%0
y 7 0%>6? _'Yp 7 ;=%;! _'Yp&`nh+ 7 ;=%>!
Z_AYY 7 0:1%0:> _'Yp&zran+ 7 ;0%2>!
@e`c{yky lm ]`rk`eia
Ylqria lm ]`rk`~kle NM YY GY M z
Nqa ~l _abrayykle 9 0?9?%60 >0:%;> 062%:: :%:::
_ayknq`c Arrlr 06 9;%1= ?%?:^l~`c ?0 0?2?%9=
Nqroke'R`~yle y~`~ky~ki 7 0%16
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 682/846
Nr% Fëi~lr Pqa}anl Qrç`y
;'>6
^@OC@ ;%02% ^`oc` gly~r`enl ac rayqgae na cly rayqc~`nly na cly nly glnacly
zrlo`nly# ay naikr# ile { yke cly }`clray av~ragly% &Ac`olr`ikùe zrlzk +
^kzl na glnacl _ ?
_ ?`hqy~`n` y Z_AYY
Glnacl ile }`clray av~ragly ;0%:! 6;%2! ;%>;0 ?2>;%;6
Glnacl yke }`clray av~ragly ;=%;! ;=%1! 0%>6? 0:1%0:
@e`ckx`enl c` ~`oc` na `rrko` ya el~` ic`r`gae~a pqë# yk fqol qe gahlr`gkae~l
ykbekmki`~k}l ae c` lo~aeikùe na cly glnacly na rabraykùe# iq`enl ya ackgke`rle cly
}`clray keqyq`cay av~ragly% Zlr ahagzcl# ac arrlr ay~àen`r na cl ay~kg`nl y# nkygkeq{ùileyknar`ocagae~a# `c avicqkr cly }`clray av~ragly# ay naikr# na ;%>; ` 0%>6?% Yk~q`ikùe
ykgkc`r liqrrkù ile c` zrankiikùe na c` yqg` na cly iq`nr`nly Z_AYY# c` iq`c
nkygkeq{ù na ?2>;%;6 ` 0:1%0:% Ae iq`e~l ac ilamkikae~a na na~argke`ikùe _ ?# ay~a
}`clr `qgae~ù na ;0! ` ;=%;!# ay naikr# `c avicqkr cly }`clray av~ragly% Kbq`cgae~a#
ac }`clr na M na c` ~`oc` na @EL]@# pqa gkna c` clebk~qn ~l~`c `qgae~ù
ileyknar`ocagae~a# `c avicqkr cly }`clray av~ragly% ^lnly ay~ly nk`beùy~kily
ay~`nçy~kily# `qe`nly ` cly bràmkily na cly rayknq`cay ay~`en`rkx`nly &pqa el ya
gqay~r`e `pqç# zarl pqa ac ay~qnk`e~a naoa `e`ckx`rcly+# kenki`e pqa c` avicqykùe na cly
}`clray keqyq`cay av~ragly# ae ac glnacl na rabraykùe# yk cl gahlr`rle
ykbekmki`e~agae~a%
@q~lilrrac`ikùe ae n`~ly na yarkay na ~kagzl
Ae cly glnacly oàykily na rabraykùe ya yqga pqa cly ~ërgkely na cly arrlray ca`~lrkly
αk yle }`rk`ocay `ca`~lrk`y yke ilrrac`ikle`r l }`rk`ocay `ca`~lrk`y elrg`cay
kenazaenkae~ay &el `q~lilrrac`ikùe+% Yke ago`rbl# z`r` yarkay na ~kagzl# c` yqzlykikùe
na arrlray yke ilrrac`ikle`r &}`clray na α kenazaenkae~ay+ el ay `zcki`oca# zlrpqa cly
~ërgkely na cly arrlray αk ay~àe zlyk~k}`gae~a ilrrac`ikle`nly ylora ac ~kagzl% O`hl
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 683/846
Nr% Fëi~lr Pqa}anl Qrç`y
;'>;
yagah`e~ay ilenkikleay# cly arrlray ca`~lrkly αk ya nkia pqa ay~àe `q~lilrrac`ikle`nly
l yark`cgae~a ilrrac`ikle`nly &`q~lilrrac`ikùe+% C` i`qy` zrkglrnk`c na lo~aear
arrlray `ca`~lrkly zlyk~k}`gae~a `q~lilrrac`ikle`nly ya naoa ` c` lgkykùe na }`rk`ocayic`}ay nac glnacl &Ea~ar a~ `c% 0;;=+% Ilgøegae~a# iq`enl cly n`~ly ay~àe `brqz`nly
yaiqaeik`cgae~a ylora qe zarklnl na ~kagzl ay naikr# ae yarkay na ~kagzl# cly }`clray
rayknq`cay ay~àe ilrrac`ikle`nly%
Zlr ahagzcl# c`y mkbqr`y ;%0? { ;%09 gqay~r`e bràmki`y na cly rayknq`cay# ae
mqeikùe nac ~kagzl# cly iq`cay avfkoae `q~lilrrac`ikùe# gkae~r`y pqa c` bràmki` na c`
Mkbqr` ;%0> kenki` kenazaenaeik` na cly rayknq`cay%
C`y g`ear`y na na~ai~`r zrlocag`y na `q~lilrrac`ikùe na zrkgar lrnae# qe`
ilenkikùe pqa kgzcki` qe` ilrrac`ikùe ae~ra cly rayknqly a~ { a~ ' 0# nlena ~ ay ac zarklnl
na ~kagzl# yle qy`enl c` ay~`nçy~ki` Nqroke'R`~yle% G`~agà~ki`gae~a# ay~` aiq`ikùe
ya namkea ilgl8
e
Ω &a~ ’ a~'0+?
~7?
N 7 ┊┊┊┊┊┊┊┊┊ &;'0>+e
Ω a?~
~70
Nlena8
N ay c` ay~`nçy~ki` na Nqroke'R`~yle
a~ { a~'0 rac`ikùe ae~ra cly rayknqly ylora ac zarklnl na ~kagzl
e ay ac eøgarl na i`yly
Ae baear`c# ` gaely pqa c`y loyar}`ikleay ya`e na yarkay na ~kagzl# c`
ay~`nçy~ki` na Nqroke'R`~yle naoarç` yar kbelr`n`# zlrpqa ay~` ay~`nçy~ki` n` qe`
zrqao` na `q~lilrrac`ikùe zlyk~k}` l eab ~k} # ylc`gae~a# z`r` yarkay na ~kagzl%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 684/846
Nr% Fëi~lr Pqa}anl Qrç`y
;'1:
Iq`enl ya ay~àe `zcki`enl yarkay na ~kagzl { avky~ae zrlocag`y na `q~lilrrac`ikùe
zqanae avky~kr qe eøgarl na kgzlr~`e~ay ileyaiqaeik`y% Zlr ahagzcl# ilamkikae~ay na
rabraykle zqanae yar keamkikae~ay# ac GYA yark`gae~a yqoay~kg`rà cly arrlray na c`}`rk`ex`# ac ytod w i`ciqc`nl zlr c` mqeikùe na cly iq`nr`nly gçekgly yark`gae~a
yqoay~kg`rà c` nay}k`ikùe ay~àen`r { cly ilamkikae~ay na rabraykùe# a~i% &Ea~ar a~ `c%
0;;=+% C`y gankn`y z`r` gk~kb`r zrlocag`y na `q~lilrrac`ikùe yle cly na `brab`r qe`
l gày }`rk`ocay zranki~lr`y `c glnacl na rabraykùe l na qy`r }`rk`ocay ~r`eymlrg`n`y
&Ea~ar a~ `c% 0;;=+%
Mkbqr` ;%0?% Bràmki` na }`clray rayknq`cay }aryqy ~kagzl gly~r`enl z`~rleay na`q~lilrrac`ikùe &m`c~` na kenazaenaeik`+%
Mkbqr` ;%09% Bràmki` na }`clray rayknq`cay }aryqy ~kagzl kenki`enl `q~lilrrac`ikùe
&m`c~` na kenazaenaeik`+%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 685/846
Nr% Fëi~lr Pqa}anl Qrç`y
;'10
Mkbqr` ;%0>% Bràmki` na }`clray rayknq`cay }aryqy ~kagzl kenki`enl kenazaenaeik` nacly n`~ly%
Ae `zcki`ikleay ae c` ailelgç` { eablikly# naoknl ` pqa ay~`y ay~kg`ikleay~kaenae ` gly~r`r ilrrac`ikùe na yarka z`rik`c# ya zqanae qy`r zrqao`y na fkzù~ayky
ilgl8
Fl8π 7 : &El f`{ `q~lilrrac`ikùe l kenazaenaeik`+ &;'01+
F`8π 5 : &`q~lilrrac`ikùe+ &;'0=+
C` zrqao` ileyky~a ae na~argke`r yk ac z`ràga~rl na `q~lilrrac`ikùe π ay kbq`c `
iarl l ay g`{lr pqa iarl% Zlr ahagzcl# yk π 7 : cly ~ërgkely nac arrlr α~ yle
kenazaenkae~ay naoknl ` pqa cly ~ërgkely q~ yle kenazaenkae~ay% El loy~`e~a# cly
}`clray irç~kily yle nkmçikcay na lo~aear# zarl c` zrqao` na Nqroke'R`~yle f` lo~aeknl
cly ckenarly yqzarklray a kemarklray nQ { nC na ~`c g`ear` pqa# qe }`clr na N mqar` na
ay~ly ckenarly cca}` ` qe` naikykùe namkek~k}`%
Na ay~` g`ear`# Ea~ar a~ `c% &0;;=+# n` c` rabc` na naikykùe z`r` zrlo`r ae~ra
ay~`y c~are`~k}`y# ay~l ay8
Yk N 5 nQ# ya ileicq{a Fl8 &;'02+Yk N 3 nC ya ileicq{a F`8 &;'06+
Yk nC ≮ N ≮ nQ# c` zrqao` ay keileicqy` &;'0;+
]`clray zapqaðly na N ilecca}`e ` c` ileicqykùe na pqa c` zrqao` na fkzl~ayky na F `8π
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 686/846
Nr% Fëi~lr Pqa}anl Qrç`y
;'1?
5 :# zlrpqa cly arrlray `ca`~lrkly `n{`iae~ay α~ { α~'0 ~kaenae ` yar na c` gkyg`
g`bek~qn iq`enl ay~àe zlyk~k}`gae~a `q~lilrrac`ikle`nly% Zlr cl ~`e~l# c` nkmaraeik`
ae cly rayqc~`nly α~ ' α~'0 ~kaenae ` yar gaelray iq`enl π 5 :# cl iq`c cca}` ` qeeqgar`nlr zapqaðl ae c` mqeikùe na N {# zlr cl ~`e~l# ` qe` zrqao` ay~`nçy~ki` na N
zapqað`%
C`y ~`oc`y na `o`hl gqay~r`e c`y zrqao`y na cly ckenarly na Nqroke'R`~yle# z`r`
qe ek}ac na ykbekmki`eik` na ΰ 7 :%:1 { :%:0% Ilgl ya }a# c` ilcqge` na c` kxpqkarn`
yað`c` cly }`clray na e% C`y ykbqkae~ay ilcqge`y n`e cly }`clray z`r` i`n` d ile yqy
ilrrayzlenkae~ay ckenarly% Ykaenl `yç# c`y ~`oc`y na `o`hl gqay~r`e c`y zrqao`y na cly
ckenarly na Nqroke'R`~yle z`r` cly ek}acay ykbekmki`eik na ΰ 7 :%:1 { ΰ 7 :%:0%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 687/846
Nr% Fëi~lr Pqa}anl Qrç`y
;'19
^`oc` ;%0;% ^`oc` gly~r`enl c`y zrqao`y na cly ckenarly na Nqroke'R`~yle z`r` qeek}ac na ykbekmki`eik` na ΰ 7 :%:1%
Mqae~a8 Daccar# B# Ork`e R`rr`id# Faer{ O`r~ac% Y~`~ky~kiy mlr G`e`bagae~ `en
Ailelgkiy &0;;:+%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 688/846
Nr% Fëi~lr Pqa}anl Qrç`y
;'1>
^`oc` ;%?:% ^`oc` gly~r`enl c`y zrqao`y na cly ckenarly na Nqroke'R`~yle z`r` qeek}ac na ykbekmki`eik` na ΰ 7 :%:0 &ile~keq`ikùe+%
Mqae~a8 Daccar# B# Ork`e R`rr`id# Faer{ O`r~ac% Y~`~ky~kiy mlr G`e`bagae~ `en
Ailelgkiy &0;;:+%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 689/846
Nr% Fëi~lr Pqa}anl Qrç`y
;'11
Ahagzcl )06% Ya n`e cly ykbqkae~ay n`~ly `n`z~`nly nac ckorl @zzckan Ckea`r _abraykùe
Glnacy na Ea~ar a~ `c% &0;;=+8
&a~ ’ a~'0+?
7 :%:;2;># a?~ 7 :%0999:06 ile qe` ~`g`ðl na gqay~r` na e 7 ?:% Zrlo`r c`y
fkzù~ayky &na `q~lilrrac`ikùe zlyk~k}`+ yað`c`n`y `o`hl qy`enl ek}acay na ykbekmki`eik`
na :%:1 { :%:08
Fl8π 7 :
F`8π 5 :
Ylcqikùe8
Qy`enl c` aiq`ikùe &;'0>+ { yqy~k~q{aenl n`8
?:
Ω &a~ ’ a~'0+?
~7? :%:;2;>N 7 ┊┊┊┊┊┊┊┊┊ 7 ┊┊┊┊┊┊┊┊ 7 :%291
?: :%0999:
Ω a?~
~70
Qy`enl c` ^`oc` ;%0; ile ΰ 7 :%:1# e 7 ?: { ile z ’ 0 7 0 &zlrpqa V 7 0# ay naikr# ileqe` ylc` }`rk`oca kenazaenkae~a+# n`8 nC 7 0%?: { nQ 7 0%>0% Naoknl ` pqa N 7 %291 ay
zapqaðl { i`a nao hl na 0%>0# ya nkia pqa N 3 nC { ya ileicq{a pqa π 5 : l ya` F`8 ay
naikr pqa f`{ `q~lilrrac`ikùe l m`c~` na kenazaenaeik`# l pqa cly ~ërgkely na arrlr α~
ay~àe zlyk~k}`gae~a `q~lilrrac`ikle`nly% Ily` ykgkc`r liqrra yk ya qy` qe ek}ac na ΰ 7
:%:0%
El~ 8 Yk ya f`ia qe` zrqao` na `q~lilrrac`ikùe eab ~k} # c` ay~`nçy~ki` qy`n` ay > ’ N#
nlena N ya n` ae c`y aiq`ikleay na `rrko`% Yk ay `yç# ae~leiay# c` zrqao` ya ilenqia na
c` gkyg` g`ear` pqa z`r` c` `q~lilrrac`ikùe zly~k}`% Ay~l pqkara naikr pqa yk c`
i`e~kn`n > ’ N i`a nao`hl na nC# ya ileicq{a π 3 :% @nagày# yk ya qy` qe` zrqao`
okc ~ar`c z`r` Fl8π 7 : }aryqy F`8π ≪ : ya f`ia qy`enl yaz`r`n`gae~a c`y zrqao`y
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 690/846
Nr% Fëi~lr Pqa}anl Qrç`y
;'1=
qekc`~ar`cay% &Ea~ar a~ `c% 0;;=+%
Fa~arlyian`y~kikn`n { flglyian`y~kikn`n
Ay~` yaiikùe n`rà qe` namkekikùe na cl pqa ya naelgke`e fa~arlyian`y~kikn`n {flglyian`y~kikn`n% Zlr ahagzcl# iq`enl c` }`rk`ex` nac arrlr# α&ω?
α+# el ay iley~`e~a#
ay~` ilenkikùe ya cc`g` fa~arlyian`y~kikn`n% Ae ile~r`y~a# iq`enl c` }`rk`ex` nac arrlr#
α&ω?α+# ay iley~`e~a# ay~` ilenkikùe ya cc`g` flglyian`y~kikn`n%
Ac gë~lnl gày ilgqe z`r` nk`bely~ki`r ac zrlocag` na fa~arlyian`y~kikn`n ay
br`mki`enl cly rayknq`cay ile~r` cly }`clray zrlely~ki`nly na {% Ykaenl `yç# ya `e`ckx` ac
ayz`rikgkae~l na cly zqe~ly br`mki`nly% Zlr ahagzcl c` Mkbqr` ;%01 nayirkoa cly
rayknq`cay gly~r`enl fa~arlyian`y~kikn`n# ay naikr# iq`enl ac arrlr ω?α el ay iley~`e~a%
Ilgl rayqc~`nl na ay~l# yk avky~ae i`gokly yky~agà~kily na cly rayknq`cay ile c`y
mqeikleay na c`y }`rk`ocay kenazaenkae~ay% Ay~` ilenkikùe ya zrqao` `e`ckx`enl c`
Mkbqr` ;%01 zlrpqa ac arrlr ω?α `z`raia zapqaðl iq`enl ac }`clr zrlely~ki`nl na { ay
zapqaðl { br`ena iq`enl ac }`clr na { ay br`ena% Ae ile~r`y~a# c` Mkbqr` ;%0= gqay~r`
qe` ilenkikle na flglyian`y~kikn`n# ay naikr# na ω?α iley~`e~a% Ilgl rayqc~`nl na
ay~l# el f`{ i`gokly `z`rae~ay ae c` }`rk`ikùe na cly rayknq`cay%
Mkbqr` ;%01% Bràmki` na rayknq`cay gly~r`enl c` ilenkikùe na fa~arlyian`y~kikn`n# ay
naikr# na c` }`rk`ex` nac arrlr# ω?α el iley~`e~a%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 691/846
Nr% Fëi~lr Pqa}anl Qrç`y
;'12
Mkbqr` ;%0=% Bràmki` na rayknq`cay gly~r`enl c` ilenkikùe na flglyian`y~kikn`n#
iq`enl c` }`rk`ex` nac arrlr# ω?α ay iley~`e~a ay naikr# iq`enl cly rayknq`cay ylekenazaenkae~ay
Zrqao` na Rfk~a z`r` ac zrlocag` na fa~arlyian`y~kikn`n
F`{ mqeikleay ay~`nçy~ki`y z`r` zrlo`r ac zrlocag` na fa~arlyian`y~kikn`n% Qe` na
ëy~`y ay c` zrqao` na Rfk~a% Na ay~` g`ear`# F`c Rfk~a zrlzqyl qe` mlrg` ykgzca z`r`
zrlo`r zlr fa~arlyian`y~kikn`n# ay naikr# na }`rk`ikleay yky~agà~ki`y na cly rayknq`cay
ile c`y }`rk`ocay rabraylr`y &Rfk~a# F`coar~# 0;6:% @ Fa~aryian`y~kik~{'Ileyky~ae~
Il}`rk`eia G`~rkx `en ` Nkrai~ ^ay~ mlr Fa~arlyian`y~kik~{% Ailelga~rki`
>68602'696+%
Z`r` avzcki`r c` zrqao` na Rfk~a z`r` fa~arlyian`y~kikn`n# yqzùeb`ya pqa ya
~kaeae d }`rk`ocay rabraylr`y keicq{aenl qe` iley~`e~a vç 7 &0# vk?# … # vkd +% Na iqarnl
` Rfk~a# nayzqëy na ay~kg`r ac glnacl na rabraykùe# ya zqanae ay~kg`r cly rayknq`cay {
c` aiq`ikùe na rabraykùe `qvkck`r8
a?k 7 ΰ¾xk $ }k &;'?:+
Nlena ΰ ay qe }ai~lr na z`ràga~rly# }k ay qe arrlr { xk ile~kaea ~lnly cly zrlnqi~ly
irqx`nly na cly acagae~ly ae vk# ay naikr8
xç 7 &0# vk?#…%%# vkd # v?k?#…%%# v?
kd # vk? vk9#…% vk#d'0 vkd + &;'?0+
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 692/846
Nr% Fëi~lr Pqa}anl Qrç`y
;'16
C`y zrqao`y na fkzl~ayky eqc`y ya zqanae f`iar na c` ykbqkae~a g`ear`8
Zlr ahagzcl# c` zrqao` na fkzù~ayky na flglyian`y~kikn`n# ay naikr# na pqa c` }`rk`ex`
nac arrlr# ω?α ay iley~`e~a ay8
Fl8ω?0 7 ω?
? 7 … 7 ω?e &;'??+
C` zrqao` na fkzù~ayky `c~are`~k}` na fa~arlyian`y~kikn`n ay8
Fl8ω?0 ≪ ω?
? ≪ … ≪ ω?e &;'?9+
Iq`enl ya qy` c` nky~rkoqikùe na c` HK iq`nr`n`# yk ac zrlnqi~l na c` ay~`nky~ki` _ ? { ac
~`g`ðl na c` gqay~r` ~kaea qe` `zrlvkg`ikùe ` ς ? ile Td&d $ 0+ , ?W '0 br`nly na
ckoar~`n# ae~leiay c` mqeikùe ya n` ilgl8e_ ? ≆ ς ? &dTd $ 0+ , ?W ’ 0+ &;'?>+
Yk ac }`clr na e_ ? ay g`{lr pqa ac }`clr irç~kil na c` HK iq`nr`n` ς ? ya raif`x` c`
fkzù~ayky eqc` ` m`}lr na c` zrqao` `c~are`~k}` na fa~arlyian`y~kikn`n%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 693/846
Nr% Fëi~lr Pqa}anl Qrç`y
;'1;
Aharikikly I`zç~qcl ;
;%0% Ay~a ay qe aharikikl rac`ikle`nl ile ac `hqy~`gkae~l nac gahlr glnacl na
rabraykùe% C` ~`oc` na `o`hl n` cly n`~ly%^`oc` gly~r`enl cly n`~ly% &Ac`olr`ikùe zrlzk`+
V s : 0 ? 9 > 1 = 2 6 ; \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ [ s ;%0 2%9 9%? >%= >%6 ?%; 1%2 2%0 6%6 0:%?
&`+ Lo~aear ac glnacl na rabraykùe gày `zrlzk`nl# ay naikr# ckea`c# iq`nrà~kil l iøokilna `iqarnl ` cly irk~arkly _ ?# _ hqy~`n # y { Z_AYY i`ciqc`nly%&o+ Ilgzcagae~`r c` naikykùe nac gahlr glnacl i`enkn`~l o`yàenlya ae ac nk`beùy~kil
yqoha~k}l nac `eàckyky bràmkil%
C` ~`oc` na `o`hl n` c`y rayzqay~`y loha~k}ky~`y%
^`oc` gly~r`enl cly n`~ly% &Ac`olr`ikùe zrlzk`+% \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ ^kzl na glnacl na rabraykùe _ ? _ ?`hqy~`n` y Z_AYYGlnacl na rabraykùe iq`nrà~kil >=%9! 9:%;! ?%0:? 0::%>:>Glnacl na rabraykùe iøokil =0%9! >?%:! 0%;?= >?0%:11Glnacl na rabraykùe ckea`c 96%?! 9:%1! ?%0:; 10%90=
;%?% Ya f`ia qe avzarkgae~l ile qe eqa}l glnacl na `q~lgù}kc# z`r` na~argke`r c`
nky~`eik`# nayzqëy na mrae`r ` }`rk`y }aclikn`nay% C` ykbqkae~a n`~` ya n`8
^`oc` gly~r`enl cly n`~ly% &Ac`olr`ikùe zrlzk`+]aclikn`n# } &dg,fr+ s 92 1? =2 6? ;2 009
Nky~`eik` nayzqëyna mrae`r ac `q~l# n &g+ s 02 ?2 >9 =9 6; 0?:
&`+ @hqy~`r ac glnacl l c` iqr}` na rabraykùe gøc~kzca zloc`ikle`c ±ns η 7 οl $ ο0}0 $
ο?}?# c` iq`c ay ay~kg`n` zlr c` aiq`ikùe na c` gqay~r` [ 7 ol $ o0v0 $ o?v?
&o+ Ay~kg`r c` nky~`eik` nayzqëy na mrae`r# iq`enl ac ilifa cca}` qe` }aclikn`n na 2:
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 694/846
Nr% Fëi~lr Pqa}anl Qrç`y
;'=:
dkcùga~rly zlr flr`%
&i+ Ay~kg`r c` nky~`eik` nayzqëy na mrae`r# iq`enl ac ilifa cca}` qe` }aclikn`n na 0?:
Dg,fr%;%9% C` }kyilykn`n na qe ~kzl na cqorki`e~a ya gknkù ile = }aclikn`nay nkmarae~ay% Ya
`yqgkù qe glnacl iq`nrà~kil na rabraykùe ilgl ac gày `zrlzk`nl { c` mqeikùe na
rabraykùe zlckelgk`c ay~kg`n` rayqc~`e~a na qe` gqay~r` na e 7 = mqa8
{ 7 '009%:;92 $ 9%9=6>v ’ :%:026:v?
&`+ Knae~kmki`r c` }`rk`oca nazaenkae~a%
&o+ Knae~kmki`r c` }`rk`oca kenazaenkae~a%
&i+ I`ciqc`r c` }kyilykn`n nac cqorki`e~a iq`enl c` }aclikn`n ay 21 rzg% &9;%>0+
;%>% Ac ~av~l na Zrlo`okckn`n { Ay~`nçy~ki` z`r` Kebaekarly na cly `q~lray _le`cn A%
R`czlca a~ `c% 0;;;# nkyiq~ae qe avzarkgae~l ile ac mke na na~argke`r yk ac mcqhl
y`ebqçeal iaraor`c ae yaray fqg`ely ya zqana zranaikr ` z`r~kr na c` zraykùe &ae gg
Fb+ nac lvçbael `r~ark`c% Z`r` ay~l ya qy`rle 01 }lcqe~`rkly ae ac ay~qnkl { ya
loyar}`rle cly ykbqkae~ay n`~ly8
^`oc` gly~r`enl cly n`~ly na ay~a zrlocag`% \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ Mcqhl y`ebqçeal &[ + s 6>%99 62%6: 6?%?: 26%?0 26%>> 6:%:0 69%19 2;%>=
21%?? 2=%16 22%;: 26%6: 6:%=2 6=%=: 26%?:
Zraykùe na lvçbael &v+ s =:9%>: 16?%1: 11=%?: 1;>%=: 116%;: 121%?: 6:%0:
>10%?: >:>%:: >6>%:: >1?%>: >>6%>: 9?:%9: 91:%9:
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ &Mqae~a8 R`czlca a~ `c% 0;;;+
Ay~kg`r c` aiq`ikùe iq`nrà~ki` l iøoki` pqa gahlr aei`ha ae cly n`~ly% Qe` }ax pqa ya
naikn` zlr ac gahlr glnacl zlckelgk`c &na yabqenl l ~ariar lrnae+# zranaikr ac mcqhl
y`ebqçeal iq`enl c` zraykùe nac lvçbael ay na 2=: ~lrr# ay naikr na 2=: gg Fb 7 0
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 695/846
Nr% Fëi~lr Pqa}anl Qrç`y
;'=0
`~gùymar`+% Yqbaraeik`8 Qy`r qe` rabraykùe zlr z`yly%
;%1% Ya n`e cly ykbqkae~ay n`~ly ae c` ~`oc` na `o`hl% &Ac`olr`ikùe zrlzk`+8
^`oc` gly~r`enl c` kemlrg`ikùe z`r` ay~a zrlocag`% \\\\\\\\\\\\\\\\\\\\ &V+ s : 0 ? 9 > 1 =
\\\\\\\\\\\\\\\\\\\\ &[+ s 0 > 1 9 ? 9 >
\\\\\\\\\\\\\\\\\\\\
&`+ ¼_a`cgae~a aei`h` qe glnacl iøokil gahlr pqa qe glnacl na rabraykùe iq`nrà~kil
l ckea`c< Hqy~kmki`r ac `rbqgae~l% &Yk# zlrpqa ac }`clr na _ ? 7 62%1! ay ac gày c~l na cly
9 glnacly zrlo`nly4 nagày ac }`clr na y 7 :%=2?= { ac }`clr na Z_AYY 7 06%>9 yle cly
}`clray gày o`hly na cly 9 glnacly zrlo`nly% @nagày# cly nk`beùy~kily bràmkily
~`gokëe `zl{`e ` c` elikùe na qe glnacl iøokil+
&o+ Yk ac glnacl iøokil ay yqzarklr &hqy~kmki`enl ac `rbqgae~l+# ae~leiay# zrlely~ki r
[ iq`enl V 7 ?% &>%>??+
;%=% Ac ckorl na Zrlo`okckn`n { Ay~`nçy~ki` @zcki`n`y ` c` Kebaekarç` na Gle~blgar{
a~ `c% 0;;=# z%169 n` qe ahagzcl rac`ikle`nl ile cly z`eacay na c`y z`ranay c`~ar`cay na
qe `}kùe mlrg`nly ae qe` zraey` na 01:: ~leac`n`y% Ac ily~l na m`orki`ikùe na i`n`
qekn`n i`gok` ile ac ~`g`ðl nac cl~a na zrlnqiikùe% C` ~`oc` na `o`hl zrlzlrikle` cly
n`~ly%
&`+ F`iar qe nk`br`g` na nkyzarykùe { naiknkr pqë br`nl nac glnacl zlckelgk`c ay
ile}aekae~a qy`r%
&o+ F`iar qe `eàckyky na }`rk`ex` { zrlo`r pqa cly ilamkikae~ay yle kbq`c ` iarl%I`ciqc`r ac }`clr na z { y`i`r ileicqykleay%
&i+ Lo~aear ac glnacl zlckelgk`c pqa gahlr aei`ha ae cly n`~ly qy`enl c` aiq`ikùe
&;';+# ile yq rayzai~k}` zrqao` na fkzù~ayky%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 696/846
Nr% Fëi~lr Pqa}anl Qrç`y
;'=?
^`oc` gly~r`enl cly n`~ly na ay~a aharikikl%
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ { s 0%60 0%2: 0%=1 0%11 0%>6 0%>: 0%9: 0%?= 0%?> 0%?0 0%?: 0%06
v s ?: ?1 9: 91 >: 1: =: =1 2: 21 6: ;:
Mqae~a8 Gle~blgar{ a~ `c% 0;;=
;%2% Ya n`e cly ykbqkae~ay n`~ly ae c` ~`oc` na `o`hl%
^`oc` gly~r`enl cly n`~ly nac zrlocag`% &Ac`olr`ikùe zrlzk`+
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ [ s ?>%=: ?>%20 ?9%;: 9;%1: 9;%=: 12%0? =2%00 =2%?> =2%01 22%62 6:%00 6>%=2 V s >%: >%: >%: 1%: 1%: =%: =%1 =%1 =%6 2%: 2%0 2%9
&`+ @hqy~`r cly n`~ly ` qe glnacl zlckelgk`c na yabqenl lrnae%
&o+ @hqy~`r cly n`~ly ` qe glnacl zlckelgk`c na ~ariar lrnae%
&i+ Qy`enl gë~lnly yqoha~k}ly { loha~k}ly# naiknkr iqàc na cly nly glnacly aei`h`
gahlr ae cly n`~ly hqy~kmki`enl ac `rbqgae~l% &Qe glnacl iq`nrà~kil ay ac gahlr
i`enkn`~l% Hqy~kmki`r c` `yarikùe+
;%6% Cly n`~ly na c` ~`oc` na `o`hl ilrrayzlenae ` qe ay~qnkl z`r` c` lo~aeikùe na ikar~l
zrlnqi~l a~çckil rac`ikle`nl ile ac ~kagzl%
^`oc` ile cly n`~ly% \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ v s 0 0 ? > > > = \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ { s ?1%: ?2%1 ?6%: 90%; 99%: 9>%= ??%: \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
&`+ Lo~aear ac glnacl zrlo`okcçy~kil &iq`nrà~kil l iøokil# yke `yqgkr ke~ar`iikùe+# gày
`naiq`nl z`r` cly n`~ly { ay~kg`r c` mqeikùe na rabraykùe ilrrayzlenkae~a%
&o+ ]`ckn`r ac glnacl na~argke`nl ae &`+ iley~rq{aenl qe` bràmki` ile cly rayknq`cay
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 697/846
Nr% Fëi~lr Pqa}anl Qrç`y
;'=9
ay~`en`rkx`nly { `e`ckx`enl# yqoha~k}`gae~a# c` ilemlrg`ikùe na cly n`~ly na c`
bràmki`%
&i+ Ay~kg`r ac ilamkikae~a na na~argke`ikùe _ ?
% Na `iqarnl ile ac irk~arkl na _ ?
# {nagày ay~`nçy~ki`y ¼aei`h`e okae cly n`~ly ae ac glnacl na rabraykùe yacaiikle`nl<
&n+ F`iar qe `eàckyky na }`rk`ex` { ay~kg`r ac ek}ac na zrlo`okckn`n z%
&a+ Ilgzcagae~`r ac zrliankgkae~l qy`enl c` aiq`ikùe &;'0?+ {# na `iqarnl ` cly
rayqc~`nly# { ` c` zrqao` na fkzù~ayky# naikr iqàc na cly nly glnacly aei`h` gahlr ae cly
n`~ly%
&m+ Qy`r ac irk~arkl na Iz na G`ccl z`r` `e`ckx`r yk fqokara gqif`y }`rk`ocay
kenazaenkae~ay l yqzarmcq`y# pqa ya zqan`e ackgke`r nac glnacl yk Iz 5 &z $ 0+% El
loy~`e~a# yk Iz 3 &z $ 0+ ay~l zqnkar` kenki`r pqa ya f`e lgk~knl }`rk`ocay
kenazaenkae~ay kgzlr~`e%
;%;% Ae qe ay~qnkl na yabqrkn`n z`r` cly gl~lrky~`y ae c`y i`rra~ar`y ay~`~`cay# ya y`oa
pqa ac eøgarl na `iiknae~ay `q~lgl}kcçy~kily# ae ikar~` z`r~a na na qe` i`rra~ar`# ay~à
rac`ikle`nl ile ac eøgarl na }afçiqcly { c` }aclikn`n na ëy~ly% Z`r` ay~l# `c aei`rb`nl
na ay~a ay~qnkl ya ca zknae cly zrlgankly na c`y ay~`nçy~ki`y na cly øc~kgly 0: `ðly# ile
ac loha~l na ay~`ocaiar qe glnacl na rabraykùe z`r` zranaikr ac eøgarl na `iiknae~ay%
Ykaenl `yç# ya naikna zlear ilgl }`rk`oca nazaenkae~a ac eøgarl na `iiknae~ay &[+%
@nagày# ilgl }`rk`ocay kenazaenkae~ay ya zleae ac eøgarl na }afçiqcly pqa z`y`e
zlr ac ~raifl &v0+ {# c` }aclikn`n zrlgankl ` pqa }k`h`e &gkcc`y zlr flr`+# ilgl &v?+% Ya
naikna zrlo`r iq`~rl glnacly na rabraykùe# ay naikr# qel ckea`c gøc~kzca yke ke~ar`iikùe
{ l~rl ile ke~ar`iikùe% Z`r` ac l~rl glnacl zrlo`nl ya naikna zlr qel iq`nrà~kil# ile{ yke ke~ar`iikùe% ^lnl ay~l ya f`ia z`r` }ar iq`c na cly glnacly aei`h` gahlr ae cly
n`~ly% F`iar cly ykbqkae~ay iàciqcly8
&`+ I`ciqc`r cly }`clray na _ ?# _ ?`hqy~`n`# y# Z_AYY# M { ac }`clr na z z`r` i`n` qel na cly
glnacly zrlo`nly%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 698/846
Nr% Fëi~lr Pqa}anl Qrç`y
;'=>
&o+ F`iar qe rayqgae na cly rayqc~`nly na cly > glnacly na rabraykùe zrlo`nly {
naiknkr iq`c yky~ag` ay yqzarklr%
^`oc` gly~r`enl cly zrlgankly `eq`cay nac eøgarl na `iiknae~ay# ae mqeikùe naceøgarl na }afçiqcly { c` }aclikn`n &gkcc`y zlr flr`+ ae pqa }k`h`e% &Ac`olr`ikùe
zrlzk`+ Eøgarl na &[ + Eøgarl na & V 0+ ]aclikn`n nac & V ?+
`iiknae~ay }afçiqcly }afçiqcl1 >: 19; 11 29
01 => ;:9 ?1 11
> ?2 =:= 9: 2:0 1 1:
0: 1= 61= 91 6:6 =: =2
&o+ Ilgzca~`r c` ~`oc` na `o`hl ile cly rayqc~`nly na cly iq`~rl glnacly zrlo`nly
{ naikr iq`c ay ac glnacl yqzarklr%
^`oc` gly~r`enl cly n`~ly nac zrlocag` &Ac`olr`ikùe zrlzk`+
^kzl na glnacl _ ? _ ?`hqy~`n` y Z_AYY M z
Glnacl ckea`c yke
ke~ar`iikùe
Glnacl ckea`c ile
ke~ar`iikùeGlnacl iq`nrà~kil
yke ke~ar`iikùe
Glnacl iq`nrà~kil
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 699/846
Nr% Fëi~lr Pqa}anl Qrç`y
;'=1
ile ke~ar`iikùe
;%0:% @e`ckx r c`y bràmki`y na `o`hl na { }aryqy v0 z`r` qe` }`rkan`n na }`clray na v? {
na~argke`r yk f`{ l el ke~ar`iikùe%
Bràmki`y &`+# &o+# &i+# &n+# &a+ { &m+ na { ae mqeikùe na }`rkly }`clray na v% &Ac`olr`ikùe zrlzk`+
;%00% Ac ~av~l na cly `q~lray Gkif`ac H% Ea~ar# F%# Dq~ear# Ifrky~lzfar H% E`if~yfakg
{ Rkcck`g R`yyarg`e# iq{l ~ç~qcl ay @zzckan Ckea r _abrayykle Glnacy &0;;=+nkyiq~a c` amkikaeik na qe ~kzl na qe gai`ekygl na ~r`eygkykùe pqa mqeikle` ` gày
na c` i`z`ikn`n elrg`c ya zrqao` z`r` ranqikr ac ileyqgl na b`ylcke` {# zlr aena# c`
ranqiikùe na c` ile~`gke`ikùe `gokae~`c &zlr c`y agkykleay na b`yay na
ke}are`narl+% Ay~l ya ay~qnkù ae 0? zrqao`y# ile qe` i`gklea~` apqkz`n` ile ay~a
~kzl na ~r`eygkykùe% C` ~`oc` na `o`hl gqay~r` c` }aclikn`n iley~`e~a & vk+# ae gkcc`y
zlr flr`# ae mqeikùe na c`y gkcc`y zlr b`cùe lo~aekn`y & {k+% @yøg`ya qe glnacl na
rabraykùe na yabqenl lrnae% Cly n`~ly ya n`e ae c` ~`oc` na `o`hl%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 700/846
Nr% Fëi~lr Pqa}anl Qrç`y
;'==
^`oc` gly~r`enl cly n`~ly nac zrlocag` nac raenkgkae~l na b`ylcke`%
El% na zrqao` s 0 ? 9 > 1 = 2 6 ; 0: 00 0?
]aclikn`n &vk+ s 91 91 >: >: >1 >1 1: 1: 11 11 =: =:
_aenkgkae~l &{k+ s ?? ?: ?6 90 92 96 >0 9; 9> 92 ?2 9:
&Mqae~a8 Ea~ar a~ `c% 0;;=+
&`+ Br`mki`r cly na n`~ly gkcc`y zlr b`cùe }aryqy }aclikn`n%
&o+ @hqy~`r ac glnacl na rabraykùe zlckelgk c na yabqenl lrnae% &[ 7 '069 $
6%;6V ’ :%:;00&V ?++
&i+ ]`ckn`r c` mqeikùe zrlo`okcçy~ki` iq`nrà~ki` br`mki`enl cly rayknqly }aryqy }`clrayloyar}`nly na {% ^`gokëe# f`iar qe fky~lbr`g` na mraiqaeik` }aryqy }`clray
rayknq`cay% ^`gokëe# zraz`r`r qe` bràmki` na zrlo`okckn`n elrg`c# ay naikr# na rayknqly
}aryqy }`clray na x%
&n+ ]`ckn`r ac glnacl iq`nrà~kil ay~kg`enl ac }`clr na YYa# _ ? { _% @ykgkygl# f`iar
qe` ~`oc` na @EL]@ { f`iar zrqao`y na fkzù~ayky ile c` ~ na ay~qnk`e~a% Y`i`r
ileicqykleay `zrlzk`n`y%
&a+ Zrlo`r qe glnacl iøokil { ilgz`r`r cly rayqc~`nly ile cly nac glnacl iq`nrà~kil%
&m+ ¼Iqàc na cly nly glnacly ay yqzarklr<
;%0?% Ae qe` ke}ay~kb`ikùe ikae~çmki` `brçilc`# ya ay~qnkù# ae 0: zrqao`y# cly amai~ly na
c` fqgan`n na c` ~karr` &vk ae zqcb`n`y+ { c` ~agzar`~qr` &v? ae lI+ ae mqeikùe nac
raenkgkae~l &ae m`eab`y+# na ikar~` }`rkan`n na zc`e~`y br`gçea`y &[+% Cly n`~ly ya n`e
`o`hl%
^`oc` gly~r`enl cly n`~ly% &Ac`olr`ikùe zrlzk`+% \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ Fqgan`n & v0+ s = = = = 0> 0> 0> 01 0= 0=
^agzar`~qr` & v?+ s ?: ?0 ?? ?? ?? ?9 ?9 ?9 ?> ?>
_aenkgkae~l &[ + s >; >6 >6 >6 >6 >? >> >> >: >:
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 701/846
Nr% Fëi~lr Pqa}anl Qrç`y
;'=2
Ac kebaekarl `brùelgl ke}ay~kb`nlr ayzar` qe glnacl na c` mlrg`8
{k7 ο
l$ ο
0v
k0$ ο
?v
k?$ ο
00v?
k0$ ο
??v?
k?$ ο
0?v
k0v
k?$ α
&`+ Br`mki`r cly }`clray na {k ile~r` cly }`clray `hqy~`nly%
&o+ I`ciqc`r _ ?%
&i+ I`ciqc`r ac }`clr na M { z%
&n+ Ay~kg`r ac raenkgkae~l zrlgankl &ae m`eab`y+# iq`enl c` fqgan`n ay kbq`c ` 6 {
iq`enl c` ~agzar`~qr` ay kbq`c ?? br`nly Iacykqy%
&a+ ¼Ya zqnkar` ackgke`r ac ~ërgkel na ke~ar`iikùe# yke gaelyi`o`r c` amkikaeik` nac
glnacl na rabraykùe# pqa ayzar` ac kebaekarl `brùelgl<
;%09% C` yqg` na cly iq`nr`nly nac arrlr na qe glnacl na rabraykùe zlckelgk`c
iq`nrà~kil ilgzca~l# ile ke~ar`iikùe ile~aekaenl nly }`rk`ocay kenazaenkae~ay ay na
YYa 7 ?::%:% C` yqg` na cly iq`nr`nly nac glnacl ykgzca# yke ke~ar`iikùe# ile qe`
}`rk`oca kenazaenkae~a ay na YY` 7 1::% @yqgkr d 0 7 ># d ? 7 1# e 7 1: { ΰ 7 :%:1%
&`+ Na~argke`r iqàc na cly nly glnacly ay yqzarklr% &Ac glnacl ilgzca~l ay
yqzarklr% Hqy~kmki`r ac `rbqgae~l+;%0>% Zrlo`r c`y ykbqkae~ay fkzù~ayky qy`enl c` mqeikùe &;';+8
&`+ Zrqao` na fkzù~ayky eqc` ay Fl8 ο9 7 ο> 7 ο1 7 : ile~r` c` fkzù~ayky `c~are`~k}` na
iq`enl gaely qel na cly ~ray ilamkikae~ay ο9# ο># ο1 el ay kbq`c ` iarl% @yqgkr# d 0 7 ?#
d ? 7 1# e 7 0::# ΰ 7 :%:1# YYA0 7 2#:::%: nac glnacl `ora}k`nl { YYA? 7 =#:::%: nac
glnacl ilgzca~l% Na `iqarnl ` ay~ly n`~ly# ¼Iqàc na cly nly glnacly ay yqzarklr<
&o+ Zrqao` na fkzù~ayky eqc` Fl8ο> 7 ο1 7 ο= 7 ο2 7 : ile~r` F08 iq`enl gaely qel na
ay~ly ilamkikae~ay el ay kbq`c ` :% @yqgkr d 0 7 9# d ? 7 2# e 7 >1# ΰ 7 :%:1# YYa0 7 0#=::#
YYa? 7 ;::%:% ¼Iqàc na cly nly glnacly ay ac gahlr<
&i+ Fl8ο9 7 ο> 7 : ile~r` F08ο9 ≪ ο> ≪ : na pqa iq`enl gaely qel na cly nly
ilamkikae~ay el ay kbq`c ` :% @yqgkr d 0 7 ?# d ? 7 ># e 7 9:# ΰ 7 :%:1# yqg` nac arrlr na c`y
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 702/846
Nr% Fëi~lr Pqa}anl Qrç`y
;'=6
iq`nr`nly nac glnacl ykgzca ay 09:%: { c` yqg` na cly iq`nr`nly nac glnacl ilgzcahl
ay na 0::%:%
;%01% Ae qe` ke}ay~kb`ikùe rac`ikle`n` ile c` ile~`gke`ikùe nac `kra zlr ac lxlel# `ek}ac nac yqacl# ya y`iù qe` gqay~r` na 1 `ðly &0;;;'?::9+ zrlianae~a na qe` ay~`ikùe
gqay~ra`nlr` cli`ckx`n` ae ac Z`rpqa If`gkx`c ae Ac Z`yl# ^av`y% Ac g`e~aekgkae~l
{ i`ckor`ikùe na cly `z`r`~ly na ay~` ay~`ikùe gqay~ra`nlr` mqa faif` zlr c` A% Z% @% na
Cly Ay~`nly Qeknly% Ac ay~qnkl ileyky~kù ae ac zrliay`gkae~l ay~`nçy~kil na }`rk`ocay#
ilgl ac lxlel &L9+# ac gleùvknl na ek~rùbael &EL+# ac okùvknl na ek~rùbael &EL?+ {
c` ~agzar`~qr` ae br`nly M`fraefak~ &lM+% Ay~l ya fkxl ile ac loha~l na lo~aear qe
glnacl na rabraykùe ay~`nçy~kil z`r` mkeay na zrankiikùe% Ac zrliankgkae~l ileyky~kù ae
y`i`r cly zrlgankly &na cly }`clray ayz`ikl'~agzlr`cay na qe` flr`+# na i`n` qe` na c`y
> }`rk`ocay kenazaenkae~ay na i`n` qe` na c`y ?> flr`y nac nç` na i`n` gay na i`n` qel
na cly 1 `ðly% @zrlvkg`n`gae~a# ya zrliay`rle 026#1=: n`~ly &?> flr`y v 90 nç`y v 0?
gayay v 1 `ðly v > }`rk`ocay+% Cly zrlgankly na cly zrlgankly# ae z`r~ay zlr okccùe
&zzo+ ya n`e o`hl% F`iar cly ykbqkae~ay iàciqcly8
&`+ Br`mki`r cly n`~ly z`r` }ar ac ~kzl na c` mqeikùe bràmki` pqa ya zqan` ayzar`r%
Yqbaraeik`8 Qy`r ac z`pqa~a na ilgzq~`nlr` Aviac%
&o+ Z`r` lo~aear ac gahlr i`enkn`~l nac glnacl na rabraykùe qy`r qe –oay~ yqoya~
rabraykùe‗ &gahlr yqoilehqe~l na rabraykùe+ { qe –Y~azkya _abraykùe‗ &rabraykùe zlr
z`yly+% A}`cq`r c` q~kckn`n nac glnacl qy`enl cly irk~arkly _ ?# y# Iz { Z_AYY { cly
irk~arkly yqoha~k}ly &bràmki`y na rayknq`cay { zrqao` na elrg`ckn`n+%
&i+ Qy`enl ac glnacl na rabraykùe yacaiikle`nl# z`r` ac gay na hqckl# zranaikr c`ileiae~r`ikùe na lxlel# yk c` ileiae~r`ikùe na EL ay kbq`c ` >%: zzo# EL? kbq`c ` 00%6
zzo# { c ~agzar ~qr ay na ?9%1 lI%
C` ~`oc` na `o`hl gqay~r` c` kemlrg`ikùe rapqarkn`%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 703/846
Nr% Fëi~lr Pqa}anl Qrç`y
;'=;
^`oc` gly~r`enl cly n`~ly nac zrlocag` na `rrko`% &Ac`olr`ikùe zrlzk`+%Gay Lxlel &zzo+ EL &zzo+ EL? &zzo+ ^agzar`~qr` &lM+
Aearl 0=%2 ?6%? ?0%: >;%=6
Maorarl 0;%> ?9%: 06%; 19%:=G`rxl 9:%: 0?%1 0=%9 16%6?@orkc 9>%> 0:%? 0>%> =6%::G`{l 91%6 =%? 0?%6 22%9=Hqekl 92%1 >%: 0:%; 6?%;>Hqckl 96%2 9%9 0?%2 69%==@bly~l 9=%> 9%; 0>%> 69%0?Yaz~kagora 9:%2 6%6 0=%= 26%>>Li~qora ?0%? ;%6 ?:%; =2%0: El}kagora 0=%= 99%: ??%2 1=%9:Nkikagora 0>%; 9>%; ?9%6 >=%06
&`+ Qy`enl ac zrlbr`g` Aviac ke~rlnqikr cly n`~ly ae c` flh` na Aviac# na c` ykbqkae~ag`ear`8 Ae c` zrkgar` ilcqge` zlear cly gayay nac `ðl# ae c` yabqen` ilcqge` zlear
cly }`clray na L9# ae c` ~ariar` ilcqge` zlear cly }`clray na EL { ae c` øc~kg` ilcqge`
zlear cly } clray na EL?% Qe` }ax faifl ay~l krya `8
If`r~ Rkx`rn ↟ Ae c` }ae~`e` na If`r~'Rkx`rn'Y~az 0 lm > If`r~ 1 ↟ If`r~ {za ↟
Ckea ↟ Eav~ ↟ N`~` _`eba &ylgora`enl cly n`~ly+↟ Ilcqge ↟ Eav~ ↟ ^ç~qcl ↟
Mkeçyf% lnly ay~ly ùrnaeay baear`e c` bràmki` gly~r`n` o`hl%
:
0:
?:
9:
>:
1:
=:
2:
6:
;:
A M G @ G H H @ Y L E N
Ilei% L9
Ilei% EL
Ilei% EL
^agzar`~qr`
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 704/846
Nr% Fëi~lr Pqa}anl Qrç`y
;'2:
Ilgl ya }a ae c` bràmki`# c`y ileiae~r`ikleay na L9 yle nkrai~`gae~a zrlzlrikle`cay `
c`y ~agzar`~qr`y# zarl ke}ary`gae~a zrlzlrikle`cay ` c`y ileiae~r`ikleay na EL {
EL?% Ile c` pqçgki` `~glymërki`# ay~`y rac`ikleay g`~agà~ki`y ay~àe na `iqarnl ` qe`cùbki` ` zly~arklrk% ¼Zlr pqë ay `yç< Ya ca zkna `c cai~lr ile~ay~`r ay~` zrabqe~`%
Z`r` ac keikyl &o+ qy`enl ac Oay~ Yqoya~y _abrayykle na c` mqeikùe nac Gkek~`o ile [
}aryqy V?# V?# …… z`r` lo~aear ac ykbqkae~a aypqag` gly~r`nl `o`hl8 &Ac`olr`ikùe
zrlzk`+%
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ V V V V
0 ? 9 >Y Y Y YV V V P P P P
]`ry _'Yp _'Yp&`nh+ Iz y ? ? > _ _ _ _
0 ;2%> ;=%; ;1%? 0%=?;> V0 ;=%2 ;=%: 0?9%1 0%6>9> V? ;6%? ;2%1 =6%? 0%>1;2 V V? ;6%? ;2%1 =6%1 0%>=9: V V9 ;;%1 ;;%9 0=%: :%22;= V V V9 ;6%; ;6%9 >:%; 0%?:?; V V V> ;;%2 ;;%1 0:%2 :%=>6; V V V V> ;;%2 ;;%> 0?%> :%=;;1 V V V V(1 ;;%; ;;%6 1%> :%>:9= V V V V V1 ;;%; ;;%6 1%; :%>?;> V V V V V= ;;%; ;;%6 2%: :%>?0> V V V V V V= ;;%; ;;%6 2%? :%>909 V V V V V V2 ;;%; ;;%2 ;%: :%>61; V V V V V V VY~azkya rabrayykle8 [ }aryqy V0# V?# V9# V>#V0YP_# V9YP_
@czf` ~l ae~ar8 :%01 @czf` ~l ragl}a8 :%01
_ayzleya ky8 le = zranki~lry# k~f E 7 0? \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ Na `iqarnl ` cl loyar}`nl `rrko`# ya zqana naikr pqa# `c hqxb`r zlr cly }`clray na _ ?# y
{ Iz# ac glnacl gày `zrlzk`nl ay ac pqa avicq{a ` V ?? { ` V ?
># zarl pqa keicq{a `
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 705/846
Nr% Fëi~lr Pqa}anl Qrç`y
;'20
&EL?+# ilgl c` gahlr `c~are`~k}`# ay naikr# qy`enl qe glnacl na rabraykùe iq`nrà~kil
na c` mlrg` na `o`hl &pqa avicq{a ` V ?? { V ?
>+%
[ 7 οl $ ο0V0 $ ο?V? $ ο9V9 $ ο>V> $ ο1V?0 $ ο=V
?9
C` q~kckn`n nac glnacl i`enkn`~l ya n` na `iqarnl ` cly }`clray na8 y 7 :%>:9=# _ ? 7
;;%;!# _ ?`nh% 7 ;;%6!# _ ? zran% 7 ;;%2! { Z_AYY 7 9%002>% @nagày# f`ikaenl qe
`eàckyky na rabraykùe zlr z`yly ya loyar}` pqa# ykagzra ya }`e nayzraik`enl cly }`clray
na V ?? { V ?
># zarl ykagzra ya yacaiikle` ` V ?9 ilgl gahlr `c~are`~k}`% Ay~` yk~q`ikùe
ay ilemkrg`n` zlr cly }`clray na ^ { na Z gly~r`nly ae c` ~`oc` na `o`hl%
^`oc` gly~r`enl cly ilamkikae~ay { cly }`clray na { Z% &Ac`olr`ikùe zrlzk +Zranki~lr Ilam% YA Ilam ^ ZIley~`e~ 02%?29 =%?==: ?%2= :%:>:V0 '?%:1>> :%01:? '09%=2 :%:::V? ':%9216 :%001; '9%?> :%:?9V9 0%>92= :%0;;0 2%?? :%::0V> '0:%691: ?%6:6: '9%6= :%:0?V0YP_ :%:9?9 :%::?; 00%29 :%:::V9YP_ ':%:0?: :%::0= '2%>2 :%::0
Z`r` ile~ay~`r c`y zrabqe~`y nac keikyl &i+ qy`r ac glnacl na rabraykle yacaiikle`nl%
;%0=% C`y ~`oc`y na `o`hl gqay~r`e n`~ly y`i`nly na qe avzarkgae~l# ac iq`c ileyky~a ae
> }`rk`ocay kenazaenkae~ay% Ya qy` qe z`pqa~a na ilgzq~`nlr`# ac iq`c yacaiikle` ~ray
na cly glnacly i`enkn`~ly gày `zrlzk`nly%
&`+ Ilemkrg`r c` yacaiikùe nac cly ~ray i`enkn`~ly glnacly na rabraykùe gày zrlzk`nly
qy`enl ac z`pqa~a Gkek~`o# EIYY l Y@Y%
&o+ Na cly ~ray glnacly mke`cky~`y yað`c`nly ae c` ~`oc` na `o`hl# yacaiikle`r ac glnacl
gày ùz~kgl o`y`enl ac irk~arkl ae cly nk`beùy~kily ay~`nçy~kily _ ?# y# Z_AYY { Iz%
Ilgzcagae~`r c` naikykùe qy`enl aemlpqay yqoha~k}ky~`y# ay naikr# `e`ckx`enl cly
bràmkily na cly rayknqly ay~`en`rkx`nly% F`iar# `nagày# qe` zrqao` na elrg`ckn`n%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 706/846
Nr% Fëi~lr Pqa}anl Qrç`y
;'2?
^`oc` gly~r`enl cly n`~ly lrkbke`cay% &Ac`olr`ikùe zrlzk`+ \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
&[ + V 0 V ? V 9 V >
\\\\\\\ 2;%9 1%1 90 0: 6
?::%0 ?%1 11 6 =0=9%? 6%: =2 0? ;?::%0 9%: 1: 2 0=0>=%: 9%: 96 6 01022%2 ?%; 20 0? 029:%; 6%: 9: 0? 6
?;0%; ;%: 1= 1 0:0=:%: >%: >? 6 >99;%> =%1 29 1 0=01;%= 1%1 =: 00 26=%9 1%: >> 0? 0?
?92%1 =%: 1: = =0:2%? 1%: 9; 0: >011%: 9%1 11 0: >
C` ~`oc` na `o`hl gqay~r` cly ~ray gahlray i`enkn`~ly na glnacly# z`r` pqa ac cai~lr
f`b` qe` naikykùe ylora iq`c na cly ~ray glnacly ay ac gahlr% F`iar ay~` naikykùe mke`c
o`yàenlya ae cly irk~arkly ay~`nçy~kily _ ?# y# Z_AYY { Iz% ¼Zqnkar` qe` ke~ar`iikùe
gahlr`r ac glnacl na rabraykùe<
^`oc` gly~r`enl cly rayqc~`nly% &Ac`olr`ikùe zrlzk`+
Glnacl na rabraykùe Mi`ci% _ ? y Z_AYY Iz Nqroke'R`~yle
V?# V9 ;;6 :%;;>: =%=2>; 26?%06;= 00%>:09 0%;0
V0# V?# V9 0?:: :%;;2: >%;2;1 =>9%9126 9%>:21 ?%:?
V0# V?# V9# V> 61? :%;;20 1%00;9 2>0%2112 1%:::: ?%:?
;%02% Ay~a zrlocag` ay~à rac`ikle`nl ile qe` kemlrg`ikùe na n`~ly na qe avzarkgae~l
rac`ikle`nl ae~ra ac zF & V + { c` ilenqi~k}kn`n acëi~rki` &[ +% Cly n`~ly ya n`e ae c` ~`oc`
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 707/846
Nr% Fëi~lr Pqa}anl Qrç`y
;'29
na `o`hl &ac`olr`ikùe zrlzk`+% O`y`enl ac r`xle`gkae~l ae cly rayqc~`nly n`nly zlr ac
z`pqa~a Gkek~`o# naiknkr yk ac glnacl na rabraykùe gày `zrlzk`nl ay qe glnacl na
iq`nrà~kil l qe glnacl na rabraykùe iøokil%
Pq`nr`~ki _abrayykle @e`c{yky8 &[+ }aryqy &V+# VYP_ ^fa rabrayykle apq`~kle ky8&[+ 7 >=%; ’ 0;%; &V+ $ ?%0? VYP
Zranki~lr Ilam YA Ilam ^ ZIley~`e~ >=%;:2 ;%>9? >%;2 :%:::&V+ '0;%;:; >%90: '>%=? :%:::VYP ?%00=0 :%>;00 >%90 :%::0
Y 7 :%:;99? _'Yp 7 ;>%:! _'Yp&`nh+ 7 ;9%0!Z_AYY 7 :%029?:0 _'Yp&zran+ 7 ;:%66!
@e`c{yky lm ]`rk`eia ^`ocaYlqria NM YY GY M Z_abrayykle ? 0%26126 :%6;?6; 0:?%19 :%:::_ayknq`c Arrlr 09 :%009?? :%::620
^l~`c 01 0%6;;::
Iqoki _abrayykle @e`c{yky8 &[+ }aryqy &V+# VYP_# VYIQOA^fa rabrayykle apq`~kle ky8&[+ 7 ?>6 ’ 016&V+ $ 99%= VYP_ ’ ?%>: VIQOA
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 708/846
Nr% Fëi~lr Pqa}anl Qrç`y
;'2>
Zranki~lr Ilam YA Ilam ^ ZIley~`e~ ?>2%; ?:=%> 0%?: :%?19&V+ '012%; 0>0%= '0%00 :%?62
VYP 99%=> 9?%91 0%:> :%90;VIQOA '?%9;2 ?%>1; ':%;2 :%9>;
Y 7 :%:;91: _'Yp 7 ;>%1! _'Yp&`nh+ 7 ;9%0!Z_AYY 7 :%02?2;; _'Yp&zran+ 7 ;:%;:!
@e`c{yky lm ]`rk`eia ^`oca \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
NM YY GY M Z_abrayykle 9 0%2;>:; :%1;6:9 =6%>0 :%:::_ayknq`c Arrlr 0? :%0:>;0 :%::62>^l~`c 01 0%6;;::
;%06% Ya n`e cly ykbqkae~ay n`~ly rac`ikle`nly ile c` g`eqm`i~qr` na ifqg`iar`y z`r`
}afçiqcly% Ya ylyzaif` pqa ikar~`y gankikleay el ay~àe nae~rl nac r`ebl zargk~knl#
zlykocagae~a# naoknl m cc`y na cly lzar`nlray l ~`c }ax na c g`pqke`rk %
^`oc` gly~r`enl cly n`~ly nac zrlocag` &Ac`olr`ikùe zrlzk`+%Gankikleay s ?%21 ?%=? ?%2> 9%61 ?%9> ?%2> 9%;9 >%?0 9%66 >%99 9%>= >%1? ?%>9 9%=1 ?%26 9%1= 9%:0
El% gqay~r`s 0 ? 9 > 1 = 2 6 ; 0: 00 0? 09 0> 01 0= 02
F`iar cly ykbqkae~ay iàciqcly8
&`+ Zlear cly n`~ly ae mlrg` yiaenae~a%
&o+ Na~argke`r ac }`clr nac iq`r~l kemarklr nac iq`r~l yqzarklr%
&i+ I`ciqc`r c` iq`r~` nkyzarykùe m y# 0%1m y { 9m y%
&n+ I`ciqc`r qe glnacl na rabraykùe pqa keicq{` ~lnly cly n`~ly%
&a+ I`ciqc`r l~rl glnacl na rabraykùe pqa avicq{` cly }`clray `~çzkily av~ragly
i`ciqc`nly ae cly keikyly `e~arklray%
&m+ Na `iqarnl ` cly nk`beùy~kily loha~k}ky~`y { yqoha~k}ky~`y# na~argke`r iq`c na cly nly
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 709/846
Nr% Fëi~lr Pqa}anl Qrç`y
;'21
glnacly ay yqzarklr%
;%0;% Ya n` c` ~`oc` na `o`hl ile n`~ly rac`ikle`nly ile c`y ileiae~r`ikleay na
gleùvknl na i`rolel &IL+ agk~kn`y zlr gl~lray na ilgoqy~kùe ke~are % Yke ago`rbl#ya `rbqgae~` pqa# ac `z`r`~l `e`ckx`nlr pqa gqay~ra`o` ac IL# zqnl f`oar ~aeknl m`cc`y
nqr`e~a ac gqay~ral na IL naoknl ` pqa ya el~`rle }`clray mqar` na cl elrg`c% Z`r`
}arkmki`r yk ae }arn`n fqol }`clray `~çzkily ae c`y ileiae~r`ikleay na IL# ya rapqkara
y`oar# iq`cay mqarle cly }`clray av~ragly% Z`r` ~`cay mkeay qy`r nk`br`g`y na i`h` pqa
knae~kmkpqae }`clray `~çzkily av~ragly% Z`r` ay~l ya n` c` ~`oc` na `o`hl%
^`oc` gly~r`enl cly }`clray na c`y ileiae~r`ikleay na gleùvknl na i`rolel &zzg+%&Ac`olr`ikùe zrlzk`+%Ileiae~r`ikùe na IL s ;1 ;: ;: 6: 21 =1 >1 =: 12 ;1 ;2 09: 09: 0?:
0:1 0:9 0:: ;; ;; El% na loyar}`ikùe s 0 ? 9 > 1 = 2 6 ; 0: 00 0? 09 0> 01 0= 02
06 0;
F`iar cly ykbqkae~ay iàciqcly8
&`+ Lrnae`r cly n`~ly ae mlrg` yiaenae~a%
&o+ Ay~kg`r c` gank`e`# ac }`clr gàvkgl# ac }`clr gçekgl# ac iq`r~l kemarklr# ac iq`r~l
yqzarklr# c` iq`r~` nkyzarykùe m y# 0%1m y# 9m y# ac iq`r~l kemarklr P0 { ac iq`r~l yqzarklr P9%
&;1# 09:# >1# 22%1# 0:># ?=%1# 21# 0:9+
&i+ F`iar qe` bràmki` ile qe nk`br`g` na i`h` { f`iar ilgae~`rkly `c rayzai~l%
&n+ Knae~kmki`r cly }`clray `~çzkily keqyq`cay av~ragly na IL%
&a+ Ilrrar qe glnacl na rabraykùe# ay naikr# `yqgkaenl qe glnacl na rabraykùe
iq`nrà~kil# ile cly }`clray lrkbke`cay {# l~rl gày# ile cly }`clray av~ragly ackgke`nly%¼F`{ qe gahlr`gkae~l ykbekmki`e~a ae ac glnacl ilrrabknl# ay naikr# na `iqarnl ` cly
}`clray na _ ?# _ ?`hqy~`n`# y { Z_AYY# na i`n` qel na cly nly glnacly na rabraykùe
iq`nrà~kily# ay~l ay# keicq{aenl { avicq{aenl cly }`clray `~çzkily av~ragly< &^`c
z`raia pqa yk f`{ qe gahlr`gkae~l ykbekmki`e~a ile ac glnacl na rabraykùe iq`nrà~kil#
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 710/846
Nr% Fëi~lr Pqa}anl Qrç`y
;'2=
pqa el keicq{a cly }`clray av~ragly% O`hl ay~`y ilenkikleay# cly }`clray na cly
nk`beùy~kily ay~`nçy~kily# z`r` ac glnacl na rabraykùe# yke cly }`clray `~çzkily av~ragly
yle8 _ ?
7 ;6#>!# _ ?`hqy~`n` 7 ;6%?!# y 7 ?%10# Z_AYY 7 091%2>% Ae ile~r`y~a# z`r` ac
glnacl na rabraykùe iq`nrà~kil# pqa keicq{a ~lnly cly }`clray `~çzkily av~ragly# cly
}`clray na cly nk`beùy~kily ay~`nçy~kily yle8 _ ? 7 ;9%=# _ ?`hqy~`n` 7 ;?%6!# y 7 =%?= {
Z_AYY 7 ;>;%22+
&m+ Na `iqarnl ` cly nk`beùy~kily loha~k}ky~`y { yqoha~k}ky~`y# na~argke`r iq`c na cly
nly glnacly ay yqzarklr%
;%?:% Ac ~av~l na H`{ C% Na}lra ke~k~qc`nl Zrlo`okckn`n { Ay~`nçy~ki` z`r` Kebaekarç`
{ Ikaeik`y &?::0+ ik~` qe` ke}ay~kb`ikùe z`r` na~argke`r c` ileiae~r`ikùe na ili`çe`
ae c` y`ebra &gb,C+ ae qe` gqay~r` na kenk}knqly pqkaeay gqrkarle na nackrkl avik~`nl
&NA+ naoknl `c qyl na c` ili`çe`% @nagày# fqol l~r` gqay~r` na ili`çe` ae c` y`ebra
na l~rl brqzl na `nki~ly ` ay~` nrlb`# pqkaeay gqrkarle zlr yloranlyky# yke nackrkl
avik~`nl% Ac ~kagzl na yqzar}k}aeik` na `goly brqzly mqa na = flr`y% Cly n`~ly
`nhqe~ly ya br`mki`rle ae qe nk`br`g` na i`h`% Ay~a ay~qnkl ya zqockiù ae c` ra}ky~`
–M`~`c Avik~an Nackrkqg Mlcclkeb Ili`kea Qya‗ &H% lm Mlraeyki Yikaeiay# 0;;2# zz%
?1'90+% Cly n`~ly na ay~a ay~qnkl ya n`e ae c` ~`oc` na `o`hl%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 711/846
Nr% Fëi~lr Pqa}anl Qrç`y
;'22
^`oc` gly~r`enl cly rayqc~`nly na ay~a zrlocag`%
Ile nackrkl avik~`nl &NA+
: : : : %0 %0 %0 %0 %? %? %9 %9 %9 %> %1 %2 %6 0%: 0%1 ?%2?%6 9%1 >%: 6%; ;%? 00%2 ?0%:
Yke nackrkl avik~`nl &Yke NA+
: : : : %0 %0 %0 %0 %? %? %? %9 %9 %9 %> %1 %= %1 %2 %= %6
0%: 0%? 0%> 0%1 0%2 ?%: 9%? 9%1 >%0 >%9 >%6 1%: 1%= 1%; =%: =%>
2%; 6%9 6%2 ;%0 ;%= ;%; 00%: 00%1 0?%? 0?%2 0>%: 0=%= 02%6
&Mqae~a8 Na}lra# ?::0+&`+ Na~argke`r c`y gank`e`y# ac iq`r~l kemarklr P0# ac iq`r~l yqzarklr P?# cly iq`r~ly
kemarklray { yqzarklray { c`y iq`r~`y nkyzarykleay m y na c`y nly gqay~r`y { ac zrlgankl%
&Z`r` AN8 %>#%0?%21#?%=1# ?%=:24 z`r` el AN8 0%=# %9# 2%;# 2%=:# >%?1+
&o+ Knae~kmki`r cly }`clray `~çzkily glnar`nly { av~ragly% &AN8 6%; { ;%? yle }`clray
`~çzkily glnar`nly { 00%2 { ?0%: yle }`clray `~çzkily av~ragly% Ae c` gqay~r` na el
AN8 el f`{ }`clray `~çzkily+%
&i+ ^r`x`r qe nk`br`g` na i`h` ilgz`r`~k}l { qy`rcl z`r` ilgz`r`r { nkmaraeik`r c`y
gqay~r`y ile { yke nackrkl avik~`nl% &Avky~a qe` `ykga~rç` zlyk~k}` `zraik`oca ae AN {
ae el AN4 gaelr }`rk`okckn`n ae loyar}`ikleay na c` gqay~r` AN# ay~l ay# gaelr m y%
@nagày# c`y loyar}`ikleay na c` gqay~r` el AN yle g`{lray pqa c`y na c` gqay~r` el
AN+
;%?0% Ac ~av~l na pqçgki` ke~k~qc`nl Ifagky~r{8 ^fa Iae~r`c Yikaeia na Orle a~% `c%
&?:::+# nkyiq~a c` m`ya b`yaly` na c` nayilgzlykikùe na EL?# c` iq`c ay n`n` zlr8 EL?&b`y+ ↟ EL&b`y+ $ · L?&b`y+
&`+ Naikr yk c` ra`iikùe ay na zrkgarl l yabqenl lrnae ile rayzai~l ` c` ileiae~r`ikùe
na EL?% Nayzqëy# r`~kmki`r c` naikykùe faif` qy`enl ~ëieki`y na rabraykùe a}`cq`n`y
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 712/846
Nr% Fëi~lr Pqa}anl Qrç`y
;'26
zlr ay~`nçy~kily loha~k}ky~ y &ilgl _ ?# y# Z_AYY { @EL]@+ { ilgzcagae~`n`y zlr
gankl na bràmkily yqoha~k}ky~`y &ilgl zrqao` na elrg`ckn`n# rayknqly }y% }`clray
`hqy~`nly# a~i%+% @nagày# i`ciqc`r ac }`clr na c` iley~`e~a na c` ra`iikùe d &zaenkae~a+%Cly }`clray ya n`e `o`hl%
^`oc` gly~r`enl cly n`~ly nac zrlocag` \\\\\\\\\\\\\\\\\\\\\\\\\\\ ^kagzl &yab+ TEL?W &G+
\\\\\\\\\\\\\\\\\\\\\\\\\\\ :%: :%0:::1%: :%:02:
0:%: :%::;:01%: :%::=??:%: :%::>2
\\\\\\\\\\\\\\\\\\\\\\\\\\\ Mqae~a8 Ifagky~r{8 ^fa Iae~r`c Yikaeia% Orle a~ `c% &?:::+
;%??% Cly `q~lray Y`{ar I% E%# Zarr{ C% GiI`r~{ nac ckorl Ifagky~r{ mlr Y`ek~`r{
Aebkeaary# ?en% Ank~kle &0;=2+ zrlzlrikle`e cly ykbqkae~ay n`~ly zrl}aekae~ay na qe
avzarkgae~l z`r` a}`cq`r c` naykemaiikùe na qe `cg`iae`gkae~l na `bq` ile qe`
nlyky na iclrl n`n` z`r` g`~`r c`y o`i~ark`y ilckmlrgay% Qy`enl ac zrlbr`g` Gkek~`o
l iq`cpqkar l~rl zrlbr`g` na ilgzq~`nlr`# ilrrar qe `eàckyky na rabraykùe
ay~`nçy~kil { f`iar cl ykbqkae~a8
&`+ Naikr ac lrnae na c` ra`iikùe na ay~ly n`~ly% &Zrkgar lrnae+
&o+ ¼Pqa ~`e okae aei`h`e cly n`~ly ae ac glnacl na rabraykùe< Z`r` ay~l# qy`r qe
irk~arkl loha~k}ky~` { qel yqoha~k}ky~` z`r` hqy~kmki`r c` `yarikùe%
&i+ I`ciqc`r c` }kn` gank`
&n+ I`ciqc`r c` ~`y` na c` ra`iikùe &:%06>6+
&a+ ¼Zranaikr ac ~kagzl pqa ya cca}`rç` z`r` `ekpqkc`r ac 1:! na c`y o`i~ark`y
ilckmlrgay<
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 713/846
Nr% Fëi~lr Pqa}anl Qrç`y
;'2;
^`oc` gly~r`enl cly n`~ly nac zrlocag`% \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ ^kagzl &gke+ Zlriae~`ha na ilckmlrgay
pqa }`e pqan`enl \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
: 0::0: 2:?: ?09: =%9=: :%=
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ &Mqae~a8 Y`{ar a~ `c% 0;=2+
;%?9% Ac ~av~l na Gleblgar{# Zaid { ]kekeb# ke~k~qc`nl Ke~rlnqiikùe `c @eàckyky na
_abraykùe Ckea`c &?::0+ n` qe ay~qnkl rac`ikle`nl ile c` kebaekarç` pqçgki` {
gaiàeki` ae c` iq`c ya eaiayk~` ileliar c` zraykùe na }`zlr na `bq` ` nk}ary`y
~agzar`~qr`y4 z`r` ay~l ya zqanae qy`r c` –kem`gay‗ ~`oc`y na }`zlr% Cly n`~ly na c`
zraykùe na }`zlr { nac bq` ` nk}ary`y ~agzar`~qr`y ya n`e `o`hl%
^`oc` gly~r`enl cly n`~ly nac zrlocag`% \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ { 7 zraykùe na }`zlr na `bq` &gg Fb+ v 7 ^agzar`~qr` &lI+ \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
;%? 0:02%1 ?:90%6 9:11%9 >:;?%1 1:
0>;%> =:Mqae~a8 Gle~blgar{ a~ `c% &?::0+
&`+ @hqy~`r qe glnacl na rabraykùe na zrkgar lrnae yqy~ae~`nl zlr ay~`nçy~kily
loha~k}ly { bràmkily%
&o+ @hqy~`r qe glnacl na rabraykùe na yabqenl lrnae yqy~ae~`nl ile ay~`nçy~kily
loha~k}ly { bràmkily%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 714/846
Nr% Fëi~lr Pqa}anl Qrç`y
;'6:
&i+ Na `iqarnl ` cly rayqc~`nly lo~aeknly ae cly keikyly &`+ { &o+ naiknkr iq`c na cly
nly glnacly ay yqzarklr# ay naikr# ac glnacl na rabraykùe pqa `hqy~` gahlr ` cly
n`~ly%;%?>% Ae qe avzarkgae~l rac`ikle`nl ile c` }aclikn`n nac }afçiqcl { ac ileyqgl na
b`ylcke` ya ay~qnk` ae qe` gqay~r` na qe ~`g`ðl 01# ay naikr# qy`enl qe ylcl
}afçiqcl% Cly n`~ly ya n`e ae c` ~`oc` na `o`hl%
^`oc` gly~r`enl cly n`~ly na ay~a avzarkgae~l% \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ ]aclikn`n &dg,fr+ s 12 12%= => == == 6: 60 6;%= ;6 ;;
Ileyqgl na b`ylcke` &C,dg+ s ?: ?0 ?1 ?=%9 ?=%1 ?; ?; ?2 ?1%1 ?1
F`iar cly ykbqkae~ay iàciqcly8
&`+ Knae~kmki`r c` }`rk`oca nazaenkae~a { c` }`rk`oca kenazaenkae~a
&o+ Br`mki`r cly n`~ly na c` }`rk`oca nazaenkae~a }aryqy c` }`rk`oca kenazaenkae~a%
&i+ @hqy~`r ac glnacl na rabraykùe pqa gahlr aei`ha ae cly n`~ly
&n+ A}`cq`r c` q~kckn`n nac glnacl i`enkn`~l gank`e~a `eàckyky loha~k}ky~`y &_ ?# _ ?`hqy~`n`#
arrlr ay~àen`r na cl ay~kg`nl y# Z_AYY# { `eàckyky na }`rk`ex`+% Ilgzcagae~`r c`naikykùe lo~aekn` qy`enl bràmkily yqoha~k}ky~`y &Bràmkily na zrqao` na elrg`ckn`n#
rayknq`cay }aryqy }`clray hqy~`nly na [# rayknq`cay }aryqy lrnaeay# a~i%+
&a+ Qe` }ax pqa ya f`{` lo~aeknl ac glnacl yqzarklr# zranaikr ac ileyqgl na b`ylcke`
iq`enl c` }aclikn`n ay na ;= dg,fr
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 715/846
Nr% Fëi~lr Pqa}anl Qrç`y
0:'0
I@ZK^QCL 0:
Ay~`nçy~ki` el z`r`gë~rki`% Ac glnacl na @EL]@ ckora
]ae~`h`y na cly gë~lnly el z`r`gë~rkily%' Nay}ae~`h`y na cly gë~lnly el
z`r`gë~rkily%' Zrqao` na F na Drqyd`c'R`ccky z`r` `eàckyky na }`rk`ex` zlr
r`ebly%' Zrqao`y na fkzù~ayky ile c`y mqeikleay el z`r`gë~rki`y%'
Zrliankgkae~ly na zrqao`y na Drqyd`c'R`ccky z`r` @EL]@ ykgzca%' Zrqao`y
na fkzù~ayky el ~r`nkikle`cay# z`r` c` zrqao` na Drqyd`c'R`ccky# ay naikr#
qy`enl ac }`clr na c` zrlo`okckn`n z%'
Iq`enl ya ay~qnk`e zrliankgkae~ly ckoray l na zrqao`y el z`r`gë~rki`y ya keicq{ae
c` zrqao` na yqg` na r`ebly na Rkcilvle# c` zrqao` na Drqyd`c'R`ccky z`r`
nkyaðly ilgzca~`gae~a `ca`~lrkx`nly# c` zrqao` na Mrkang`e# c` zrqao` na
Dlcglblrl}'Ygkrel}# a~i% Avky~ae gqif`y `zcki`ikleay ae c` ikaeik` { ae c`
kebaekarç` nlena cly n`~ly ya razlr~`e# el ilgl }`clray ile~keqly# ykel ae qe`
ayi`c` lrnke`c na ~`c g`ear` pqa ya zqan`e `ykbe`r r`ebly ` cly n`~ly lo~aeknly%
^lnly cly gë~lnly nkyiq~knly `e~arklrgae~a# ilgl c` nky~rkoqikùe elrg`c# c`
nky~rkoqikùe na ~ na ay~qnk`e~a# c` nky~rkoqikùe na M# ac glnacl na rabraykùe# a~i%# ya
cc`g`e gë~lnly ay~`nçy~kily z`r`gë~rkily% Ay~l ya naoa ` pqë# ay~`y nky~rkoqikleay
ile~keq`y `yqgae pqa c` }`rk`ikùe `ca`~lrk` na cly n`~ly naoa na yabqkr ` c`
yqzlykikùe na elrg`ckn`n% Yke ago`rbl# avky~ae yk~q`ikleay ae pqa c`y
yqzlykikleay na elrg`ckn`n el ya y`~kym`iae z`r` c`y zrqao`y na fkzù~ayky% Z`r`
raylc}ar ay~a zrlocag`# cly ay~`nçy~kily f`e nkyað`nl }`rk`y `c~are`~k}`y z`r`
`pqaccly ke}ay~kb`nlray pqa ay~ëe raeqae~ay ` `iaz~`r c`y yqzlykikleay na
elrg`ckn`n# ay naikr# na mqeikleay el z`r`gë~rki`y% Ay~ly zrliankgkae~ly el
z`r`gë~rkily ya `zcki`e kbq`cgae~a ` nky~rkoqikleay z`r`gë~rki`y { ` nky~rkoqikleay
el z`r`gë~rki`y%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 716/846
Nr% Fëi~lr Pqa}anl Qrç`y
0:'?
Ae ac qyl na c`y ay~`nçy~ki`y el z`r`gë~rki`y# ilgl c` zrqao` na Drqyd`c'
R`ccky# c` zrqao` na ykbely# c` zrqao` na Rkcilvle# c` zrqao` na r`ebly na ykbely#
a~i%# ay eaiay`rkl `ic`r`r pqë# ay~ly zrliankgkae~ly el z`r`gë~rkily el yle ~`e
zlnarlyly ilgl yqy ile~r`z`r~ay# ay naikr# cly gë~lnly z`r`gë~rkily# ilgl c`
nky~rkoqikùe elrg`c# c` ~ na ay~qnk`e~a# a~i% Ay~l ya naoa ` pqë# z`r` qe`
zrlo`okckn`n na arrlr K# c`y zrqao`y el z`ragë~rki`y n`ràe qe` zrlo`okckn`n gày
`c~` nac arrlr ~kzl KK% Ay~l ~`gokëe liqrra# zlrpqa qe` zrqao` pqa kbelra c`
elrg`ckn`n na cly n`~ly &ilgl cl f`iae c`y zrqao`y el z`r`gë~rki`y+# el yarà ~`e
oqae` ilgl `pqacc`y zrqao`y pqa c` ykb`e &ilgl c`y z`r`gë~rki`y+% L~r` ckgk~`ikùe
na c`y zrqao`y el z`r`gë~rki`y ay pqa c`y zloc`ikleay gqay~ra`n`y naoae ya yar
kenazaenkae~ay% Ay~l pqkara naikr pqë# qe brqzl el naoa na ~aear kemcqaeik` ylora
ac l~rl% Yke ago`rbl# f`ikaenl ` qe c`nl c` ilenkikùe na elrg`ckn`n# ae pqa ya
o`y`e c`y ay~`nçy~ki`y z`r`gë~rki`y# c`y zrqao`y el z`r`gë~rki`y ~kaeae gqif`y
`zcki`ikleay ae ac i`gzl na c` kebaekarç`%
]ae~`h`y na cly gë~lnly el z`r`gë~rkily
0% Ay~ly gë~lnly zqanae `zcki`rya ` qe br`e eøgarl na yk~q`ikleay# zlrpqa el
rapqkarae na c`y ilenkikleay na elrg`ckn`n rapqarkn`y zlr yqy ile~r`z`r~ay
z`r`gë~rki`y { yle gày ykgzcay pqa yq ile~r`z`r~a# cly gë~lnly z`r`gë~rkily%
?% Ae ile~r`y~a ile cly gë~lnly z`r`gë~rkily# cly gë~lnly el z`r`gë~rkily zqanae
yar `zcki`nly ` n`~ly el eqgërkily%
Nay}ae~`h`y na cly gë~lnly el z`r`gë~rkily
0% Cly gë~lnly el z`r`gë~rkily ~kaenae nayzarnkik`r kemlrg`ikùe# zlrpqa cly n`~ly
eqgërkily av`i~ly qyq`cgae~a ya ranqiae ` mlrg` iq`ck~`~k}`% Zlr ahagzcl# ae qe`
zrqao` el z`r`gë~rki`# nkb`gly na zrqao`y na ykbely# c` zërnkn` na zayl zlr
nka~ky~`y ya rabky~r`e ykgzcagae~a ykbely eab`~k}ly% Ile ay~a gë~lnl na ykbely# c`
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 717/846
Nr% Fëi~lr Pqa}anl Qrç`y
0:'9
zërnkn` na zayl na qe` ylc` ckor`# raikoa c` gkyg` razrayae~`ikùe pqa c` zërnkn` na
1: ckor`y%
?% C`y zrqao`y el z`ragë~rki`y el ~kaeae c` amkikaeik` na c`y zrqao`y z`r`gë~rki`y%
Ay~l ya naoa ` pqë# ile cly gë~lnly el z`r`gë~rkily# ae c`y zrqao`y na fkzù~ayky ya
eaiayk~` qe` mqar~a a}knaeik`# `e~ay na pqa ya zqan` raif`x`r c` fkzù~ayky%
C` ^@OC@ 0:%: gqay~r` qe` ilgz`r`ikùe ae~ra cly gë~lnly z`r`gë~rkily { cly
el z`r`gë~rkily%
^@OC@ 0:%:% ^`oc` gly~r`enl qe` ilgz`r`ikùe ae~ra cly gë~lnly z`r`gë~rkily {
cly gë~lnly el z`r`gë~rkily% &Ac`olr`ikùe zrlzk`+
@zcki`ikùe Zrqao` Zrqao` el Amkikaeik`
z`r`gë~rki` z`r`gë~rki`N`~ly z`ra`nly z`r` Zrqao` na x Zrqao` na :%=9
gqay~r`y nazaenkae~ay l na ~( ykbel
N`~ly z`r` gqay~r`y Zrqao` na x Zrqao` na ykbely na :%;1
kenazaenkae~ay l na ~(( r`ebly na Rkcilvle
]`rk`y gqay~r`y kenazaenkae~ay @eàckyky na }`rk`ex` Zrqao` na Drqyd`c' :%;1
&@EL]@+ &zrqao` M+ R`ccky
Ilrrac`ikùe Ilrrac`ikùe ckea`c Zrqao` na ilrrac`ikùe :%;0
na r`ebly
@ca`~lrkan`n Zrqao` el z`r`gë~rki` Zrqao`y ilrrkn`y El f`{
o`ya \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
( ~ 7 & N ' ±l+, yN,∘ e
(( ~ 7 & V 0 ’ V ?+ ’ &±0 ’ ±?+ , +00
&
?0
?
ee y z
$
(( x 7 & V 0 ’ V ?+ ’ &±0 ’ ±?+ ,ee ?
?
?
0
?
0 ω ω $ ile ω0 { ω? ilelikn`y
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 718/846
Nr% Fëi~lr Pqa}anl Qrç`y
0:'>
Zrqao` na F na Drqyd`c'R`ccky z`r` `eàckyky na }`rk`ex` zlr r`ebly
C` øeki` zrqao` pqa ya nkyiq~krà ae ay~a i`zç~qcl ay c` zrqao` na Drqyd`c'R`ccky# c`
iq`c ay~à rac`ikle`n` ile c` ay~`nçy~ki` z`r`gë~rki` nac `eàckyky na }`rk`ex`% Ac
`eàckyky na }`rk`ex` l @EL]@ z`r`gë~rkil ya `zcki` z`r` }ar yk ~ray l gày
zrlgankly zloc`ikle`cay yle kbq`cay# ay naikr# pqa el f`{ nkmaraeik`y ae cly
zrlgankly% Yke ago`rbl# `pqç ya `yqga pqë# c`y zloc`ikleay gqay~ra`n`y# ay~àe
elrg`cgae~a nky~rkoqkn`y {# c`y nay}k`ikleay ay~àen`r na ay~`y nky~rkoqikleay yle
kbq`cay%
El loy~`e~a# yk el ya zqanae yabqkr c`y yqzlykikleay na c` @EL]@
z`r`gë~rki`# cl `zrlzk`nl ay qy`r c`y zrqao`y el z`r`gë~rki`y# ilgl c` zrqao` na
Drqyd`c'R`ccky# zrqao` na ykbely na r`ebly na Rkcilvle l zrqao` Q na G`ee'
Rfk~ea{% Agzarl# ilgl {` ya nkhl# `pqç naoa avky~kr kenazaenaeik` ae~ra c`y
zloc`ikleay gqay~ra`n`y% Na iq`cpqkar g`ear`# yk ya raøea ay~` ilenkikùe#
ae~leiay yk zlnagly qy`r c` mqeikùe na Drqyd`c'R`ccky z`r` f`iar `eàckyky na
}`rk`ex`# z`r` }ar yk avky~ae nkmaraeik`y ae~ra cly zrlgankly zloc`ikle`cay%
Zrliankgkae~ly na zrqao`y na Drqyd`c'R`ccky z`r` @EL]@ ykgzca
0% ^lnly cly }`clray na c` gqay~r` yle ilgoke`nly%
?% Cly r`ebly lrnae`nly yle nac gày `c~l `c gày o`hl%
9% Cly }`clray lrnae`nly ya raagzc`x`e zlr r`ebly agzax`enl zlr qel z`r` ac gày
zapqaðl f`y~` ac gày `c~l%
>% Z`r` c` zrqao` na fkzù~ayky ya qy` c` nky~rkoqikùe na c` HK iq`nr`n` { ay qekc`~ar`c
naraif`% Z`r` c` zrqao` el ~r`nkikle`c ya qy` ac iàciqcl na c` zrlo`okckn`n z qy`enl
c` ~`oc` na c` nky~rkoqikùe na c` HK iq`nr`n` &ς ?+%
C` mqeikùe na Drqyd`c'R`ccky ya naykbe` zlr c` mqeikùe F pqa ay~à gq{
iari`e` ` c` nky~rkoqikùe na c` HK iq`nr`n`% Ay~` mqeikùe ya n` ilgl8
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 719/846
Nr% Fëi~lr Pqa}anl Qrç`y
0:'1
0? &Ω_ 0+?
&Ω_ ?+?
&Ω_ d +?
F 7 ┊┊┊┊┊┊┊ T ┊┊┊┊┊┊┊ $ ┊┊┊┊┊┊ $ …… $ ┊┊┊┊┊┊ W ’ 9&E $ 0+ &0:'0+
E&E$0+ e0 e? ed
Ile η 7 d ’ 0 br`nly na ckoar~`n
Nlena d ay ac eøgarl na zloc`ikleay gqay~ra`n`y l eøgarl na gqay~r`y
Nlena8
Ω_ 0# Ω_ ?#… Ω_ d 7 yqg` na cly r`ebly z`r` c` d 'ëykg` gqay~r` e0# e?# …#ed 7
~`g`ðly na gqay~r`y 0# ?# %%# d
E 7 eøgarl ~l~`c na c`y loyar}`ikleay z`r` ~ln`y c`y gqay~r`y ilgoke`n`y%
Yk ac }`clr ilgzq~`nl na c` ay~`nçy~ki` F i`a ae c` rabkùe irç~ki` naraif` &c`y
zrqao`y ykagzra yle qekc`~ar`cay naraif`y+# ay naikr# F 5 ς ?ΰ4 η# ile η 7 d ’ 0 br`nly
na ckoar~`n# ae~leiay# ya raif`x` Fl8 `c ek}ac na ykbekmki`eik` qy`nl% Na l~r`
g`ear` ya ra~kaea Fl8
Zrqao`y na fkzù~ayky ile c`y mqeikleay el z`r`gë~rki`y
Z`r` c`y zrqao`y na fkzù~ayky ~r`nkikle`cay ya qy`e cly gkygly ~ërgkely pqa ae c`y
zrqao`y z`r`gë~rki`y na @EL]@# ay naikr# yk F 5 ς ?ΰ# ile η 7 d ’ 0 br`nly na
ckoar~`n i`a ae c` rabkùe irç~ki` naraif`# ya raif`x` c` fkzù~ayky yqy~ae~`n`# Fl8% Na
l~r` g`ear` ya ra~kaea# ya `iaz~` l el ya f`ia ekebqe` naikykùe%
C` zrqao` na fkzù~ayky eqc` z`r` c` zrqao` na Drqyd`c'R`ccky ay c`
~r`nkikle`c# ay naikr8
Fl8 ±0 7 ±? 7 ±9 7 …… 7 ±d &0:'?+
C` fkzù~ayky `c~are`~k}` ay8
F08±0 ≪ ±? ≪ ±9 ≪ …%≪ ±d &0:'9+
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 720/846
Nr% Fëi~lr Pqa}anl Qrç`y
0:'=
Cly ek}acay na ykbekmki`eik` yle cly gkygly pqa c`y zrqao`y z`r`gë~rki`y# ay naikr#
ΰ 7 %:1 { ΰ 7 %:0% Ay~ly }`clray ya oqyi`e ae c` ~`oc` na c` HK iq`nr`n` ile ς ?ΰ#
nlena ΰ ay kbq`c ` %:1 l %:0 &av~ragl naraifl na c` ~`oc`+%
Zrqao`y na fkzù~ayky el ~r`nkikle`cay# z`r` c` zrqao` na Drqyd`c'R`ccky# ay
naikr# qy`enl ac }`clr na c` zrlo`okckn`n z
Z`r` f`iar ay~`y zrqao`y na fkzù~ayky el ~r`nkikle`cay qy`enl ac }`clr na z# ya
ykbqae cly gkygly irk~arkly qy`nl `e~arklrgae~a% Ac zrliankgkae~l ya f`ia
oqyi`enl ac }`clr na c` ay~`nçy~ki` i`ciqc`n` F ae c` ~`oc` na c` nky~rkoqikùe na HK
iq`nr`n`# { ya f`ia qe` ke~arzlc`ikùe qy`enl c` gkyg` mùrgqc` qy`n` ile c`y
zrqao`y z`r`gë~rki`y%
Ahagzcly qy`enl c` zrqao` na Drqyd`c'R`ccky
Ahagzcl )0% Ya pqkara zrlo`r yk avky~ae nkmaraeik`y ae c`y ileiae~r`ikleay na
ùvknly na ek~rùbael &EL?+ zrl}aekae~ay na ~ray gqay~ra`nlray &0# ? { 9+
cli`ckx`nly ae nkmarae~ay cqb`ray% Zrlo`r pqa el f`{ nkmaraeik`y ae~ra c`y
ileiae~r`ikleay na ùvknl na ek~rùbael# ae~ra c`y ~ray zloc`ikleay gqay~ra`n`y%
Qy`r ΰ 7 :%:1% C`y ileiae~r`ikleay na EL? ya n`e ae c` ~`oc` na `o`hl%
^@OC@ 0:%0% ^`oc` gly~r`enl c`y ileiae~r`ikleay na ùvknly na ek~rùbael &ELv+
ae zzg zrl}aekae~ay na cly ~ray gqay~ra`nlray% &Ac`olr`ikùe zrlzk`+
Gqay~ra`nlr 0 Gqay~ra`nlr ? Gqay~ra`nlr 9
10 0> 6;
9? 90 ?:
02 =6 =:
=; 62 2?
6= ?: 1=
=? ?6 ??
;= 22
;2
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 721/846
Nr% Fëi~lr Pqa}anl Qrç`y
0:'2
Ylcqikùe8
Zrkgar`gae~a# ya ~kaeae pqa lrnae`r cly r`ebly# z`r` i`n` qel na cly ~ray
gqay~ra`nlray% @pqç# yke ago`rbl# f`{ pqa ~aear iqkn`nl na ~lg`r ae
ileyknar`ikùe yk~q`ikleay nlena f`{ raza~kikleay% Ae ay~a i`yl f`{ nly
raza~kikleay ae cly gqay~ra`nlray ? { 9% Ay~`y yk~q`ikleay ya glnkmki`e ilgl ya }a
ae c` ^@OC@ 0:%? na `o`hl% @e`ckx`enl c` ~`oc` na `o`hl# }agly pqa# ac g`ri`nlr
gày o`hl# ay ac 0> na c` ilcqge` nly# ac 02 na c` ilcqge` qel {# ac ?: na c`
ilcqge` nly { ~ray% C` ~`oc` na `o`hl gqay~r` ac lrnae na cly r`ebly%
^@OC@ 0:%?% ^`oc` gly~r`enl cly n`~ly na cly g`ri`nlray ile yqy rayzai~k}ly
r`ebly% &Ac`olr`ikùe zrlzk`+
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
Gay~ra`nlr 0 Gqay~ra`nlr ? Gqay~ra`nlr 9
G`ri`nlr _`ebl G`ri`nlr _`ebl G`ri`nlr _`ebl
10 ; 0> 0 6; 0;
9? 6 90 2 ?: (9%1
02 ? =6 09 =: 00
=; 0> 62 06 2? 016= 02 ?: (9%1 1= 0:
=? 0? ?6 = ?? 1
;= ?: 22 0=
;2 ?0
(Naoknl ` pqa f`{ nly eøgarly ?: ae~leiay# &9 $ >+,? 7 9%1
@flr` ya zrliana ` yqg`r cly r`ebly z`r` i`n` qe` na c`y ~ray ilcqge`y8
Ω_ 0 7 6? Ω_ ? 7 61%1 Ω_ 9 7 =9%1e0 7 2 e? 7 6 e9 7 =
C` rabkùe irç~ki` naraif` ya i`ciqc` qy`enl c` nky~rkoqikùe na c` HK iq`nr`n`% Ac
}`clr na ς ?ΰ4 η# ay naikr# ς ?:%:14 η pqa# ae ay~a i`yl# ay na ς ?
:%:14? 7 1%;;0# ay~l ay# ile η 7
d ’ 0 7 9 ’ 0 7 ? br`nly na ckoar~`n%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 722/846
Nr% Fëi~lr Pqa}anl Qrç`y
0:'6
Aeyabqkn`# yqy~k~q{aenl cly }`clray na c` mùrgqc` na Drqyd`c'R`ccky &0:'0+ {
raylc}kaenl zlr F n`8
0? &6?+?
&61%1+?
&=9%1+?
F 7 ┊┊┊┊┊┊┊┊ T ┊┊┊┊┊ $ ┊┊┊┊┊┊ $ ┊┊┊┊┊ W ’ 9 &?0$0+
?0&?0$0+ 2 6 =
7 &:%:?=+T&;=:%12+ $ &;09%26+ $ &=2?%:>+W ' ==
7 :%?0
Ileicqykùe8 Naoknl ` pqa ac }`clr na c` ay~`nçy~ki` ς ? 7 F 7 :%?0 ay gaelr pqa
ς ?:%:14? 7 1%;;0# el ya zqana raif`x`r c` fkzù~ayky eqc` na kbq`cn`n na zrlgankly {
ya nkia pqa c` zrqao` el ay ykbekmki`e~a% Ay~l nkia pqa ~aeagly qe` a}knaeik` gq{
keyqmkikae~a z`r` raif`x`r c` fkzù~ayky eqc` na pqa c`y ileiae~r`ikleay na c`y ~ray
zloc`ikleay na EL? zrl}aekae~ay na cly ~ray gqay~ra`nlray yle kbq`cay%
Ahagzcl )?% Ae qe ay~qnkl na ~lvkilclbç`# ile ac loha~l na }arkmki`r ac ile~aeknl na
`cpqk~ràe ya zrqao`e iq`~rl gqay~r`y `ca`~lrk`y na ikb`rrkccly% Ilgl el ya y`oa yk c`
zloc`ikùe gqay~ra`n` ay elrg`c# ae cqb`r na qy`r qe `eàckyky na }`rk`ex`
z`r`gë~rkil# ya naikna qy`r qe gë~lnl el z`r`gë~rkil# ay naikr# ac na Drqyd`c'
R`ccky% Qy`enl qe ek}ac na ykbekmki`eik` na ΰ 7 %:1 zrlo`r pqa el f`{ nkmaraeik`y
ae~ra c`y iq`~rl zloc`ikleay na g`ri`y na ikb`rrkccly% Cly n`~ly na c`y iq`~rl
g`ri`y na ikb`rrkccly ile yqy rayzai~k}`y ileiae~r`ikleay na `cpqk~ràe ya n`e ae c`
~`oc` na `o`hl% ^`gokëe i`ciqc`r ac }`clr na c` zrlo`okckn`n z%
^@OC@ 0:%9% ^`oc` gly~r`enl cly n`~ly% &Ac`olr`ikùe zrlzk`+
G`ri` @ G`ri` O G`ri` I G`ri` N
0: 06 01 ?:
00 0> 0> 0;
09 01 0? ?0
0> 0= 0= 02
Ylcqikùe8
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 723/846
Nr% Fëi~lr Pqa}anl Qrç`y
0:';
0% Zrkgarl y`i`gly cly r`ebly ilrrayzlenkae~ay# ilgl ya }a ae c` ~`oc` na `o`hl%
^@OC@ 0:%>% ^`oc` gly~r`enl cly r`ebly% &Ac`olr`ikùe zrlzk`+
G`ri` @ _`ebl G`ri` O _`ebl G`ri` I _`ebl G`ri` N _`ebl
0: 0 06 09 01 6%1 ?: 01
00 ? 0> = 0> = 0; 0>
09 > 01 6%1 0? 9 ?0 0=
0> = 0= 0:%1 0= 0:%1 02 0?
?% Aeyabqkn`# ay~`ocaiagly c` rabkùe irç~ki` qekc`~ar`c naraif` &el f`{ gày pqa
ay`# ¼zlr pqë<+%
ς ?Tΰ4d'0W 7 ς ?T%:14>'0W 7 ς ?T%:149W 7 2%6? &na c` ~`oc` na c` HK iq`nr`n`+
9% Qy`enl c` mùrgqc` na Drqyd`c'R`ccky &0:'0+ { yqy~k~q{aenl8
E 7 0=# &Ω_ 0+?
7 &09+?
7 0=;# &Ω_ ?+?
7 &96+?
7 0>>># &Ω_ 9+?
7 &?6+?
7 26># &Ω_ >+? 7
&12+?
7 9?>;# e0 7 e? 7 e9 7 e> 7 ># n`8
F 7 0?,0=&0=$0+ T0=;,> $ 0>>>,> $ 26>,> $ 9?>;,>W ’ 9&0=$0+
7 00%:=%
>% Ileicqykùe8 Naoknl ` pqa 00%?2 5 2%6? ya raif`x` c` fkzù~ayky na kbq`cn`n na
zloc`ikleay# { ya nkia pqa yk f`{ nkmaraeik`y ae~ra cly zrlgankly na c`y
ileiae~r`ikleay na `cpqk~ràe ae cly ikb`rrkccly%
1% Ac }`clr na c` zrlo`okckn`n z ya y`i` oqyi`enl 00%:=# ile 9 br`nly na ckoar~`n ae
c` ~`oc` na c` HK iq`nr`n` { ay~à ae~ra %:?1 { %:0% Yk ya rapqkara g`y zraikykùe ya
zqana qy`r c` mùrgqc` na ke~arzlc`ikùe &1'?6+# ay naikr# oqyi`enl ac }`clr na 00%?2%
Cly }`clray ke~arzlc`nly yle8
δ? 7 :%;;# ς ?
7 00%9>1# δ0 7 :%;21# ς ? 7 ;%9>6# F 7 00%:=Yqy~k~q{aenl ay~ly }`clray ae c` mùrgqc` na ke~arzlc`ikùe8
&δ? ’ δ0+ , &ς ?? ’ ς ?0+ 7 &δ? ’ V+ , &δ? ’ ς ?i`ci%+
&:%;; ’ :%;21+,&00%9>1 ’ ;%9>6+ 7 &:%;; ’ V+,&00%9>1 ’ 00%:=+
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 724/846
Nr% Fëi~lr Pqa}anl Qrç`y
0:'0:
@flr`# raylc}kaenl zlr V n` V 7 %;62;% Ac }`clr na z ay kbq`c ` z 7 0 ' %;62; 7
:%:0% Ay~a }`clr na c` zrlo`okckn`n z ay ykbekmki`e~a%
Ahagzcl )9% Ac ckorl Y~`~ky~kiy mlr Glnare Oqykeayy Naikykle na C`raeia C%
C`zke# gaeikle` qe ay~qnkl na pqçgki` `e`cç~ki`# rac`ikle`nl ile c`y kgzqrax`y
pqa ile~kaeae cly ra`i~k}ly pqçgkily# c`y iq`cay zqanae ke~armarkr ae c`y ra`iikleay
pqçgki`y# ay naikr# ae iq`e~l ` c` i`e~kn`n na ~kagzl rapqarknl z`r` pqa ya clbra c`
ra`iikùe pqçgki`% Cly n`~ly ya n`e ae c` ~`oc` na `o`hl%
^@OC@ 0:%1% ^`oc` gly~r`enl cly ek}acay na kgzqrax`y ae cly ra`i~k}ly pqçgkily
ae mqeikùe nac ~kagzl z`r` pqa ya f`b` c` ra`iikùe%
Ek}acay na kgzqrax`y
%::0 %:0 %:1 %0:
^kagzl na ra`iikùe ae gkeq~ly
0:9 0:> 019 ?:2
000 009 0?2 069
0:2 002 0>9 029
0:1 0?: 00;
009 096
0>9
Mqae~a8 C`zke &0;6?+
&`+ I`ciqc`r c` ay~`nçy~ki` F na Drqyd`c'R`ccky%
&o+ Qy`enl qe ek}ac na ykbekmki`eik` naΰ 7 :%:1 zrlo`r c` fkzù~ayky eqc` na pqa cly
ek}acay na kgzqrax`y ae cly ra`i~k}ly pqçgkily el `mai~`e ac ~kagzl na ra`iikùe%
F`iar ay~l# qy`enl c` zrqao` na fkzù~ayky ~r`nkikle`c { c` zrqao` na fkzù~ayky el
~r`nkikle`c%
Ylcqikùe8
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 725/846
Nr% Fëi~lr Pqa}anl Qrç`y
0:'00
C` ~`oc` na `o`hl aecky~` cly g`ri`nlray { yqy r`ebly ilrrayzlenkae~ay%
^@OC@ 0:%= ^`oc` gly~r`enl cly }`clray na cly g`ri`nlray ae mlrg` `yiaenae~a
na cly iq`~rl ek}acay na kgzqrax`y ae cly ra`i~k}ly pqçgkily { ac ~kagzl%
G`ri`nlr s 0:9 0:> 0:1 0:2 000 009 009 002 00; 0?: 0?2 096 0>9 0>9 019 029 069 ?:2
_`ebl s 0 ? 9 > 1 = 2 6 ; 0: 00 0? 09 0> 01 0= 02 006
^@OC@ 0:%2% ^`oc` gly~r`enl cly iq`~rl g`ri`nlray ile yqy ilrrayzlenkae~ay
r`ebly%
&0+ &?+ &9+ &>+
%::0 %:0 %:1 %0: \\\\\\\\\\\\\\\ \\\\\\\\\\\\\\ \\\\\\\\\\\\\\ \\\\\\\\\\\\\\\
G`ri`nlr _`ebl G`ri`nlr _`ebl G`ri`nlr _`ebl G`ri`nlr _`ebl
0:9 0 0:> ? 019 01 ?:2 06
000 1 009 =%1( 0?2 00 069 02
0:2 > 002 6 0>9 09%1( 029 0=
0:1 9 0?: 0: 00; ;
009 =%1( 096 0?0>9 09%1(
(Naoknl ` pqa f`{ nly 009 { nly 0>9# ae~leiay ac r`ebl ilrrayzlenkae~a ` 009 ay
&=$2+,? 7 =%1 { ac r`ebl ilrrayzlenkae~a ` 0>9 ay &09 $ 0>+,? 7 09%1
@flr` ya zrliana ` yqg`r cly r`ebly z`r` i`n` qe` na c`y iq`~rl ilcqge`y%
Ω_ 0 7 09 Ω_ ? 7 99 Ω_ 9 7 2> Ω_ > 7 10
e0 7 > e? 7 1 e 7 = e 7 9
C` rabkùe irç~ki` naraif` ya i`ciqc` qy`enl c` nky~rkoqikùe na HK iq`nr`n`% Ac }`clr
na ς ?ΰ4 η 7 ς ?:%:149 7 2%6?# ay naikr# nlena η 7 d ’ 0 7 > ’ 0 7 9%
Aeyabqkn` yqy~k~q{aenl cly }`clray na `rrko` ae c` aiq`ikùe &0:'0+
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 726/846
Nr% Fëi~lr Pqa}anl Qrç`y
0:'0?
0? &09+?
&99+?
&2>+?
&10+?
F 7 ┊┊┊┊┊┊┊ T ┊┊┊┊┊┊┊ $ ┊┊┊┊┊┊ $ ┊┊┊┊┊┊ $ ┊┊┊┊┊┊ W ’ 9&06 $ 0+06&06$0+ > 1 = 9
7 :%:91 T>?%?1 $ ?02%6 $ ;0?%=2 $ 6=2%:W ' 12
7 :%:91&?:9;%2?+ ’ 12
7 0>%9;
Ileicqykùe8 naoknl ` pqa ac }`clr na c` ay~`nçy~ki` ς ? 7 F 7 0>%9; ay g`{lr pqa
2%6? ya raif`x` c` fkzù~ayky eqc` { ya nkia pqa cly ek}acay na kgzqrax`y yk ay~àe
`mai~`enl ac ~kagzl na c`y ra`iikleay pqçgki`y%
@flr` okae# z`r` f`iar c` zrqao` na fkzù~ayky na z l el ~r`nkikle`c# ya f`ia
qy`enl c` mùrgqc` na ke~arzlc`ikùe na `o`hl8
&δ?’ δ0+ , &ς
?? ’ ς ?0+ 7 &δ? ’ V+ , &ς ?? ' ς ?i`ci%+
Ya oqyi` ae c` ~`oc` na HK iq`nr`n` ac }`clr na ς ?i`ci% 7 0>%9; ile > b%c% { ay~à ae~ra
0>%6= ile }`clr zlriae~q`c na %::1 { 09%?22 ile }`clr zlriae~q`c na %:0% Ay naikr#
ile cly }`clray na δ? 7 %::1# δ07 %:0# ς ?? 7 0>%6=# ς ?0 7 09%?22# { ς ?i`ci% 7 0>%9;%
Yqy~k~q{aenl ~lnly ay~ly }`clray ae c` mùrgqc` na `rrko` { raylc}kaenl zlr V n`8
&%::1 ' %:0+ , &0>%6= ’ 09%?22+ 7 &%::1 ’ V+ , &0>%6= ’ 0>%9;+
V 7 :%::2? 7 z
Ay~a }`clr na :%::2? ay gqifl gq{ ykbekmki`e~a { ramraen` c` naikykùe ~lg`n` ae
`mkrg`r pqa yk f`{ nkmaraeik`y ae~ra cly ek}acay na kgzqrax`y pqa ra~`rn`e c`y
ra`iikleay pqçgki`y%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 727/846
Nr% Fëi~lr Pqa}anl Qrç`y
0:'09
Aharikikly I`zç~qcl 0:
0:%0% Ae qe avzarkgae~l z`r` na~argke`r# iqàc na ~ray ~kzly na gl~lray qy`o`e
gaely b`ylcke`# ya fkxl qe ay~qnkl ~r`~`enl na g`e~aear ~ln`y c`y nagày }`rk`ocay
iley~`e~ay% Qy`r qe ek}ac ykbekmki`e~a naΰ 7 :%:1%
&`+ Ay~`ocaiar Fl8 { F08 { i`ciqc`r F% &Fl8±0 7 ±? 7 ±94 F08±0 ≪ ±? ≪ ±9#
F 7 0%== { ya ra~kaea Fl8+
&o+ Ay~`ocaiar c` rabkùe irç~ki`% &f 5 ς ?:%:14? 7 1%;;0+
&i+ Qy`r c` zrqao` el z`r`gë~rki` na Drqyd`c'R`ccky { zrlo`r pqa el f`{
nkmaraeik`y ae ac ileyqgl na b`ylcke` na cly ~ray gl~lray# ae iq`e~l `c gkcc`ha
lo~aeknl% &F 7 0%== { ya ra~kaea Fl8+
^`oc` gly~r`enl cly n`~ly% &Ac`olr`ikùe zrlzk`+
Gl~lr 0 Gl~lr ? Gl~lr 9
?>%: ?9%? 06%>
0=%2 0;%6 0;%0
??%6 06%0 02%90;%6 02%= 02%9
06%; ?:%? 0;%2
02%6 06%;
06%6
0;%9
0:%?% C` ~`oc` na `o`hl n` c`y ~agzar`~qr`y na 1 yqha~ly yacaiikle`nly#
`ca`~lrk`gae~a# na ~ray brqzly nkmarae~ay% Qy`enl qe ek}ac na ykbekmki`eik` na ΰ 7:%:1 zrlo`r pqa c`y ~ray zloc`ikleay na ~agzar`~qr`y yle kbq`cay% I`ciqc`r z%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 728/846
Nr% Fëi~lr Pqa}anl Qrç`y
0:'0>
^`oc` gly~r`enl c`y ~agzar`~qr`y nac iqarzl &lM+ ic`ykmki`n`y zlr an`nay%
&Ac`olr`ikùe zrlzk`+
06'?: `ðly ?0'?; `ðly 59: `ðly
;6%: ;;%= ;6%=
;6%> ;6%? ;6%=
;2%2 ;;%: ;2%:
;6%1 ;6%? ;2%1
;2%0 ;2%; ;2%1
0:%9% Qe z`eac na yka~a avzar~ly mqa ileyqc~`nl z`r` i`ckmki`r ` ikeil kenqy~rk`y &@#O# I# N# A+ ae iq`e~l ` c` zrlo`okckn`n na pqa i`gokly ~aielcùbkily zrlnqxi`e
gahlr`y ae ac ile~rlc na c` ile~`gke`ikùe `gokae~`c# ae ac iqryl na cly zrùvkgly
0: `ðly% C`y i`ckmki`ikleay ae ya n`e ae c` ~`oc` na `o`hl%
^`oc` gly~r`enl cly n`~ly% &Ac`olr`ikùe zrlzk`+
Kenqy~rk`y
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
@ O I N A \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
Avzar~l
0 :%01 :%21 :%0: :%:: :%9:
? :%9: :%=: :%?: :%:1 :%?1
9 :%?: :%6: :%9: :%:: :%1:
> :%:: :%1: :%?1 :%0: :%=:
1 :%0: :%11 :%01 :%01 :%>:
= :%?1 :%2: :%91 :%?1 :%>1
2 :%>: :%;1 :%>1 :%?: :%91
&`+ Zrlo`r ile ac ek}ac na ykbekmki`eik` na ΰ 7 :%:1# pqa c`y zloc`ikleay yle
knëe~ki`y%
&o+ I`ciqc`r ac }`clr na c` zrlo`okckn`n z%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 729/846
Nr% Fëi~lr Pqa}anl Qrç`y
0:'01
0:%>% Ac ckorl Zrlo`okckn`n { Ay~`nçy~ki` z`r` Kebaekarç` { Ikaeik`y na H`{ C%
Na}lra &z% ==?+ zrlzlrikle` cly ykbqkae~ay n`~ly# cly iq`cay ya ramkarae ` c`
ileiae~r`ikùe nac kyù~lzl ay~rleikl ;:# ae gqay~r`y na caifa lo~aekn`y na 1
caifarç`y yacaiikle`n`y# `ca`~lrk`gae~a# ae i`n` qe` na iq`~rl rabkleay nkmarae~ay%
^`oc` gly~r`enl cly n`~ly na c`y ileiae~r`ikleay na ay~rleikl ae caifa%
_abkùe 0 =%> 1%6 =%1 2%2 =%0
_abkùe ? 2%0 ;%; 00%? 0:%1 6%6
_abkùe 9 1%2 1%; 6%? =%= 1%0
_abkùe > ;%1 0?%0 0:%9 0?%> 00%2
&Mqae~a8 Na}lra# ?::0+
F`iar cly ykbqkae~ay iàciqcly8
&`+ Zrlo`r ile ac ek}ac na ykbekmki`eik` na :%0: z`r` }arkmki`r yk ac zrlgankl na
ileiae~r`ikleay na ay~rleikl ;: nkmkara# `c gaely ae nly na c`y rabkleay%
&o+ I`ciqc`r ac }`clr na z%&i+ F`iar qe `eàckyky na }`rk`ex` z`r`gë~rkil { ilgz`r`r cly rayqc~`nly%
0:%1% Cly n`~ly na `o`hl gqay~r`e > ~r`~`gkae~ly z`r` na~argke`nl zrliayl# ae ac
iq`c el ya y`oa yk c` zloc`ikùe gqay~ra`n` ay elrg`c% Qy`r c` @EL]@ z`r`gë~rki`
{# cqabl# qy`r ac gë~lnl el z`r`gë~rkil na Drqyd`c'R`ccky% ¼F`{ yqmkikae~a
a}knaeik`# ile ΰ 7 :%:1# pqa ely zargk~` ileicqkr pqa avky~ae nkmaraeik`y ae~ra cly
> ~r`~`gkae~ly< Ae `goly i`yly ilgz`r`r cly rayqc~`nly { av`gke`r ac }`clr na M {
na z% ¼Iqàc na cly nly gë~lnly &z`r`gë~rkily l el z`ragë~rkily+ yarç` ac gày
zraikyl# yk ya yqzkar` pqa c` zloc`ikùe gqay~ra`n` mqar` elrg`c<
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 730/846
Nr% Fëi~lr Pqa}anl Qrç`y
0:'0=
N`~ly nac zrlocag` na `rrko`% &Ac`olr`ikùe zrlzk`+
^r`~`gkae~ly
0 ? 9 >
0? 0: 0: ;01 0? 6 =
09 00 0? 6
06 0> 01 2
?: 0: 09 ;
0; 00 00 2
01 0? 09 =
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 731/846
Nr% Fëi~lr Pqa}anl Qrç`y
00'0
I@ZK^QCL 00
Yarkay na ~kagzl
Ic`ykmki`ikùe na cly gl}kgkae~ly na c`y yarkay na ~kagzl%' ^aenaeik`y ` c`rbl
zc`xl%' Ilgzleae~ay içickily na yarkay na ~kagzl%' ]`rk`ikleay ay~`ikle`cay%'
]`rk`ikùe krrabqc`r%' Gë~lnly z`r` aeile~r`r cçea`y na ~aenaeik`%' Cçea` na cly
iq`nr`nly gçekgly { z`ràolc`y na cly iq`nr`nly gçekgly%'
Iq`cpqkar }`rk`oca ae mqeikùe nac ~kagzl# ae yqiaykùe# ya cc`g` yarkay na ~kagzl% C`y
yarkay na ~kagzl yle qe` yaiqaeik` na }`clray na }`rk`ocay ~lg`n`y ae zarklnly na
~kagzl yqiayk}ly% C` bràmki` na qe` yarka na ~kagzl ay qe nk`br`g`# ile ac aha }ar~ki`cgly~r`enl ac }`clr loyar}`nl & {+ {# ile ac aha flrkxle~`c nael~`enl ac ~kagzl
&gkeq~ly# nç`y# `ðly# a~i%+%
C`y bràmki`y ilgl cly fky~lbr`g`y l nk`br`g`y na ~`ccl { flh` yle gë~lnly
}kyq`cay ø~kcay z`r` gly~r`r c` }`rk`ikùe ae cly n`~ly% Yke ago`rbl# ac ~kagzl ay qe
m`i~lr gq{ kgzlr~`e~a pqa ile~rkoq{a ` c` }`rk`ikùe loyar}`n` na cly n`~ly# pqa cly
fky~lbr`g`y l c`y bràmki`y na i`h` el cly ~lg`e ae iqae~`%
C`y yarkay na ~kagzl yle qe ilehqe~l na loyar}`ikleay ~lg`n`y ` ~kagzly
ayzaiçmkily# qyq`cgae~a# ` ke~ar}`cly kbq`cay ae qe lrnae irlelcùbkil% C`y yarkay na
~kagzl l yaiqaeik`y na ~kagzl ya namkeae ilgl n`~ly ay~`nçy~kily pqa yle
ilcaiikle`nly# rabky~r`nly q loyar}`nly ae keiragae~ly na ~kagzly yqiayk}ly% Ac
`eàckyky na cly n`~ly na c`y yarkay na ~kagzl ay na ke~arëy z`r` `pqëccly pqkaeay nayaae
ae~aenar c` e`~qr`cax` na cly n`~ly z`y`nly { zrayae~ay% ^`gokëe# c`y yarkay na ~kagzl
yle na ke~arëy z`r` `pqaccly ke}ay~kb`nlray# pqkaeay nayaae qy`r ac ilelikgkae~l na
n`~ly z`y`nly z`r` zranaikr ac mq~qrl%
C`y `zcki`ikleay na c`y yarkay na ~kagzl yle gq{ ilgqeay ae c` ailelgç`#
zarl ~`gokëe ae c` ailelgç` l c` kebaekarç`% Zlr ahagzcl8
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 732/846
Nr% Fëi~lr Pqa}anl Qrç`y
00'?
0% Cly blokarely na c`y e`ikleay kenqy~rk`ckx`n`y pqkarae y`oar cly }`clray mq~qrly na
c`y ~`y` na ke~arayay glea~`rkly% C`y e`ikleay ae nay`rrlccl pqkarae y`oar c`y
~aenaeik`y na c`y ~`y`y na na}`cq`ikùe na c` glean`% ^`gokëe ay naya`oca zranaikr c`y
~`y`y na agzcaly l na nayagzcaly% Kbq`cgae~a# ay kgzlr~`e~a y`oar cly zlriae~`hay na
cly keiragae~ly na cly ily~ly na c` }kn`%
?% L~r`y `zcki`ikleay na c`y yarkay na ~kagzl yle cly zrleùy~kily na c`y ~`y`y na ke~arëy
z`r` c` iley~rqiikùe na }k}kaen`y { ac ily~l na cly g ~ark cay na iley~rqiikùe%
9% ^`gokëe c`y ilgz`ðç`y g`eqm`i~qr`nlr`y pqkarae zrlely~ki`r c` nag`en` na yqy
zrlnqi~ly { yqy iikleay ae ac gari`nl%
>% Ae kebaekarç` `gokae~`c# cly `i~k}ky~`y { zrl~ai~lray nac gankl `gokae~a pqkarae
y`oar iqàcay yle c`y ~aenaeik`y ae cly `qgae~l na cly b`yay na ke}are`narl# ilgl ac
okùvknl na i`rolel &li`ykle`nl zlr c` agkykleay }afkiqc`ray# c` kenqy~rk`# mqably
mlray~`cay# a~i%+ pqa ay~àe i`cae~`enl c` ~karr`# mqenkaenl cly bc`ik`ray gle~`ðlyly {
c`y i`z`y zlc`ray { i`gok`enl ac ickg` gqenk`c% ^`gokëe ay ke~aray`e~a y`oar c`y
~aenaeik`y { cly `qgae~ly na c` r`nk`ikùe qc~r`}klca~`# pqa ~`e~l n`ðl ay~à i`qy`enl `c
yar fqg`el# zlr c` nay~rqiikùe nac lxlel e`~qr`c ay~r`~lymërkil# i`qy`nl zlr c`
krr`ikle`ckn`n nac flgora glnarel%
1% C`y yarkay na ~kagzl ~`gokëe `zcki`e z`r` y`oar c`y ~aenaeik`y { zrleùy~kily na cly
keiragae~ly na c` zloc`ikùe gqenk`c# a~i%
Iq`enl ya br`mki`e c`y gankikleay na yarkay na ~kagzl# ` gaeqnl ya loyar}`e
~aenaeik`y# ikicly l }`rk`ikleay kgzlr~`e~ay na cly n`~ly pqa# na l~r` g`ear`# z`y`rç`e
ke`n}ar~knly%Namkekikùe8 G`~agà~ki`gae~a# qe` yarka na ~kagzl ya namkea zlr }`clray[ 0# [ ?#%%%%%%% na
qe` }`rk`oca [ # ilgl c` ~agzar`~qr`# ileiae~r`ikleay na ile~`gke`e~ay# ilgl IL?#
YL?# z`r~çiqc`y `~glymërki`y# a~i%# ` ~kagzly ~0# ~?#%%%% Zlr cl ~`e~l# [ ay qe` mqeikùe na ~
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 733/846
Nr% Fëi~lr Pqa}anl Qrç`y
00'9
ykgolckx`n` zlr [ 7 M&~ +%
Ic`ykmki`ikùe na cly gl}kgkae~ly na c`y yarkay na ~kagzl
Cly gl}kgkae~ly i`r`i~arçy~kily na c`y yarkay na ~kagzl zqanae ic`ykmki`rya ae iq`~rl
~kzly cc`g`nly ilgzleae~ay na yarkay na ~kagzl% Ay~ly ilgzleae~ay na c`y yarkay na
~kagzl ya nayirkoae ilgl ykbqa8
0% ^aenaeik`y ` c`rbl zc`xl l gl}kgkae~ly yaiqc`ray%
?% Gl}kgkae~ly l mcqi~q`ikleay içicki`y%
9% ]`rk`ikleay ay~`ikle`cay l gl}kgkae~ly ay~`ikle`cay%
>% ]`rk`ikleay l gl}kgkae~ly krrabqc`ray l `ca`~lrkly%
^aenaeik`y c`rbl zc`xl
C` ~aenaeik` ` c`rbl zc`xl l ~aenaeik` yaiqc`r na qe` yarka na ~kagzl ay ac
ilgzleae~a qekmlrga na c`y yarkay pqa razrayae~`e ac iraikgkae~l l nairagae~l na
~kagzly# ylora qe zarklnl br`ena na ~kagzl% C` ~aenaeik` yaiqc`r ya ramkara ` c`
nkraiikùe baear`c ae c` iq`c c` bràmki` na qe`y yarkay `z`raiae gl}arya nqr`e~a qe
ke~ar}`cl na ~kagzl% Zlr ahagzcl# c` zloc`ikùe na cly Ay~`nly Qeknly nqr`e~a cly
øc~kgly >: `ðly f` gly~r`nl qe` ~aenaeik` na iraikgkae~l na 092 gkccleay na
zaryle`y ` ?>= ae 0;66% C y ~aenaeik y ` c`rbl zc`xl ya }a ae c` Mkbqr` 00%:& +%
C` na~argke`ikùe na c`y ~aenaeik`y na c`y cçea`y { na c`y iqr}`y ya zqana f`iar
qy`enl ac gë~lnl na `hqy~`gkae~l na iqr}`y% ^`gokëe ya zqana f`iar zlr gankl nac
`eàckyky na cly nk`br`g`y ayz`riknly# z`r` aeile~r`r c` mqeikùe g`~agà~ki` pqa gahlr
aei`ha ae cly n`~ly%
Ilgzleae~ay içickily na yarkay na ~kagzlCly ilgzleae~ay içickily ya ramkarae ` cly gl}kgkae~ly raiqrrae~ay na `rrko` { `o`hl
na c`y ~aenaeik`y na c`y yarkay na ~kagzl% Ay~`y mcqi~q`ikleay na len`# cc`g`n`y ikicly
na cly eablikly# yle nkmarae~ay na c`y mcqi~q`ikleay ay~`ikle`cay% Ay naikr# ae ac
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 734/846
Nr% Fëi~lr Pqa}anl Qrç`y
00'>
yae~knl na pqa iqorae zarklnly c`rbly na ~kagzl# ~kaeae i`qy`y nkmarae~ay { yle
gaely zranaikocay% C`y mcqi~q`ikleay nqr`e na ? ` 0: `ðly# l gày# iq`enl ya gknae
c`y len`y na iray~` ` iray~` l na i`e`c ` i`e`c% Ahagzcly na ilgzleae~ay ikicly yle
zarklnly na raiaykleay aileùgki`y l na zarklnly na kemc`ikùe# nag`en` na zrlnqi~ly
` c`rbl zc`xl# a~i% Ay~` yk~q`ikùe ya }a ae c` Mkbqr` 00%: &o+
]`rk`ikleay ay~`ikle`cay
Ay~a ~kzl na yarkay na ~kagzl ya ramkarae ` }`rk`ikleay zarkùnki`y# zarl el ay~àe
ckgk~`n`y ` }`rk`ikleay ile c` ay~`ikùe nac `ðl% Ay~ly yle z`~rleay na zarklnly ae c`y
yarkay na ~kagzl pqa ya ilgzca~`e ae qe `ðl {# cqabl ya razk~ae na `iqarnl `c gkygl
z`~rùe na zarklnl ae `ðly# yqoyaiqae~ay% Zlr ahagzcl# cly zraikly na cly gari`nly
mke`eikarly zqanae gly~r`r ~aenaeik`y `c~`y l o`h`y ae qe nç` l ae qe` yag`e`%
Ae ay~qnkly `gokae~`cay# c`y mcqi~q`ikleay na cly ile~`gke`e~ay gqay~r`e
~aenaeik`y içicki`y nqr`e~a ac nç`# ilgl ae ac i`yl nac ay~qnkl na c`y ileiae~r`ikleay
na lxlel ~rlzlymarkil% L~rly ahagzcly yle c` zrlnqiikùe na ikar~ly zrlnqi~ly na
br`eh`y `brçilc`y# ac eøgarl na }afçiqcly pqa z`y`e zlr ikar~l zqe~l# ae~ra nly yk~kly#
a~i% C` qekn`n na ~kagzl ae }`rk`ikleay ay~`ikle`cay ay gaely pqa qe `ðl# zarl
zqanae yar na qe gay# qe` yag`e`# l z`r~a nac nç`% Ay~` yk~q`ikùe ya }a ae c` Mkbqr`
00%:&i+%
]`rk`ikùe krrabqc`r
Ay~a ay qe ~kzl na }`rk`ikùe pqa el ay~à ileyknar`nl zlr ~aenaeik`y# ikicly l m`i~lray
ay~`ikle`cay# ykel pqa ya ilgzlea na mqarx`y el raiqrrae~ay# ayzlrànki`y pqa el ya
nayirkoae ilgl l `~rkoqkn`y ` m`i~lray na ~aenaeik`y# ikicly# l ay~`ikle`cay% Ahagzclyna }`rk`ikleay krrabqc`ray yle gl}kgkae~ly ayzlrànkily na yarkay na ~kagzl naoknl `
keqen`ikleay# br`ekx`n`y# fac`n`y# ~lre`nly# fqr`i`eay# yapqç`y# mqably mlray~`cay#
a~i%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 735/846
Nr% Fëi~lr Pqa}anl Qrç`y
00'1
@nagày# c`y mkbqr`y na `o`hl gqay~r`e ahagzcly na `cbqely zlykocay z`~rleay
na ~aenaeik` ae yarkay na ~kagzl% Zlr ahagzcl# c` Mkbqr` 00%0&`+ gqay~r` qe`
~aenaeik` el ckea`c% C` Mkbqr` 00%0&o+ gqay~r` qe` ~aenaeik` ckea`c pqa nkygkeq{a% C`
Mkbqr` 00%0&i+ gqay~r` qe` bràmki` yke ~aenaeik`%
Mkbqr` 00%:% Bràmki`y gly~r`enl cly ~kzly na ~aenaeik`y% C` bràmki` &`+ gqay~r` qe`
~aenaeik` na cçea` ` c`rbl zc`xl l na gl}kgkae~l yaiqc`r% C` bràmki` &o+ gqay~r` qe`
cçea` na ~aenaeik` ` c`rbl zc`xl ile qe gl}kgkae~l içickil ylorazqay~l% C` bràmki` &i+
gqay~r` ~aenaeik`y içicki`y c`rbl zc`xl { gl}kgkae~ly ay~`ikle`cay% &Yzkabac# 0;=0+%
Mkbqr` 00%0% Ahagzcly na `cbqely z`~rleay na ~aenaeik`y ae yarkay na ~kagzl%
@zcki`ikleay na c`y mqeikleay na yarkay na ~kagzl
C`y `zcki`ikleay ae kebaekarç` na c`y ~aenaeik`y ` c`rbl zc`xl l ~aenaeik`y yaiqc`ray
yle }`rk`y% Ahagzcly na ay~`y `zcki`ikleay yle cly keiragae~ly na c` ile~`gke`ikùe
`gokae~`c% Qe ahagzcl icàykil ay ac `qgae~l iley~`e~a na c`y ileiae~r`ikleay na
okùvknl na i`rolel# b`y ga~`el# }`zlr na `bq # a~i%# ` ek}ac gqenk c# pqa f`e ay~ nl
liqrrkaenl nayna ac kekikl na c` ar` kenqy~rk`c f`y~` `c zrayae~a% Ay~l# ilgl ay okae
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 736/846
Nr% Fëi~lr Pqa}anl Qrç`y
00'=
y`oknl# ay~à ilrrlgzkaenl ac ickg` na eqay~rl zc`ea~`# `c ay~`rya i`cae~`enl c` ~karr` {
c`y `bq`y g`rke`y%
L~rly ahagzcly# ` cly iq`cay ya cay zqanae `zcki`r c`y yarkay na ~kagzl# yle cly
keiragae~ly ae c` r`nk`ikùe qc~r`}klca~` &ae yqy mlrg`y na Q]'@ { Q]'O+# pqa ay~àe
i`qy`enl iàeiar ae c` zkac &ae yqy ~ray mlrg`y# gac`elg`# o`y`c { ayi`glyl+ { n`ðly
ae c` }kykùe { `c~ar`ikleay ae c` ay~rqi~qr` nac NE@%
L~r`y `zcki`ikleay na c`y yarkay na ~kagzl ay~àe rac`ikle`n`y ile cly
iraikgkae~ly zloc`ikle`cay l naglbràmkily% L~rly gày ay~àe rac`ikle`nly ile c`
zrlnqiikùe kenqy~rk`c# c` zrlnqiikùe na aearbç`# c` ailelgç # a~i%
^kzly na mqeikleay g`~agà~ki`y z`r` cçea`y na ~aenaeik`
C`y aiq`ikleay l mqeikleay g`~agà~ki`y gày ilgqeay z`r` `zrlvkg`r cly n`~ly
bràmkily na cçea`y na ~aenaeik` na qe nk`br`g` ayz`riknl ya n`e ac c` ~`oc` na `o`hl%
@pqç# c`y ca~r`y `# o { i razrayae~`e }`clray iley~`e~ay {# c`y ca~r`y V a [ # razrayae~`e
c`y }`rk`ocay kenazaenkae~ay { nazaenkae~ay# rayzai~k}`gae~a%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 737/846
Nr% Fëi~lr Pqa}anl Qrç`y
00'2
^@OC@ 00%:% ^`oc` gly~r`enl cly ~kzly na mqeikleay g`~agà~ki`y gày ilgqeay
qy`n`y z`r` cçea`y na ~aenaeik`% &Ac`olr`ikùe zrlzk`+
Mqeikùe g`~agà~ki` Nayirkzikùe
&0+ { 7 ` $ ov Cçea` rai~`
&?+ { 7 m& v+ 7 `v?
$ ov $ i Iqr}` iq`nrà~ki` l z`r`oùcki`
&9+ { 7 v9
$ o v?
$ iv $ n Iqr}` iøoki`
&>+ { 7 v>
$ o v9
$ iv?
$ nv $ a Iqr}` iqàr~ki`
&1+ { 7 ` $ o v $ i v?
$ %%% $ `e ve Zlckelgk`c baear`ckx`nl
&=+ { 7 `o v
l Clb { 7 Clb ` $ &Clb o+ v Iqr}` avzleaeik`c
&2+ { 7 v o
l Clb { 7 Clb ` $ o Clb v Iqr}` balgë~rki`
&6+ { 7 0,` $ o v l 0, { 7 ` $ o v Mqeikùe fkzërolc`&;+ { 7 zp o v
l Clb { 7 Clb z $ o v
Clb p Iqr}` na Blgzar~x
&0:+ { 7 `o v
$ b Iqr}` avzleaeik`c glnkmki`n`
&00+ { 7 v o
$ b Iqr}` balgë~rki` glnkmki`n`
&0?+ { 7 Ce v Mqeikùe clb`rç~gki`
&09+ { 7 ` ’ &` ’ o+ avz&'&i+s vs+n
Mqeikùe na Rakoqcc
&0>+ { 7 ` ’ &` ’ o+,&0 $ &is vs+n
Mqeikùe na Glrb`e'Gariar'Mclnkeb
&01+ { 7 `&0 $ &o ’ 0+ avz&'i& v ’ n++0,&0 ' o+
Mqeikùe na _kif`rny
Z`r` naiknkr# iqàc mqeikùe g`~agà~ki` ay c` gày `zrlzk`n`# z`r` `hqy~`r cly
n`~ly ya zqana f`iar }kaenl qe nk`br`g` ayz`riknl na c` bràmki` na cly n`~ly% Zlr
ahagzcl# yk ac nk`br`g` ayz`riknl ae z`zac yagkclb`rk~gl na Clb & {+ }y% v gqay~r`
qe` rac`ikùe ckea`c# c` aiq`ikùe ~kaea c` mlrg` na c` iqr}` avzleaeik`c &=+% Yk ya qy` ac
z`zac clb`rk~gl ilgzca~l# Clb {'Clb v# { cly n`~ly gqay~r`e qe` rac`ikùe ckea`c# c`
aiq`ikùe ~kaea c` mlrg` na qe` iqr}` balgë~rki` &2+% Na iq`cpqkar g`ear`# cly
zrlbr`g`y na ilgzq~`nlr # ilgl ac Y@Y# Gkek~`o# EIYY# a~i%# yle c`y gahlray
farr`gkae~`y z`r` aeile~r`r c` mqeikùe pqa gahlr `hqy~a cly n`~ly%
Gë~lnly z`r` aeile~r`r cçea`y na ~aenaeik`
0% Ac gë~lnl ` g`el ckora l }kyq`c%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 738/846
Nr% Fëi~lr Pqa}anl Qrç`y
00'6
?% Ac gë~lnl na cly iq`nr`nly gçekgly%
9% Ac gë~lnl na yagkzrlgankly%
>% Ac gë~lnl na zrlgankly ae gl}kgkae~l%
0% Ac gë~lnl ` g`el ckora ileyky~a ae `hqy~`r c` ~aenaeik` na qe` cçea` l iqr}`#
av`gke`enl c` bràmki` }kyq`cgae~a% Yke ago`rbl# ay~a gë~lnl yqoha~k}l nazaena
gqifl nac hqkikl kenk}knq`c%
?% Ac gë~lnl na cly iq`nr`nly gçekgly zqana qy`rya z`r` aeile~r`r c` aiq`ikùe na c`
~aenaeik` na c` iqr}`% F`{ gqifly zrlbr`g`y na ilgzq~`nlr` pqa `{qn`e ` ay~l%
9% Ac gë~lnl nac zrlgankl nac gl}kgkae~l% Qy`enl cly ùrnaeay `zrlzk`nly nac
gl}kgkae~l na zrlgankly# cly z`~rleay içickily# ay~`ikle`cay l krrabqc`ray zqanae yar
ackgke`nly nah`enl# ylc`gae~a# c` ~aenaeik` nac gl}kgkae~l%
>% Gë~lnl na yagkzrlgankly% Ay~a gë~lnl ileyky~a ae yaz`r`r cly n`~ly ae nly z`r~ay
&zramarae~agae~a kbq`cay+ zrlgank`enl cly n`~ly ae i`n` z`r~a# lo~aekaenl nly
zqe~ly ae c` bràmki` na c`y yarkay na ~kagzl% Aeyabqkn`# qe` cçea` na ~aenaeik` ya
nkoqh` lo~aekaenl nly zqe~ly ae c` bràmki` na c`y yarkay na ~kagzl% Ay~a gë~lnl ay
ykgzca# zarl zqana n`r rayqc~`nly zloray% Ay~a gë~lnl ay `zcki`oca# ylc`gae~a#
iq`enl c` ~aenaeik` ay ckea`c l `zrlvkg`n`gae~a ckea`c%
Gë~lnl ` g`el ckora z`r` ac `hqy~`gkae~l na iqr}`y
Ay~a ay ac gë~lnl gày ykgzca z`r` c`y yarkay na ~kagzl% Ileyky~a ae br`mki`r c`y yarkay
na ~kagzl {# zlr gankl na loyar}`ikùe }kyq`c# ~r`x`r qe` cçea` rai~` ylora cly zqe~ly%
Qe` }ax faifl ay~l# ya ay~kg` c` aiq`ikùe na c` cçea` rai~` z`r` nayzqëy i`ciqc`r
iq`cpqkar }`clr na [ i yqy~k~q{aenl ac }`clr na V %Gë~lnl na cly iq`nr`nly gçekgly
Ay~a gë~lnl ay ac gày qy`nl { zraikyl z`r` aeile~r`r c` aiq`ikùe na qe` yarka na
~kagzl% Ileyknëraya c` Mkbqr` 00%? na `o`hl%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 739/846
Nr% Fëi~lr Pqa}anl Qrç`y
00';
Mkbqr` 00%?% Bràmki` gly~r`enl ac gë~lnl na cly iq`nr`nly gçekgly% Mqae~a8 Yzkabac
&0;=0+%
Cly zqe~ly na cly n`~ly ya n`e zlr & V 0# [ 0+# & V ?# [ ?+# %%%# & V e# [ e+% Z`r` qe }`clr n`nl na
V # nkb`gly# V 0# f`orà qe` nkmaraeik` ae~ra ac }`clr [ 0 { ac }`clr ilrrayzlenkae~a ilgl
ya na~argkeù na c` iqr}` I% Ilgl ya }a ae c` bràmki`# nael~`gly ay~` nkmaraeik` zlr
N0# c` iq`c# ae `cbqe`y li`ykleay ya ramkara ilgl c` nay}k`ikùe# arrlr l rayknq`c {
zqana yar zlyk~k}l# eab`~k}l l iarl% Ykgkc rgae~a# ilrrayzlenkaenl ` cly }`clray na
V 0 # V ?#%% V e# lo~aeagly c`y nay}k`ikleay N?# N9# %%%# Ne% C` gankikùe na –olen`n na
`hqy~a‗ ya n` zlr c` rac`ikùe N?0 $ N
?? $ %%% $ N
?e% Na ay~` g`ear`# yk c` yqg` na ay~ly
iq`nr`nly N?0# N
??# N
?9# a~i%# ay zapqað`# ac `hqy~a ay oqael% Zarl# yk c` yqg` ay
br`ena# ac `hqy~a ay g`cl# cl iq`c pqkara naikr pqa# ac arrlr l rayknq`c yarà br`ena#
kenki`enl gqif` }`rk`ikùe ae~ra cly n`~ly &Yzkabac# 0;=0+%
Namkekikùe% Na ~ln`y c`y iqr}`y pqa `zrlvkg`e qe brqzl na n`~ly ae ac yae~knl na cly
iq`nr`nly gçekgly# c` iqr}` pqa ~kaea c` zrlzkan`n na8 N?0 $ N
?? $%%% $ N
?e ay qe
gçekgl { ya cc`g` c` iqr}` pqa gahlr `hqy~` cly n`~ly% Qe` iqr}` pqa ~aeb` ay~`
zrlzkan`n ya nkia pqa `hqy~ cly n`~ly ae ac yae~knl nac gahlr iq`nr nl gçekgl { ya
cc`g` c` iqr}` na cly iq`nr`nly gçekgly% Zlr cl ~`e~l# qe` cçea` pqa ~aeb` ay~`
zrlzkan`n ya cc`g` c` cçea` na cly iq`nr nly gçekgly# z`ràolc` na cly iq`nr`nly
gçekgly# a~i%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 740/846
Nr% Fëi~lr Pqa}anl Qrç`y
00'0:
Cçea` na cly iq`nr`nly gçekgly { z`ràolc`y na cly iq`nr`nly gçekgly
C` cçea` na cly iq`nr`nly gçekgly# pqa `zrlvkg` ac ilehqe~l na zqe~ly & V 0 # [ 0+# & V ?#
[ ?+# %%% & V e # [ e+# {` ya nkyiq~kù ae ac I`zç~qcl 6# nlena ya f`ocù na rabraykùe {
ilrrac`ikùe ykgzca { gøc~kzca% Ae aya i`zç~qcl ya nayirkokarle glnacly na rabraykùe
na qe` cçea` rai~`% @eàclb`gae~a# ae ac I`zç~qcl ;# ya nayirkokarle glnacly
zlckelgk cay# ile qe` l gày na nly }`rk`ocay kenazaenkae~ay% ^`gokëe# ae aya
i`zç~qcl ya nayirkokarle glnacly iøokily% Ykaenl `yç# ae~leiay# el ya raza~kràe cly
gai`ekygly qy`nly z`r` `hqy~`r cly n`~ly ` cly glnacly gày `zrlzk`nly%
Ahagzcly qy`enl c`y yarkay na ~kagzl
Ahagzcl )0% Ya n`e cly n`~ly na c`y ykbqkae~ay ileiae~r`ikleay na okùvknl na i`rolel
&IL?+ &[ + ae mqeikùe nac ~kagzl & V + ae c` ~`oc` na `o`hl%
^@OC@ 00%0% ^`oc` gly~r`enl cly n`~ly na IL? ae mqeikùe nac ~kagzl% &Ac`olr`ikùe zrlzk +
Ilei% na IL? &[ + s 0 ? > > 1 2 6 ; 0: 00%1
&Gkccleay na ~leac`n`y+
^kagzl & V + s 0;:: 0;?: 0;9: 0;>: 0;1: 0;=: 0;2: 0;6: 0;;: ?:::
&@ðly ilnkmki`nly+ s 0 9 > = 6 ; 00 0> 01 0=
F`iar cly ykbqkae~ay iàciqcly8
&`+ ^r`x`r ` g`el ae c` bràmki` lo~aekn` cly n`~ly { qe` cçea` rai~`%
&o+ Aeile~r`r c` aiq`ikùe na ay~` cçea`%
&i+ Qy`enl ay~`nçy~ki` aeile~r`r c` cçea` `hqy~`n` na cly iq`nr`nly gçekgly {
ilgz`r`r cly }`clray na c` zaenkae~a { nac ke~ariaz~l[ aeile~r`nly ae ac keikyl &o+%
&n+ ^r`x`r ae c` bràmki` c` cçea` na c` aiq`ikùe aeile~r`n` &`+%
&a+ Qy`enl c`y aiq`ikleay aeile~r`n`y ae cly keikyly &o+ { &i+# ay~kg`r c`y
ileiae~r`ikleay na IL? z`r` ac `ðl ?:0:%
Ylcqikùe8
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 741/846
Nr% Fëi~lr Pqa}anl Qrç`y
00'00
&`+ Br`mki`gly cly zqe~ly &0#0+# &?#9+# &>#>+# &=#>+# &6#1+# &;#2+# &00#6+# &0>#;+ ae qe
yky~ag` na illrnae`n`y rai~`ebqc`ray ilgl ya gqay~r` ae c` Mkbqr` 00%9 na `o`hl%
^kagzl &V+
I l e i %
n a I L ? & [ +
0=0>0?0:6=>?:
0?
0:
6
=
>
?
:
Bràmki` gly~r`enl c`y ileiae~r`ikleay na okùvknl na i`rolel }y% ~kagzl
Mkbqr` 00%9% Bràmki` gly~r`enl c`y ileiae~r`ikleay na IL? ae mqeikùe nac ~kagzl%
&Ac`olr`ikùe zrlzk`+
&o+ Z`r` lo~aear c` aiq`ikùe na c` cçea`# qy`gly c` bràmki` na `rrko` { yacaiikle`gly
iq`caypqkar` na nly zqe~ly ylora c` cçea` ~r`x`n` ` g`el# ay~l ay# ilgl zqe~l Z {
zqe~l P% Aeyabqkn`# ay~kg`gly c`y illrnae`n`y na ay~ly nly z`ray zqe~ly pqa yle &0#
0+ { &0?# 2%1+% @flr` qy`enl c` aiq`ikùe na cly iq`nr`nly gçekgly n`n` zlr c`
mqeikùe [ 7 ` $ o&V+ { yqy~k~q{aenl cly }`clray na [ 0 7 0# V 0 7 :# [ ? 7 2%1 { V ? 7 0?
ely n`8
0%: 7 ` $ o&:+
2%1 7 ` $ o&0?+
_aylc}kaenl n`
` 7 0% 2%1 7 0 $ o &0?+ { o 7 %1>?% Zlr cl ~`e~l# c` aiq`ikùe ay8
[ i 7 0 $ %1>?& V +
L~r` mlrg` na f`iar cl gkygl ay ile c` aiq`ikùe na c` mlrg` nac zqe~l na c`
zaenkae~a na qe` cçea # [ 7 [ 0 7 g& V ’ V 0+# nlena g 7 &[ ? ’ [ 0+,& V ? ’ V 0+# z`r` n`r8
[ ’ [ 0 7 &[ ? ’ [ 0+,& V ? ’ V 0+ & V ’ V 0+ &00'0+
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 742/846
Nr% Fëi~lr Pqa}anl Qrç`y
00'0?
@flr` yqy~k~q{aenl cly }`clray ae &00'0+ n`8
[ ’ 0 7 &2%1 ’ 0+,&0? ’ :+ &V ’ :+# [ ’ 0 7 %1>? V # ay~l ay
[ i 7 0 $ %1>? & V +
&i+ Z`r` aeile~r`r c` aiq`ikùe na c` cçea` rai~` qy`gly gë~lnly ay~`nçy~kily# ay naikr#
qy`enl c`y aiq`ikleay pqa ay~kg`e ac ke~ariaz~l ae [ { c` zaenkae~a na c` cçea` &]ar
i`zç~qcl na rabraykùe+%
&Ω [ +&Ω V ?+ ’ &Ω V +&Ω V[ +
` 7 ┊┊┊┊┊┊┊┊┊┊┊┊┊┊┊┊ &00'?+
e Ω V ?
’ &Ω V +?
e Ω V[ ’ &Ω V +&Ω[ +
o 7 ┊┊┊┊┊┊┊┊┊┊┊┊ &00'9+
e Ω V ?
’ &Ω V +?
Z`r` ay~l# zlnagly qy`r qe` i`ciqc`nlr` na olcykccl l qe zrlbr`g` na ilgzq~`nlr` {
ay~kg`gly c`y ykbqkae~ay yqg`~lrk`y8
Ω V 7 1=# Ω[ 7 >:# Ω V ?
7 1?># Ω[ ?
7 ?1=# &Ω[ +?
7 0=::# Ω V[ 7 9=># Ω V Ω[ 7 ??>:#
&Ω V +? 7 909=# e 7 6%
@flr`# yqy~k~q{aenl ~lnly ay~ly }`clray ae c`y aiq`ikleay &00'?+ { &00'9+# z`r` ` { o
n`e cly ykbqkae~ay rayqc~`nly8
&>:+&1?>+ ’ &1=+&9=>+
Ke~ariaz~l ae [ 7 ` 7 ┊┊┊┊┊┊┊┊┊┊┊┊┊┊ 7 %1>1
&6+&1?>+ ’ &1=+?
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 743/846
Nr% Fëi~lr Pqa}anl Qrç`y
00'09
&6+&9=>+ ’ &1=+&>:+
Zaenkae~a 7 o 7 ┊┊┊┊┊┊┊┊┊┊┊┊┊ 7 %=9=
&6+&1?>+ ’ &1=+?
Zlr cl ~`e~l# c` aiq`ikùe na cly iq`nr`nly gçekgly ay8
[ i 7 %1>1 $ %=9=& V +
@pqç# ya zqana }ar pqa ay~` aiq`ikùe ay gày zraiky`# pqa c` lo~aekn` zlr gankl nac
hqkikl kenk}knq`c%
@flr`# z`r` ~r`x`r c` cçea` ae c` bràmki` ilrrayzlenkae~a ` c` aiq`ikùe na `rrko`#
zleagly [ 7 : { raylc}agly zlr V z`r` n r V 7 ':%612% Aeyabqkn`# zleagly V 7 : {
raylc}agly zlr [ z`r` n`r [ 7 :%1>1% Aeyabqkn`# qy`enl ay~ly nly z`ray na
illrnae`n`y# ay naikr# &’:%612# :+ { &:# :%1>1+ zlnagly ~r`x`r ae c` bràmki` qe` cçea`
gày zraiky` pqa `pqëcc` faif` ` g`el%
&a+ Qy`enl c`y aiq`ikleay [ 7 0 $ %1>?& V + a [ 7 %1>1 $ %=9=& V +# iq`enl V 7 02 &`ðl
?:0:+# ely n`# rayzai~k}`gae~a# [ 7 0:%?0 { 00%9=# ay~a øc~kgl }`clr ykaenl gày
zraikyl pqa ac e~arklr%
Ahagzcl )?% Qe` ilgz`ðç` na zrlbr`g`y na ilgzq~`nlr` razlr~` c` nag`en` z`r` qe
na~argke`nl z`pqa~a na ilgzq~`nlr`# ylora qe zarklnl na ~ray `ðly% Cly n`~ly ya n`e
ae c` ~`oc` na `o`hl8
^@OC@ 00%?% ^`oc` gly~r`enl cly n`~ly nac zrlocag`% &Ac`olr`ikùe zrlzk`+
Nag`en` ~rkgay~r`c &[ + s 92 ?? =? 6: 22 ;1 ;> 090 0>6 011 0?= 0=0
Zarklnly na ~kagzl & V + s 0 ? 9 > 1 = 2 6 ; 0: 00 0?
F`iar cly ykbqkae~ay iàciqcly8
&`+ ]kyq`cgae~a `hqy~`r qe` cçea` rai~` ` cly n`~ly na c` bràmki`%
&o+ Qy`enl gë~lnly ay~`nçy~kily ay~kg`r c` aiq`ikùe ckea`c na c`y yarkay na ~kagzl# ay
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 744/846
Nr% Fëi~lr Pqa}anl Qrç`y
00'0>
naikr# [ i 7 ` $ o&V+ Eù~aya pqa ~`gokëe ya zqana qy`r [ ae cqb`r na [ i+%
&i+ ^r`x`r qe` cçea` rai~` qy`enl ac z`r na illrnae`n`y nark}`nly na ay~` aiq`ikùe%
&n+ Ilgz`r`r c` cçea` faif` ` g`el ile c` cçea` lo~aekn` ae &i+%
&a+ Zranaikr ac }`clr na [ i iq`enl V 7 0>:
Ylcqikùe8
&`+ C` bràmki` na `o`hl gqay~r` ac nk`br`g` ayz`riknl na cly n`~ly%
Zarklnly na ~kagzl &V+
N a g ` e n ` ~ r k g a y ~ r ` c & [ +
0?0:6=>?:
06:
0=:
0>:
0?:
0::
6:
=:
>:
?:
Bràmki` gly~r`enl c` nag`en` ~rkgay~r`c }y % zarklnly na ~kagzl
Mkbqr` 00%>% Bràmki` gly~r`enl cly n`~ly nac ahagzcl na `rrko`% &Ac`olr`ikùe zrlzk`+
&`+ Z`r` lo~aear c` aiq`ikùe na c` cçea`# qy`gly c` bràmki` na `rrko` { yacaiikle`glyiq`caypqkar` na nly zqe~ly ylora c` cçea` ~r`x`n` ` g`el# ay~l ay# ilgl zqe~l Z {
zqe~l P% Aeyabqkn`# ay~kg`gly c`y illrnae`n`y na ay~ly z`ray na zqe~ly pqa yle &0#
0+ { &0?# 2%1+% @flr`# qy`enl c` aiq`ikùe na cly iq`nr`nly gçekgly n`n` zlr8
[ 7 ` $ o&V+ &00'9+
@flr`# yqy~k~q{aenl ~lnly cly }`clray na [ 0 7 0# V ? 7 : { [ ? 7 2%1 { V ? 7 0?# ae c`
aiq`ikùe &00'9+ na `rrko`# ely n`8
0%: 7 ` $ o&:+ { 2%1 7 ` $ o&0?+%
&o+ Qy`enl gë~lnly ay~`nçy~kily i`ciqc`gly c`y yqg`~lrk`y8
Ω V 7 26# Ω V ?
7 =1:# &Ω V +?,e 7 1:2# Ω[ 7 0066# Ω[
?7 0>:22># &Ω[ +
?,e 7 002#=0?
Ke~ariaz~l [ 7 ?:%11# Zaenkae~a 7 0?%:2# _ 7 :%;1%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 745/846
Nr% Fëi~lr Pqa}anl Qrç`y
00'01
Zlr cl ~`e~l# c` aiq`ikùe na ckea`c na c`y yarkay na ~kagzl z`r` ay~a zrlocag` ay8
[ i 7 ?:%11 $ 0?%:2& V +
Z`r` ~r`x`r qe` cçea` rai~` ylora cly n`~ly# yacaiikle`gly nly zqe~ly# nkb`gly V 7 9 {
V 7 ; { raylc}kaenl zlr [ i ae c` aiq`ikùe ckea`c na c`y yarkay na ~kagzl# n` c`y
illrnae`n`y &9# 1=%2=+ { &;# 0?;%06+% @flr`# hqe~`gly cly zqe~ly na ay~`y nly
illrnae`n`y { ~r`x`gly c` cçea` ilgl ya }a ae c` mkbqr` na `rrko`# c` iq`c gqay~r` c`
nag`en` ~rkgay~r`c &[ + zlr qe zarklnl na 9 `ðly & V +%
&i+ Z`r` zranaikr ac }`clr na [ i iq`enl V 7 0>: cly zlnagly f`iar zlr ke~arzlc`ikùe
qy`enl c` Mkbqr` 00%> l# ykgzcagae~a# yqy~k~q{aenl ac }`clr na V 7 0>: ae c` aiq`ikùe
na c`y yarkay na ~kagzl# ay~l ay8
[ i 7 ?:%11 $ 0?%:2& V +
7 ?:%11 $ 0?%:2&0>:+
7 0#20:%91
Ahagzcl )9% Ae qe ay~qnkl fkzl~ë~kil rac`ikle`nl ile cly i`yly na YKN@# ya n` c`
~`oc` na `o`hl% Ay~kg`r c` mqeikùe `hqy~`n` na ~aenaeik` qy`enl `eàckyky na ~aenaeik`
ile qe` mqeikùe avzleaeik`c%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 746/846
Nr% Fëi~lr Pqa}anl Qrç`y
00'0=
^@OC@ 00%9% ^`oc` gly~r`enl cly i`yly na YKN@ na qe ay~qnkl fkzl~ë~kil%
@ðly Eøgarl fkzl~ë~kil na i`yly na YKN@
0;60 0;;0 0#?:: 9;:#:::
0;6? 0;;? =#1:: 1::#:::
0;69 0;;9 0?#::: ;::#:::
0;6> 0;;> 01#::: 0#1::#:::
0;61 0;;1 ?2#::: ?#;::#:::
0;6= 0;;= 1:#::: >#:::#:::
0;62 0;;2 =9#::: 2#:::#:::
0;66 0;;6 00:#::: 0:#:::#:::
0;6; 0;;; 02:#::: 01#:::#:::
0;;: ?::: ?0:#::: ?1#:::#:::
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
F`iar cl ykbqkae~a8
&`+ Nark}`r c` aiq`ikùe nac glnacl zrlzqay~l% Ile ay~a glnacl zranaikr ac eøgarl na
i`yly na YKN@ z`r` ac `ðl ?::6% &Ac cai~lr cl naoarà f`iar+%
Kenav
I ` y l y n a Y K N @
?:060=0>0?0:6=>?
9:::::::
?1::::::
?:::::::
01::::::
0:::::::
1::::::
:
@ii qr`i{ Ga`yqray
G@ZA ?%:>==?A$:0
G@N ?%69;21A$:1
GYN >%0962>A$00
]`rk`oca
@i~q`c
Mk~y
El% na i`yly na YKN@ z`r` zarklnl &0;60'?:::+Brl~f Iqr}a Glnac
[~ 7 ?:;2%;: ( &0%=:102(( ~+
Mkbqr` 00%1% Bràmki` gly~r`enl c` rac`ikùe na cly i`yly na YKN@ ae mqeikùe nac
~kagzl%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 747/846
Nr% Fëi~lr Pqa}anl Qrç`y
00'02
Aharikikly I`zç~qcl 00%
00%0% Cly n`~ly na `o`hl gqay~r`e cly gkccleay na ~leac`n`y na okùvknl na i`rolel
agk~knly ` c` `~gùymar` nqr`e~a cly `ðly na 0;1: ` 0;11 ae ikar~` rabkùe kenqy~rk`c%
^`oc` gly~r`enl cly n`~ly nac zrlocag`% &Ac`olr`ikùe zrlzk`+
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
@ðly &V+ Ilnkmki`nly Gkccleay na ~leac`n`y na IL? &[+
0;1: &1:+ 10;10 &10+ 6
0;1? &1?+ 0?0;19 &19+ 01
0;1> &1>+ ?:0;11 &11+ ?9
F`iar cly ykbqkae~ay iàciqcly8
&`+ Br`mki`r cly n`~ly nac nk`br`g` ayz`riknl%
&o+ ^r`x`r qe` cçea` }kyq`cgae~a pqa gahlr ileai~a cly n`~ly { nark}`r c` aiq`ikùe {
ay~kg`r c` aiq`ikùe zlr hqkikl kenk}knq`c%
&i+ Ay~kg`r c` aiq`ikùe na cly iq`nr`nly gçekgly# ay naikr# qy`enl gë~lnlyay~`nçy~kily% &[ 7 1 $ 9%= V+
&n+ Zranaikr cly gkccleay na ~leac`n`y na IL? &ac }`clr na [+ z`r` ac `ņl ?::1%
Yqbaraeik`8 Z`r` raylc}ar ay~ly zrlocag`y qy`r ac zrlbr`g` na ilgzq~`nlr` Gkek~`o
l Y@Y%
00%?% Ay~a zrlocag` ay~à rac`ikle`nl ile qe ay~qnkl na ile~`gke`ikùe `~glymërki`# na
z`r~çiqc y gaelray pqa 0: gkir y agk~kn`y ae ikar~ rabkùe kenqy~rk`c# nqr`e~a ac
zarklnl na 0;6> ` 0;;;% Cly n`~ly na c`y ileiae~r`ikleay zrlgankl na c`y z`r~çiqc`y
ya gqay~r`e ae c` ~`oc` na `o`hl%
&`+ F`iar qe` bràmki` ile cly n`~ly
&o+ Qy`enl gë~lnly ay~`nçy~kily &el na hqkikl kenk}knq`c+ ay~kg`r c` aiq`ikùe na c`
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 748/846
Nr% Fëi~lr Pqa}anl Qrç`y
00'06
cçea` na c`y yarkay na ~kagzl &[i+ { ~r`x`r qe` cçea` rai~` ylora cly n`~ly bràmkily
&i+ Ay~kg`r c`y ileiae~r`ikleay zrlgankl na z`r~çiqc`y z`r` ac `ðl ?::9 zlr gankl na
ke~arzlc`ikùe { zlr gankl na c` aiq`ikùe%
^`oc` gly~r`enl cly n`~ly nac zrlocag`% &Ac`olr`ikùe zrlzk`+ \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
@ðly Ilei% zrlgankl @ðly Ilei% zrlgankl&zzg+ &zzg+
0;6> &6>+ 0:: 0;;? &;?+ 00=
0;61 &61+ 00: 0;;9 &;9+ 002
0;6= &6=+ 00? 0;;> &;>+ 006
0;62 &62+ 001 0;;1 &;1+ 0?:
0;66 &66+ 009 0;;= &;=+ 0?9
0;6; &6;+ 00= 0;;2 &;2+ 0?1
0;;: &;:+ 002 0;;6 &;6+ 0?>
0;;0 &;0+ 002 0;;; &;;+ 0?1
00%9% Naikr yk c` bràmki` na `o`hl gqay~r` ~aenaeik` l ay~`ikle`ckn`n%
]ae~`y
^kagzl
00%>% Pqë ~kzl na ~aenaeik gqay~r` c` bràmki` na `o`hl# ay naikr# ¼~aenaeik` l
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 749/846
Nr% Fëi~lr Pqa}anl Qrç`y
00'0;
ay~`ikle`ckn`n<
00%1% C` bràmki` na `o`hl gqay~r`# ¼~aenaeik` l ay~`ikle`ckn`n< &~aenaeik`+
]ae~`y `eq`cay &qekn`nay+
Zraikl
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 750/846
Nr% Fëi~lr Pqa}anl Qrç`y
00'0
I@ZK^QCL 00
Yarkay na ~kagzl
Ic`ykmki`ikùe na cly gl}kgkae~ly na c`y yarkay na ~kagzl%' ^aenaeik`y ` c`rbl
zc`xl%' Ilgzleae~ay içickily na yarkay na ~kagzl%' ]`rk`ikleay ay~`ikle`cay%'
]`rk`ikùe krrabqc`r%' Gë~lnly z`r` aeile~r`r cçea`y na ~aenaeik`%' Cçea` na cly
iq`nr`nly gçekgly { z`ràolc`y na cly iq`nr`nly gçekgly%'
Iq`cpqkar }`rk`oca ae mqeikùe nac ~kagzl# ae yqiaykùe# ya cc`g` yarkay na ~kagzl% C`y
yarkay na ~kagzl yle qe` yaiqaeik` na }`clray na }`rk`ocay ~lg`n`y ae zarklnly na
~kagzl yqiayk}ly% C` bràmki` na qe` yarka na ~kagzl ay qe nk`br`g`# ile ac aha }ar~ki`cgly~r`enl ac }`clr loyar}`nl & {+ {# ile ac aha flrkxle~`c nael~`enl ac ~kagzl
&gkeq~ly# nç`y# `ðly# a~i%+%
C`y bràmki`y ilgl cly fky~lbr`g`y l nk`br`g`y na ~`ccl { flh` yle gë~lnly
}kyq`cay ø~kcay z`r` gly~r`r c` }`rk`ikùe ae cly n`~ly% Yke ago`rbl# ac ~kagzl ay qe
m`i~lr gq{ kgzlr~`e~a pqa ile~rkoq{a ` c` }`rk`ikùe loyar}`n` na cly n`~ly# pqa cly
fky~lbr`g`y l c`y bràmki`y na i`h` el cly ~lg`e ae iqae~`%
C`y yarkay na ~kagzl yle qe ilehqe~l na loyar}`ikleay ~lg`n`y ` ~kagzly
ayzaiçmkily# qyq`cgae~a# ` ke~ar}`cly kbq`cay ae qe lrnae irlelcùbkil% C`y yarkay na
~kagzl l yaiqaeik`y na ~kagzl ya namkeae ilgl n`~ly ay~`nçy~kily pqa yle
ilcaiikle`nly# rabky~r`nly q loyar}`nly ae keiragae~ly na ~kagzly yqiayk}ly% Ac
`eàckyky na cly n`~ly na c`y yarkay na ~kagzl ay na ke~arëy z`r` `pqëccly pqkaeay nayaae
ae~aenar c` e`~qr`cax` na cly n`~ly z`y`nly { zrayae~ay% ^`gokëe# c`y yarkay na ~kagzl
yle na ke~arëy z`r` `pqaccly ke}ay~kb`nlray# pqkaeay nayaae qy`r ac ilelikgkae~l na
n`~ly z`y`nly z`r` zranaikr ac mq~qrl%
C`y `zcki`ikleay na c`y yarkay na ~kagzl yle gq{ ilgqeay ae c` ailelgç`#
zarl ~`gokëe ae c` ailelgç` l c` kebaekarç`% Zlr ahagzcl8
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 751/846
Nr% Fëi~lr Pqa}anl Qrç`y
00'?
0% Cly blokarely na c`y e`ikleay kenqy~rk`ckx`n`y pqkarae y`oar cly }`clray mq~qrly na
c`y ~`y` na ke~arayay glea~`rkly% C`y e`ikleay ae nay`rrlccl pqkarae y`oar c`y
~aenaeik`y na c`y ~`y`y na na}`cq`ikùe na c` glean`% ^`gokëe ay naya`oca zranaikr c`y
~`y`y na agzcaly l na nayagzcaly% Kbq`cgae~a# ay kgzlr~`e~a y`oar cly zlriae~`hay na
cly keiragae~ly na cly ily~ly na c` }kn`%
?% L~r`y `zcki`ikleay na c`y yarkay na ~kagzl yle cly zrleùy~kily na c`y ~`y`y na ke~arëy
z`r` c` iley~rqiikùe na }k}kaen`y { ac ily~l na cly g ~ark cay na iley~rqiikùe%
9% ^`gokëe c`y ilgz`ðç`y g`eqm`i~qr`nlr`y pqkarae zrlely~ki`r c` nag`en` na yqy
zrlnqi~ly { yqy iikleay ae ac gari`nl%
>% Ae kebaekarç` `gokae~`c# cly `i~k}ky~`y { zrl~ai~lray nac gankl `gokae~a pqkarae
y`oar iqàcay yle c`y ~aenaeik`y ae cly `qgae~l na cly b`yay na ke}are`narl# ilgl ac
okùvknl na i`rolel &li`ykle`nl zlr c` agkykleay }afkiqc`ray# c` kenqy~rk`# mqably
mlray~`cay# a~i%+ pqa ay~àe i`cae~`enl c` ~karr`# mqenkaenl cly bc`ik`ray gle~`ðlyly {
c`y i`z`y zlc`ray { i`gok`enl ac ickg` gqenk`c% ^`gokëe ay ke~aray`e~a y`oar c`y
~aenaeik`y { cly `qgae~ly na c` r`nk`ikùe qc~r`}klca~`# pqa ~`e~l n`ðl ay~à i`qy`enl `c
yar fqg`el# zlr c` nay~rqiikùe nac lxlel e`~qr`c ay~r`~lymërkil# i`qy`nl zlr c`
krr`ikle`ckn`n nac flgora glnarel%
1% C`y yarkay na ~kagzl ~`gokëe `zcki`e z`r` y`oar c`y ~aenaeik`y { zrleùy~kily na cly
keiragae~ly na c` zloc`ikùe gqenk`c# a~i%
Iq`enl ya br`mki`e c`y gankikleay na yarkay na ~kagzl# ` gaeqnl ya loyar}`e
~aenaeik`y# ikicly l }`rk`ikleay kgzlr~`e~ay na cly n`~ly pqa# na l~r` g`ear`# z`y`rç`e
ke`n}ar~knly%Namkekikùe8 G`~agà~ki`gae~a# qe` yarka na ~kagzl ya namkea zlr }`clray[ 0# [ ?#%%%%%%% na
qe` }`rk`oca [ # ilgl c` ~agzar`~qr`# ileiae~r`ikleay na ile~`gke`e~ay# ilgl IL?#
YL?# z`r~çiqc`y `~glymërki`y# a~i%# ` ~kagzly ~0# ~?#%%%% Zlr cl ~`e~l# [ ay qe` mqeikùe na ~
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 752/846
Nr% Fëi~lr Pqa}anl Qrç`y
00'9
ykgolckx`n` zlr [ 7 M&~ +%
Ic`ykmki`ikùe na cly gl}kgkae~ly na c`y yarkay na ~kagzl
Cly gl}kgkae~ly i`r`i~arçy~kily na c`y yarkay na ~kagzl zqanae ic`ykmki`rya ae iq`~rl
~kzly cc`g`nly ilgzleae~ay na yarkay na ~kagzl% Ay~ly ilgzleae~ay na c`y yarkay na
~kagzl ya nayirkoae ilgl ykbqa8
0% ^aenaeik`y ` c`rbl zc`xl l gl}kgkae~ly yaiqc`ray%
?% Gl}kgkae~ly l mcqi~q`ikleay içicki`y%
9% ]`rk`ikleay ay~`ikle`cay l gl}kgkae~ly ay~`ikle`cay%
>% ]`rk`ikleay l gl}kgkae~ly krrabqc`ray l `ca`~lrkly%
^aenaeik`y c`rbl zc`xl
C` ~aenaeik` ` c`rbl zc`xl l ~aenaeik` yaiqc`r na qe` yarka na ~kagzl ay ac
ilgzleae~a qekmlrga na c`y yarkay pqa razrayae~`e ac iraikgkae~l l nairagae~l na
~kagzly# ylora qe zarklnl br`ena na ~kagzl% C` ~aenaeik` yaiqc`r ya ramkara ` c`
nkraiikùe baear`c ae c` iq`c c` bràmki` na qe`y yarkay `z`raiae gl}arya nqr`e~a qe
ke~ar}`cl na ~kagzl% Zlr ahagzcl# c` zloc`ikùe na cly Ay~`nly Qeknly nqr`e~a cly
øc~kgly >: `ðly f` gly~r`nl qe` ~aenaeik` na iraikgkae~l na 092 gkccleay na
zaryle`y ` ?>= ae 0;66% C y ~aenaeik y ` c`rbl zc`xl ya }a ae c` Mkbqr` 00%:& +%
C` na~argke`ikùe na c`y ~aenaeik`y na c`y cçea`y { na c`y iqr}`y ya zqana f`iar
qy`enl ac gë~lnl na `hqy~`gkae~l na iqr}`y% ^`gokëe ya zqana f`iar zlr gankl nac
`eàckyky na cly nk`br`g`y ayz`riknly# z`r` aeile~r`r c` mqeikùe g`~agà~ki` pqa gahlr
aei`ha ae cly n`~ly%
Ilgzleae~ay içickily na yarkay na ~kagzlCly ilgzleae~ay içickily ya ramkarae ` cly gl}kgkae~ly raiqrrae~ay na `rrko` { `o`hl
na c`y ~aenaeik`y na c`y yarkay na ~kagzl% Ay~`y mcqi~q`ikleay na len`# cc`g`n`y ikicly
na cly eablikly# yle nkmarae~ay na c`y mcqi~q`ikleay ay~`ikle`cay% Ay naikr# ae ac
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 753/846
Nr% Fëi~lr Pqa}anl Qrç`y
00'>
yae~knl na pqa iqorae zarklnly c`rbly na ~kagzl# ~kaeae i`qy`y nkmarae~ay { yle
gaely zranaikocay% C`y mcqi~q`ikleay nqr`e na ? ` 0: `ðly# l gày# iq`enl ya gknae
c`y len`y na iray~` ` iray~` l na i`e`c ` i`e`c% Ahagzcly na ilgzleae~ay ikicly yle
zarklnly na raiaykleay aileùgki`y l na zarklnly na kemc`ikùe# nag`en` na zrlnqi~ly
` c`rbl zc`xl# a~i% Ay~` yk~q`ikùe ya }a ae c` Mkbqr` 00%: &o+
]`rk`ikleay ay~`ikle`cay
Ay~a ~kzl na yarkay na ~kagzl ya ramkarae ` }`rk`ikleay zarkùnki`y# zarl el ay~àe
ckgk~`n`y ` }`rk`ikleay ile c` ay~`ikùe nac `ðl% Ay~ly yle z`~rleay na zarklnly ae c`y
yarkay na ~kagzl pqa ya ilgzca~`e ae qe `ðl {# cqabl ya razk~ae na `iqarnl `c gkygl
z`~rùe na zarklnl ae `ðly# yqoyaiqae~ay% Zlr ahagzcl# cly zraikly na cly gari`nly
mke`eikarly zqanae gly~r`r ~aenaeik`y `c~`y l o`h`y ae qe nç` l ae qe` yag`e`%
Ae ay~qnkly `gokae~`cay# c`y mcqi~q`ikleay na cly ile~`gke`e~ay gqay~r`e
~aenaeik`y içicki`y nqr`e~a ac nç`# ilgl ae ac i`yl nac ay~qnkl na c`y ileiae~r`ikleay
na lxlel ~rlzlymarkil% L~rly ahagzcly yle c` zrlnqiikùe na ikar~ly zrlnqi~ly na
br`eh`y `brçilc`y# ac eøgarl na }afçiqcly pqa z`y`e zlr ikar~l zqe~l# ae~ra nly yk~kly#
a~i% C` qekn`n na ~kagzl ae }`rk`ikleay ay~`ikle`cay ay gaely pqa qe `ðl# zarl
zqanae yar na qe gay# qe` yag`e`# l z`r~a nac nç`% Ay~` yk~q`ikùe ya }a ae c` Mkbqr`
00%:&i+%
]`rk`ikùe krrabqc`r
Ay~a ay qe ~kzl na }`rk`ikùe pqa el ay~à ileyknar`nl zlr ~aenaeik`y# ikicly l m`i~lray
ay~`ikle`cay# ykel pqa ya ilgzlea na mqarx`y el raiqrrae~ay# ayzlrànki`y pqa el ya
nayirkoae ilgl l `~rkoqkn`y ` m`i~lray na ~aenaeik`y# ikicly# l ay~`ikle`cay% Ahagzclyna }`rk`ikleay krrabqc`ray yle gl}kgkae~ly ayzlrànkily na yarkay na ~kagzl naoknl `
keqen`ikleay# br`ekx`n`y# fac`n`y# ~lre`nly# fqr`i`eay# yapqç`y# mqably mlray~`cay#
a~i%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 754/846
Nr% Fëi~lr Pqa}anl Qrç`y
00'1
@nagày# c`y mkbqr`y na `o`hl gqay~r`e ahagzcly na `cbqely zlykocay z`~rleay
na ~aenaeik` ae yarkay na ~kagzl% Zlr ahagzcl# c` Mkbqr` 00%0&`+ gqay~r` qe`
~aenaeik` el ckea`c% C` Mkbqr` 00%0&o+ gqay~r` qe` ~aenaeik` ckea`c pqa nkygkeq{a% C`
Mkbqr` 00%0&i+ gqay~r` qe` bràmki` yke ~aenaeik`%
Mkbqr` 00%:% Bràmki`y gly~r`enl cly ~kzly na ~aenaeik`y% C` bràmki` &`+ gqay~r` qe`
~aenaeik` na cçea` ` c`rbl zc`xl l na gl}kgkae~l yaiqc`r% C` bràmki` &o+ gqay~r` qe`
cçea` na ~aenaeik` ` c`rbl zc`xl ile qe gl}kgkae~l içickil ylorazqay~l% C` bràmki` &i+
gqay~r` ~aenaeik`y içicki`y c`rbl zc`xl { gl}kgkae~ly ay~`ikle`cay% &Yzkabac# 0;=0+%
Mkbqr` 00%0% Ahagzcly na `cbqely z`~rleay na ~aenaeik`y ae yarkay na ~kagzl%
@zcki`ikleay na c`y mqeikleay na yarkay na ~kagzl
C`y `zcki`ikleay ae kebaekarç` na c`y ~aenaeik`y ` c`rbl zc`xl l ~aenaeik`y yaiqc`ray
yle }`rk`y% Ahagzcly na ay~`y `zcki`ikleay yle cly keiragae~ly na c` ile~`gke`ikùe
`gokae~`c% Qe ahagzcl icàykil ay ac `qgae~l iley~`e~a na c`y ileiae~r`ikleay na
okùvknl na i`rolel# b`y ga~`el# }`zlr na `bq # a~i%# ` ek}ac gqenk c# pqa f`e ay~ nl
liqrrkaenl nayna ac kekikl na c` ar` kenqy~rk`c f`y~` `c zrayae~a% Ay~l# ilgl ay okae
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 755/846
Nr% Fëi~lr Pqa}anl Qrç`y
00'=
y`oknl# ay~à ilrrlgzkaenl ac ickg` na eqay~rl zc`ea~`# `c ay~`rya i`cae~`enl c` ~karr` {
c`y `bq`y g`rke`y%
L~rly ahagzcly# ` cly iq`cay ya cay zqanae `zcki`r c`y yarkay na ~kagzl# yle cly
keiragae~ly ae c` r`nk`ikùe qc~r`}klca~` &ae yqy mlrg`y na Q]'@ { Q]'O+# pqa ay~àe
i`qy`enl iàeiar ae c` zkac &ae yqy ~ray mlrg`y# gac`elg`# o`y`c { ayi`glyl+ { n`ðly
ae c` }kykùe { `c~ar`ikleay ae c` ay~rqi~qr` nac NE@%
L~r`y `zcki`ikleay na c`y yarkay na ~kagzl ay~àe rac`ikle`n`y ile cly
iraikgkae~ly zloc`ikle`cay l naglbràmkily% L~rly gày ay~àe rac`ikle`nly ile c`
zrlnqiikùe kenqy~rk`c# c` zrlnqiikùe na aearbç`# c` ailelgç # a~i%
^kzly na mqeikleay g`~agà~ki`y z`r` cçea`y na ~aenaeik`
C`y aiq`ikleay l mqeikleay g`~agà~ki`y gày ilgqeay z`r` `zrlvkg`r cly n`~ly
bràmkily na cçea`y na ~aenaeik` na qe nk`br`g` ayz`riknl ya n`e ac c` ~`oc` na `o`hl%
@pqç# c`y ca~r`y `# o { i razrayae~`e }`clray iley~`e~ay {# c`y ca~r`y V a [ # razrayae~`e
c`y }`rk`ocay kenazaenkae~ay { nazaenkae~ay# rayzai~k}`gae~a%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 756/846
Nr% Fëi~lr Pqa}anl Qrç`y
00'2
^@OC@ 00%:% ^`oc` gly~r`enl cly ~kzly na mqeikleay g`~agà~ki`y gày ilgqeay
qy`n`y z`r` cçea`y na ~aenaeik`% &Ac`olr`ikùe zrlzk`+
Mqeikùe g`~agà~ki` Nayirkzikùe
&0+ { 7 ` $ ov Cçea` rai~`
&?+ { 7 m& v+ 7 `v?
$ ov $ i Iqr}` iq`nrà~ki` l z`r`oùcki`
&9+ { 7 v9
$ o v?
$ iv $ n Iqr}` iøoki`
&>+ { 7 v>
$ o v9
$ iv?
$ nv $ a Iqr}` iqàr~ki`
&1+ { 7 ` $ o v $ i v?
$ %%% $ `e ve Zlckelgk`c baear`ckx`nl
&=+ { 7 `o v
l Clb { 7 Clb ` $ &Clb o+ v Iqr}` avzleaeik`c
&2+ { 7 v o
l Clb { 7 Clb ` $ o Clb v Iqr}` balgë~rki`
&6+ { 7 0,` $ o v l 0, { 7 ` $ o v Mqeikùe fkzërolc`&;+ { 7 zp o v
l Clb { 7 Clb z $ o v
Clb p Iqr}` na Blgzar~x
&0:+ { 7 `o v
$ b Iqr}` avzleaeik`c glnkmki`n`
&00+ { 7 v o
$ b Iqr}` balgë~rki` glnkmki`n`
&0?+ { 7 Ce v Mqeikùe clb`rç~gki`
&09+ { 7 ` ’ &` ’ o+ avz&'&i+s vs+n
Mqeikùe na Rakoqcc
&0>+ { 7 ` ’ &` ’ o+,&0 $ &is vs+n
Mqeikùe na Glrb`e'Gariar'Mclnkeb
&01+ { 7 `&0 $ &o ’ 0+ avz&'i& v ’ n++0,&0 ' o+
Mqeikùe na _kif`rny
Z`r` naiknkr# iqàc mqeikùe g`~agà~ki` ay c` gày `zrlzk`n`# z`r` `hqy~`r cly
n`~ly ya zqana f`iar }kaenl qe nk`br`g` ayz`riknl na c` bràmki` na cly n`~ly% Zlr
ahagzcl# yk ac nk`br`g` ayz`riknl ae z`zac yagkclb`rk~gl na Clb & {+ }y% v gqay~r`
qe` rac`ikùe ckea`c# c` aiq`ikùe ~kaea c` mlrg` na c` iqr}` avzleaeik`c &=+% Yk ya qy` ac
z`zac clb`rk~gl ilgzca~l# Clb {'Clb v# { cly n`~ly gqay~r`e qe` rac`ikùe ckea`c# c`
aiq`ikùe ~kaea c` mlrg` na qe` iqr}` balgë~rki` &2+% Na iq`cpqkar g`ear`# cly
zrlbr`g`y na ilgzq~`nlr # ilgl ac Y@Y# Gkek~`o# EIYY# a~i%# yle c`y gahlray
farr`gkae~`y z`r` aeile~r`r c` mqeikùe pqa gahlr `hqy~a cly n`~ly%
Gë~lnly z`r` aeile~r`r cçea`y na ~aenaeik`
0% Ac gë~lnl ` g`el ckora l }kyq`c%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 757/846
Nr% Fëi~lr Pqa}anl Qrç`y
00'6
?% Ac gë~lnl na cly iq`nr`nly gçekgly%
9% Ac gë~lnl na yagkzrlgankly%
>% Ac gë~lnl na zrlgankly ae gl}kgkae~l%
0% Ac gë~lnl ` g`el ckora ileyky~a ae `hqy~`r c` ~aenaeik` na qe` cçea` l iqr}`#
av`gke`enl c` bràmki` }kyq`cgae~a% Yke ago`rbl# ay~a gë~lnl yqoha~k}l nazaena
gqifl nac hqkikl kenk}knq`c%
?% Ac gë~lnl na cly iq`nr`nly gçekgly zqana qy`rya z`r` aeile~r`r c` aiq`ikùe na c`
~aenaeik` na c` iqr}`% F`{ gqifly zrlbr`g`y na ilgzq~`nlr` pqa `{qn`e ` ay~l%
9% Ac gë~lnl nac zrlgankl nac gl}kgkae~l% Qy`enl cly ùrnaeay `zrlzk`nly nac
gl}kgkae~l na zrlgankly# cly z`~rleay içickily# ay~`ikle`cay l krrabqc`ray zqanae yar
ackgke`nly nah`enl# ylc`gae~a# c` ~aenaeik` nac gl}kgkae~l%
>% Gë~lnl na yagkzrlgankly% Ay~a gë~lnl ileyky~a ae yaz`r`r cly n`~ly ae nly z`r~ay
&zramarae~agae~a kbq`cay+ zrlgank`enl cly n`~ly ae i`n` z`r~a# lo~aekaenl nly
zqe~ly ae c` bràmki` na c`y yarkay na ~kagzl% Aeyabqkn`# qe` cçea` na ~aenaeik` ya
nkoqh` lo~aekaenl nly zqe~ly ae c` bràmki` na c`y yarkay na ~kagzl% Ay~a gë~lnl ay
ykgzca# zarl zqana n`r rayqc~`nly zloray% Ay~a gë~lnl ay `zcki`oca# ylc`gae~a#
iq`enl c` ~aenaeik` ay ckea`c l `zrlvkg`n`gae~a ckea`c%
Gë~lnl ` g`el ckora z`r` ac `hqy~`gkae~l na iqr}`y
Ay~a ay ac gë~lnl gày ykgzca z`r` c`y yarkay na ~kagzl% Ileyky~a ae br`mki`r c`y yarkay
na ~kagzl {# zlr gankl na loyar}`ikùe }kyq`c# ~r`x`r qe` cçea` rai~` ylora cly zqe~ly%
Qe` }ax faifl ay~l# ya ay~kg` c` aiq`ikùe na c` cçea` rai~` z`r` nayzqëy i`ciqc`r
iq`cpqkar }`clr na [ i yqy~k~q{aenl ac }`clr na V %Gë~lnl na cly iq`nr`nly gçekgly
Ay~a gë~lnl ay ac gày qy`nl { zraikyl z`r` aeile~r`r c` aiq`ikùe na qe` yarka na
~kagzl% Ileyknëraya c` Mkbqr` 00%? na `o`hl%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 758/846
Nr% Fëi~lr Pqa}anl Qrç`y
00';
Mkbqr` 00%?% Bràmki` gly~r`enl ac gë~lnl na cly iq`nr`nly gçekgly% Mqae~a8 Yzkabac
&0;=0+%
Cly zqe~ly na cly n`~ly ya n`e zlr & V 0# [ 0+# & V ?# [ ?+# %%%# & V e# [ e+% Z`r` qe }`clr n`nl na
V # nkb`gly# V 0# f`orà qe` nkmaraeik` ae~ra ac }`clr [ 0 { ac }`clr ilrrayzlenkae~a ilgl
ya na~argkeù na c` iqr}` I% Ilgl ya }a ae c` bràmki`# nael~`gly ay~` nkmaraeik` zlr
N0# c` iq`c# ae `cbqe`y li`ykleay ya ramkara ilgl c` nay}k`ikùe# arrlr l rayknq`c {
zqana yar zlyk~k}l# eab`~k}l l iarl% Ykgkc rgae~a# ilrrayzlenkaenl ` cly }`clray na
V 0 # V ?#%% V e# lo~aeagly c`y nay}k`ikleay N?# N9# %%%# Ne% C` gankikùe na –olen`n na
`hqy~a‗ ya n` zlr c` rac`ikùe N?0 $ N
?? $ %%% $ N
?e% Na ay~` g`ear`# yk c` yqg` na ay~ly
iq`nr`nly N?0# N
??# N
?9# a~i%# ay zapqað`# ac `hqy~a ay oqael% Zarl# yk c` yqg` ay
br`ena# ac `hqy~a ay g`cl# cl iq`c pqkara naikr pqa# ac arrlr l rayknq`c yarà br`ena#
kenki`enl gqif` }`rk`ikùe ae~ra cly n`~ly &Yzkabac# 0;=0+%
Namkekikùe% Na ~ln`y c`y iqr}`y pqa `zrlvkg`e qe brqzl na n`~ly ae ac yae~knl na cly
iq`nr`nly gçekgly# c` iqr}` pqa ~kaea c` zrlzkan`n na8 N?0 $ N
?? $%%% $ N
?e ay qe
gçekgl { ya cc`g` c` iqr}` pqa gahlr `hqy~` cly n`~ly% Qe` iqr}` pqa ~aeb` ay~`
zrlzkan`n ya nkia pqa `hqy~ cly n`~ly ae ac yae~knl nac gahlr iq`nr nl gçekgl { ya
cc`g` c` iqr}` na cly iq`nr`nly gçekgly% Zlr cl ~`e~l# qe` cçea` pqa ~aeb` ay~`
zrlzkan`n ya cc`g` c` cçea` na cly iq`nr nly gçekgly# z`ràolc` na cly iq`nr`nly
gçekgly# a~i%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 759/846
Nr% Fëi~lr Pqa}anl Qrç`y
00'0:
Cçea` na cly iq`nr`nly gçekgly { z`ràolc`y na cly iq`nr`nly gçekgly
C` cçea` na cly iq`nr`nly gçekgly# pqa `zrlvkg` ac ilehqe~l na zqe~ly & V 0 # [ 0+# & V ?#
[ ?+# %%% & V e # [ e+# {` ya nkyiq~kù ae ac I`zç~qcl 6# nlena ya f`ocù na rabraykùe {
ilrrac`ikùe ykgzca { gøc~kzca% Ae aya i`zç~qcl ya nayirkokarle glnacly na rabraykùe
na qe` cçea` rai~`% @eàclb`gae~a# ae ac I`zç~qcl ;# ya nayirkokarle glnacly
zlckelgk cay# ile qe` l gày na nly }`rk`ocay kenazaenkae~ay% ^`gokëe# ae aya
i`zç~qcl ya nayirkokarle glnacly iøokily% Ykaenl `yç# ae~leiay# el ya raza~kràe cly
gai`ekygly qy`nly z`r` `hqy~`r cly n`~ly ` cly glnacly gày `zrlzk`nly%
Ahagzcly qy`enl c`y yarkay na ~kagzl
Ahagzcl )0% Ya n`e cly n`~ly na c`y ykbqkae~ay ileiae~r`ikleay na okùvknl na i`rolel
&IL?+ &[ + ae mqeikùe nac ~kagzl & V + ae c` ~`oc` na `o`hl%
^@OC@ 00%0% ^`oc` gly~r`enl cly n`~ly na IL? ae mqeikùe nac ~kagzl% &Ac`olr`ikùe zrlzk +
Ilei% na IL? &[ + s 0 ? > > 1 2 6 ; 0: 00%1
&Gkccleay na ~leac`n`y+
^kagzl & V + s 0;:: 0;?: 0;9: 0;>: 0;1: 0;=: 0;2: 0;6: 0;;: ?:::
&@ðly ilnkmki`nly+ s 0 9 > = 6 ; 00 0> 01 0=
F`iar cly ykbqkae~ay iàciqcly8
&`+ ^r`x`r ` g`el ae c` bràmki` lo~aekn` cly n`~ly { qe` cçea` rai~`%
&o+ Aeile~r`r c` aiq`ikùe na ay~` cçea`%
&i+ Qy`enl ay~`nçy~ki` aeile~r`r c` cçea` `hqy~`n` na cly iq`nr`nly gçekgly {
ilgz`r`r cly }`clray na c` zaenkae~a { nac ke~ariaz~l[ aeile~r`nly ae ac keikyl &o+%
&n+ ^r`x`r ae c` bràmki` c` cçea` na c` aiq`ikùe aeile~r`n` &`+%
&a+ Qy`enl c`y aiq`ikleay aeile~r`n`y ae cly keikyly &o+ { &i+# ay~kg`r c`y
ileiae~r`ikleay na IL? z`r` ac `ðl ?:0:%
Ylcqikùe8
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 760/846
Nr% Fëi~lr Pqa}anl Qrç`y
00'00
&`+ Br`mki`gly cly zqe~ly &0#0+# &?#9+# &>#>+# &=#>+# &6#1+# &;#2+# &00#6+# &0>#;+ ae qe
yky~ag` na illrnae`n`y rai~`ebqc`ray ilgl ya gqay~r` ae c` Mkbqr` 00%9 na `o`hl%
^kagzl &V+
I l e i %
n a I L ? & [ +
0=0>0?0:6=>?:
0?
0:
6
=
>
?
:
Bràmki` gly~r`enl c`y ileiae~r`ikleay na okùvknl na i`rolel }y% ~kagzl
Mkbqr` 00%9% Bràmki` gly~r`enl c`y ileiae~r`ikleay na IL? ae mqeikùe nac ~kagzl%
&Ac`olr`ikùe zrlzk`+
&o+ Z`r` lo~aear c` aiq`ikùe na c` cçea`# qy`gly c` bràmki` na `rrko` { yacaiikle`gly
iq`caypqkar` na nly zqe~ly ylora c` cçea` ~r`x`n` ` g`el# ay~l ay# ilgl zqe~l Z {
zqe~l P% Aeyabqkn`# ay~kg`gly c`y illrnae`n`y na ay~ly nly z`ray zqe~ly pqa yle &0#
0+ { &0?# 2%1+% @flr` qy`enl c` aiq`ikùe na cly iq`nr`nly gçekgly n`n` zlr c`
mqeikùe [ 7 ` $ o&V+ { yqy~k~q{aenl cly }`clray na [ 0 7 0# V 0 7 :# [ ? 7 2%1 { V ? 7 0?
ely n`8
0%: 7 ` $ o&:+
2%1 7 ` $ o&0?+
_aylc}kaenl n`
` 7 0% 2%1 7 0 $ o &0?+ { o 7 %1>?% Zlr cl ~`e~l# c` aiq`ikùe ay8
[ i 7 0 $ %1>?& V +
L~r` mlrg` na f`iar cl gkygl ay ile c` aiq`ikùe na c` mlrg` nac zqe~l na c`
zaenkae~a na qe` cçea # [ 7 [ 0 7 g& V ’ V 0+# nlena g 7 &[ ? ’ [ 0+,& V ? ’ V 0+# z`r` n`r8
[ ’ [ 0 7 &[ ? ’ [ 0+,& V ? ’ V 0+ & V ’ V 0+ &00'0+
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 761/846
Nr% Fëi~lr Pqa}anl Qrç`y
00'0?
@flr` yqy~k~q{aenl cly }`clray ae &00'0+ n`8
[ ’ 0 7 &2%1 ’ 0+,&0? ’ :+ &V ’ :+# [ ’ 0 7 %1>? V # ay~l ay
[ i 7 0 $ %1>? & V +
&i+ Z`r` aeile~r`r c` aiq`ikùe na c` cçea` rai~` qy`gly gë~lnly ay~`nçy~kily# ay naikr#
qy`enl c`y aiq`ikleay pqa ay~kg`e ac ke~ariaz~l ae [ { c` zaenkae~a na c` cçea` &]ar
i`zç~qcl na rabraykùe+%
&Ω [ +&Ω V ?+ ’ &Ω V +&Ω V[ +
` 7 ┊┊┊┊┊┊┊┊┊┊┊┊┊┊┊┊ &00'?+
e Ω V ?
’ &Ω V +?
e Ω V[ ’ &Ω V +&Ω[ +
o 7 ┊┊┊┊┊┊┊┊┊┊┊┊ &00'9+
e Ω V ?
’ &Ω V +?
Z`r` ay~l# zlnagly qy`r qe` i`ciqc`nlr` na olcykccl l qe zrlbr`g` na ilgzq~`nlr` {
ay~kg`gly c`y ykbqkae~ay yqg`~lrk`y8
Ω V 7 1=# Ω[ 7 >:# Ω V ?
7 1?># Ω[ ?
7 ?1=# &Ω[ +?
7 0=::# Ω V[ 7 9=># Ω V Ω[ 7 ??>:#
&Ω V +? 7 909=# e 7 6%
@flr`# yqy~k~q{aenl ~lnly ay~ly }`clray ae c`y aiq`ikleay &00'?+ { &00'9+# z`r` ` { o
n`e cly ykbqkae~ay rayqc~`nly8
&>:+&1?>+ ’ &1=+&9=>+
Ke~ariaz~l ae [ 7 ` 7 ┊┊┊┊┊┊┊┊┊┊┊┊┊┊ 7 %1>1
&6+&1?>+ ’ &1=+?
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 762/846
Nr% Fëi~lr Pqa}anl Qrç`y
00'09
&6+&9=>+ ’ &1=+&>:+
Zaenkae~a 7 o 7 ┊┊┊┊┊┊┊┊┊┊┊┊┊ 7 %=9=
&6+&1?>+ ’ &1=+?
Zlr cl ~`e~l# c` aiq`ikùe na cly iq`nr`nly gçekgly ay8
[ i 7 %1>1 $ %=9=& V +
@pqç# ya zqana }ar pqa ay~` aiq`ikùe ay gày zraiky`# pqa c` lo~aekn` zlr gankl nac
hqkikl kenk}knq`c%
@flr`# z`r` ~r`x`r c` cçea` ae c` bràmki` ilrrayzlenkae~a ` c` aiq`ikùe na `rrko`#
zleagly [ 7 : { raylc}agly zlr V z`r` n r V 7 ':%612% Aeyabqkn`# zleagly V 7 : {
raylc}agly zlr [ z`r` n`r [ 7 :%1>1% Aeyabqkn`# qy`enl ay~ly nly z`ray na
illrnae`n`y# ay naikr# &’:%612# :+ { &:# :%1>1+ zlnagly ~r`x`r ae c` bràmki` qe` cçea`
gày zraiky` pqa `pqëcc` faif` ` g`el%
&a+ Qy`enl c`y aiq`ikleay [ 7 0 $ %1>?& V + a [ 7 %1>1 $ %=9=& V +# iq`enl V 7 02 &`ðl
?:0:+# ely n`# rayzai~k}`gae~a# [ 7 0:%?0 { 00%9=# ay~a øc~kgl }`clr ykaenl gày
zraikyl pqa ac e~arklr%
Ahagzcl )?% Qe` ilgz`ðç` na zrlbr`g`y na ilgzq~`nlr` razlr~` c` nag`en` z`r` qe
na~argke`nl z`pqa~a na ilgzq~`nlr`# ylora qe zarklnl na ~ray `ðly% Cly n`~ly ya n`e
ae c` ~`oc` na `o`hl8
^@OC@ 00%?% ^`oc` gly~r`enl cly n`~ly nac zrlocag`% &Ac`olr`ikùe zrlzk`+
Nag`en` ~rkgay~r`c &[ + s 92 ?? =? 6: 22 ;1 ;> 090 0>6 011 0?= 0=0
Zarklnly na ~kagzl & V + s 0 ? 9 > 1 = 2 6 ; 0: 00 0?
F`iar cly ykbqkae~ay iàciqcly8
&`+ ]kyq`cgae~a `hqy~`r qe` cçea` rai~` ` cly n`~ly na c` bràmki`%
&o+ Qy`enl gë~lnly ay~`nçy~kily ay~kg`r c` aiq`ikùe ckea`c na c`y yarkay na ~kagzl# ay
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 763/846
Nr% Fëi~lr Pqa}anl Qrç`y
00'0>
naikr# [ i 7 ` $ o&V+ Eù~aya pqa ~`gokëe ya zqana qy`r [ ae cqb`r na [ i+%
&i+ ^r`x`r qe` cçea` rai~` qy`enl ac z`r na illrnae`n`y nark}`nly na ay~` aiq`ikùe%
&n+ Ilgz`r`r c` cçea` faif` ` g`el ile c` cçea` lo~aekn` ae &i+%
&a+ Zranaikr ac }`clr na [ i iq`enl V 7 0>:
Ylcqikùe8
&`+ C` bràmki` na `o`hl gqay~r` ac nk`br`g` ayz`riknl na cly n`~ly%
Zarklnly na ~kagzl &V+
N a g ` e n ` ~ r k g a y ~ r ` c & [ +
0?0:6=>?:
06:
0=:
0>:
0?:
0::
6:
=:
>:
?:
Bràmki` gly~r`enl c` nag`en` ~rkgay~r`c }y % zarklnly na ~kagzl
Mkbqr` 00%>% Bràmki` gly~r`enl cly n`~ly nac ahagzcl na `rrko`% &Ac`olr`ikùe zrlzk`+
&`+ Z`r` lo~aear c` aiq`ikùe na c` cçea`# qy`gly c` bràmki` na `rrko` { yacaiikle`glyiq`caypqkar` na nly zqe~ly ylora c` cçea` ~r`x`n` ` g`el# ay~l ay# ilgl zqe~l Z {
zqe~l P% Aeyabqkn`# ay~kg`gly c`y illrnae`n`y na ay~ly z`ray na zqe~ly pqa yle &0#
0+ { &0?# 2%1+% @flr`# qy`enl c` aiq`ikùe na cly iq`nr`nly gçekgly n`n` zlr8
[ 7 ` $ o&V+ &00'9+
@flr`# yqy~k~q{aenl ~lnly cly }`clray na [ 0 7 0# V ? 7 : { [ ? 7 2%1 { V ? 7 0?# ae c`
aiq`ikùe &00'9+ na `rrko`# ely n`8
0%: 7 ` $ o&:+ { 2%1 7 ` $ o&0?+%
&o+ Qy`enl gë~lnly ay~`nçy~kily i`ciqc`gly c`y yqg`~lrk`y8
Ω V 7 26# Ω V ?
7 =1:# &Ω V +?,e 7 1:2# Ω[ 7 0066# Ω[
?7 0>:22># &Ω[ +
?,e 7 002#=0?
Ke~ariaz~l [ 7 ?:%11# Zaenkae~a 7 0?%:2# _ 7 :%;1%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 764/846
Nr% Fëi~lr Pqa}anl Qrç`y
00'01
Zlr cl ~`e~l# c` aiq`ikùe na ckea`c na c`y yarkay na ~kagzl z`r` ay~a zrlocag` ay8
[ i 7 ?:%11 $ 0?%:2& V +
Z`r` ~r`x`r qe` cçea` rai~` ylora cly n`~ly# yacaiikle`gly nly zqe~ly# nkb`gly V 7 9 {
V 7 ; { raylc}kaenl zlr [ i ae c` aiq`ikùe ckea`c na c`y yarkay na ~kagzl# n` c`y
illrnae`n`y &9# 1=%2=+ { &;# 0?;%06+% @flr`# hqe~`gly cly zqe~ly na ay~`y nly
illrnae`n`y { ~r`x`gly c` cçea` ilgl ya }a ae c` mkbqr` na `rrko`# c` iq`c gqay~r` c`
nag`en` ~rkgay~r`c &[ + zlr qe zarklnl na 9 `ðly & V +%
&i+ Z`r` zranaikr ac }`clr na [ i iq`enl V 7 0>: cly zlnagly f`iar zlr ke~arzlc`ikùe
qy`enl c` Mkbqr` 00%> l# ykgzcagae~a# yqy~k~q{aenl ac }`clr na V 7 0>: ae c` aiq`ikùe
na c`y yarkay na ~kagzl# ay~l ay8
[ i 7 ?:%11 $ 0?%:2& V +
7 ?:%11 $ 0?%:2&0>:+
7 0#20:%91
Ahagzcl )9% Ae qe ay~qnkl fkzl~ë~kil rac`ikle`nl ile cly i`yly na YKN@# ya n` c`
~`oc` na `o`hl% Ay~kg`r c` mqeikùe `hqy~`n` na ~aenaeik` qy`enl `eàckyky na ~aenaeik`
ile qe` mqeikùe avzleaeik`c%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 765/846
Nr% Fëi~lr Pqa}anl Qrç`y
00'0=
^@OC@ 00%9% ^`oc` gly~r`enl cly i`yly na YKN@ na qe ay~qnkl fkzl~ë~kil%
@ðly Eøgarl fkzl~ë~kil na i`yly na YKN@
0;60 0;;0 0#?:: 9;:#:::
0;6? 0;;? =#1:: 1::#:::
0;69 0;;9 0?#::: ;::#:::
0;6> 0;;> 01#::: 0#1::#:::
0;61 0;;1 ?2#::: ?#;::#:::
0;6= 0;;= 1:#::: >#:::#:::
0;62 0;;2 =9#::: 2#:::#:::
0;66 0;;6 00:#::: 0:#:::#:::
0;6; 0;;; 02:#::: 01#:::#:::
0;;: ?::: ?0:#::: ?1#:::#:::
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
F`iar cl ykbqkae~a8
&`+ Nark}`r c` aiq`ikùe nac glnacl zrlzqay~l% Ile ay~a glnacl zranaikr ac eøgarl na
i`yly na YKN@ z`r` ac `ðl ?::6% &Ac cai~lr cl naoarà f`iar+%
Kenav
I ` y l y n a Y K N @
?:060=0>0?0:6=>?
9:::::::
?1::::::
?:::::::
01::::::
0:::::::
1::::::
:
@ii qr`i{ Ga`yqray
G@ZA ?%:>==?A$:0
G@N ?%69;21A$:1
GYN >%0962>A$00
]`rk`oca
@i~q`c
Mk~y
El% na i`yly na YKN@ z`r` zarklnl &0;60'?:::+Brl~f Iqr}a Glnac
[~ 7 ?:;2%;: ( &0%=:102(( ~+
Mkbqr` 00%1% Bràmki` gly~r`enl c` rac`ikùe na cly i`yly na YKN@ ae mqeikùe nac
~kagzl%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 766/846
Nr% Fëi~lr Pqa}anl Qrç`y
00'02
Aharikikly I`zç~qcl 00%
00%0% Cly n`~ly na `o`hl gqay~r`e cly gkccleay na ~leac`n`y na okùvknl na i`rolel
agk~knly ` c` `~gùymar` nqr`e~a cly `ðly na 0;1: ` 0;11 ae ikar~` rabkùe kenqy~rk`c%
^`oc` gly~r`enl cly n`~ly nac zrlocag`% &Ac`olr`ikùe zrlzk`+
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
@ðly &V+ Ilnkmki`nly Gkccleay na ~leac`n`y na IL? &[+
0;1: &1:+ 10;10 &10+ 6
0;1? &1?+ 0?0;19 &19+ 01
0;1> &1>+ ?:0;11 &11+ ?9
F`iar cly ykbqkae~ay iàciqcly8
&`+ Br`mki`r cly n`~ly nac nk`br`g` ayz`riknl%
&o+ ^r`x`r qe` cçea` }kyq`cgae~a pqa gahlr ileai~a cly n`~ly { nark}`r c` aiq`ikùe {
ay~kg`r c` aiq`ikùe zlr hqkikl kenk}knq`c%
&i+ Ay~kg`r c` aiq`ikùe na cly iq`nr`nly gçekgly# ay naikr# qy`enl gë~lnlyay~`nçy~kily% &[ 7 1 $ 9%= V+
&n+ Zranaikr cly gkccleay na ~leac`n`y na IL? &ac }`clr na [+ z`r` ac `ņl ?::1%
Yqbaraeik`8 Z`r` raylc}ar ay~ly zrlocag`y qy`r ac zrlbr`g` na ilgzq~`nlr` Gkek~`o
l Y@Y%
00%?% Ay~a zrlocag` ay~à rac`ikle`nl ile qe ay~qnkl na ile~`gke`ikùe `~glymërki`# na
z`r~çiqc y gaelray pqa 0: gkir y agk~kn`y ae ikar~ rabkùe kenqy~rk`c# nqr`e~a ac
zarklnl na 0;6> ` 0;;;% Cly n`~ly na c`y ileiae~r`ikleay zrlgankl na c`y z`r~çiqc`y
ya gqay~r`e ae c` ~`oc` na `o`hl%
&`+ F`iar qe` bràmki` ile cly n`~ly
&o+ Qy`enl gë~lnly ay~`nçy~kily &el na hqkikl kenk}knq`c+ ay~kg`r c` aiq`ikùe na c`
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 767/846
Nr% Fëi~lr Pqa}anl Qrç`y
00'06
cçea` na c`y yarkay na ~kagzl &[i+ { ~r`x`r qe` cçea` rai~` ylora cly n`~ly bràmkily
&i+ Ay~kg`r c`y ileiae~r`ikleay zrlgankl na z`r~çiqc`y z`r` ac `ðl ?::9 zlr gankl na
ke~arzlc`ikùe { zlr gankl na c` aiq`ikùe%
^`oc` gly~r`enl cly n`~ly nac zrlocag`% &Ac`olr`ikùe zrlzk`+ \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
@ðly Ilei% zrlgankl @ðly Ilei% zrlgankl&zzg+ &zzg+
0;6> &6>+ 0:: 0;;? &;?+ 00=
0;61 &61+ 00: 0;;9 &;9+ 002
0;6= &6=+ 00? 0;;> &;>+ 006
0;62 &62+ 001 0;;1 &;1+ 0?:
0;66 &66+ 009 0;;= &;=+ 0?9
0;6; &6;+ 00= 0;;2 &;2+ 0?1
0;;: &;:+ 002 0;;6 &;6+ 0?>
0;;0 &;0+ 002 0;;; &;;+ 0?1
00%9% Naikr yk c` bràmki` na `o`hl gqay~r` ~aenaeik` l ay~`ikle`ckn`n%
]ae~`y
^kagzl
00%>% Pqë ~kzl na ~aenaeik gqay~r` c` bràmki` na `o`hl# ay naikr# ¼~aenaeik` l
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 768/846
Nr% Fëi~lr Pqa}anl Qrç`y
00'0;
ay~`ikle`ckn`n<
00%1% C` bràmki` na `o`hl gqay~r`# ¼~aenaeik` l ay~`ikle`ckn`n< &~aenaeik`+
]ae~`y `eq`cay &qekn`nay+
Zraikl
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 769/846
Nr% Fëi~lr Pqa}anl Qrç`y
0?'0
I@ZK^QCL 0?
Yacaiikùe nac ~`g`ðl na c` gqay~r` Nark}`ikùe na c` mùrgqc` z`r` ay~kg`r ac ~`g`ðl gày `zrlzk`nl na c`
gqay~r` z`r` ac zrlgankl%' Yacaiikùe nac ~`g`ðl na c` gqay~r` z`r` nly
zloc`ikleay%'
Ae ay~qnkly na nkyaðly avzarkgae~`cay ay~`nçy~kily ay eaiay`rkl ay~kg`r ac ~`g`ðl
na c` gqay~r` gày `zrlzk`nl z`r` c` ay~kg`ikùe na zrlgankly# zrlzlrikleay# a~i%
C` yacaiikùe gày `zrlzk`n` nac ~`g`ðl na c` gqay~r` ay kgzlr~`e~a# zlrpqa el
pqaragly y`i`r qe ~`g`ðl na gqay~r` aviayk}`gae~a br`ena# pqa }` ` yar gq{
ily~lyl% Zlr c` gkyg` r`xùe# ~`gzlil pqaragly y`i`r qe ~`g`ðl na gqay~r`
zapqaðl# pqa ely keickea ` `iaz~`r fkzù~ayky eqc`y# ay naikr# na ilga~ar ac arrlr KK%
Na ay~` g`ear`# ac ~`g`ðl `zrlzk`nl na c` gqay~r` ay kgzlr~`e~a# zlrpqa ~`g`ðly
na gqay~r`y keeaiay`rk`gae~a br`enay yle ily~lyly { nayzarnkik`e nkearl { ~kagzl
{# ~`gokëe# zlrpqa ~`g`ðly na gqay~r`y zapqaðly n`e rayqc~`nly zloray%
Avky~ae }`rk`y mqeikleay ay~`nçy~ki`y z`r` na~argke`r ac ~`g`ðl gày
`zrlzk`nl na c` gqay~r` ay~`nçy~ki`# ay naikr# z`r` ay~kg`r ac zrlgankl zloc`ikle`c
±# c` }`rk`ex` ω?# c` nay}k`ikùe ay~àen`r ω# c` zrlzlrikùe π# a~i% Z`r` ay~kg`r ay~ly
z`ràga~rly qy`gly c` nky~rkoqikùe elrg`c# zarl ay eaiay`rkl y`oar yk c` zloc`ikùe
gqay~ra`n` ay elrg`c l `zrlvkg`n`gae~a elrg`c% Ay~l ya f`ia z`r` c`y zrqao`y na
fkzù~ayky qy`enl cly ek}acay na ykbekmki`eik` na :%:1 { :%:0# pqa n`e cly
ilamkikae~ay irç~kily na 0%;= { ?%16# ay naikr# ilrrayzlenkae~ay ` cly ek}acay na
ilemk`ex` na ;1! { ;;!%
Ae yk~q`ikleay nlena zqana ile~rlc`rya ac ~`g`ðl na c` gqay~r` ay zlykoca
acabkr qe ~`g`ðl na gqay~r` e# na glnl pqa ya ~aeb` qe` ilemk`ex` nac 0::&0 ’ ΰ+
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 770/846
Nr% Fëi~lr Pqa}anl Qrç`y
0?'?
zlr ikae~l na pqa ac arrlr# `c ay~kg`r# nkb`gly ±# ya` gaelr pqa ac arrlr
ayzaikmki`nl A# ay~l ay# cl pqa pqaragly `rrkayb`r%
Ae c` na~argke`ikùe nac ~`g`ðl na c` gqay~r` ae qe avzarkgae~l ay~`nçy~kil
~aeagly pqa y`oar nly ily`y8
0% Pqë ~`e iari` naya`gly pqa eqay~r` ay~kg`ikùe ay~ë nac }arn`narl }`clr nac
z`ràga~rl zloc`ikle`c%
?% Pqë ~`e~` iar~ax` naya`gly pqa eqay~r` ay~kg`ikùe ay~ë nae~rl nac eøgarl na
qekn`nay yacaiikle`n`y nac }`clr nac z`ràga~rl%
Nark}`ikùe na c` mùrgqc` z`r` ay~kg`r ac ~`g`ðl gày `zrlzk`nl na c`gqay~r` z`r` ac zrlgankl
Z`r` nark}`r c` mùrgqc` z`r` ay~kg`r ac ~`g`ðl ùz~kgl na c` gqay~r`# qy`gly c`
nky~rkoqikùe na c` ay~`nçy~ki` nac zrlgankl V % Zlr ahagzcl# y`oagly pqë# na c`
nky~rkoqikùe nac zrlgankl V gly~r`n` `o`hl# ac ke~ar}`cl ± ° ?ω V ile~kaea#
`zrlvkg`n`gae~a# ac ;1! na cly }`clray na c` ay~`nçy~ki` nac zrlgankl V %
Mkbqr` 0?%:% Bràmki` gly~r`enl c` nky~rkoqikùe na c` ay~`nçy~ki` nac zrlgankl%
&Ac`olr`ikùe zrlzk`+
@ilrnagae~a# yk naya`gly ay~`r# ` el gày na A qekn`nay na ± ile eqay~rl
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 771/846
Nr% Fëi~lr Pqa}anl Qrç`y
0?'9
ay~kg`nlr ay~`nçy~kil nac zrlgankl V # ae~leiay# nah`gly pqa A 7 ?ω V ay~l ay#
A 7 ? ω , ∘e &0?'0+
@flr`# raylc}kaenl zlr e n`8
e 7 >ω?, A
?&0?'?+
Ay~` mqeikùe &0?'?+ ~kaea qe ilamkikae~a na ilemk`ex` na &0 ’ ΰ+ 7 :%;1>>% Yk
pqaragly qe ilamkikae~a na ilemk`ex` na &0 ’ ΰ+# ae~leiay# ya nah` pqa8
xΰ,? ω V 7 A l okae xΰ,? ω,∘e 7 A &0?'9+
Pqa rayqc~` ae c` mùrgqc`8
e 7 xΰ,? ω?
,A
?
&0?'>+7 &xΰ,? ω,A+
?&0?'1+
Nlena8
xΰ,? 7 }`clr na c` nky~rkoqikùe elrg`c ay~àen`r na ~`c g`ear` pqa# Z&X≡ xΰ,?+ 7 ΰ,?%
@pqç# qyq`cgae~a# cly }`clray irç~kily na
xΰ,? yle na 0%;2 { ?%16#ω 7 nay}k`ikùe ay~àen`r zloc`ikle`c%
A 7 arrlr gàvkgl na c` ay~kg`ikùe
Na `iqarnl ` c` aiq`ikùe `e~arklr# ac arrlr A ay n`nl zlr8
A 7 xΰ,?&ω∘e+ &0?'=+
Z`r` zlnar qy`r c` mùrgqc` &0?'>+ eaiayk~`gly ileliar &0 ’ ΰ+# A { ω% Yk ac
~`g`ðl na c` gqay~r` ay e ≡ 9: i`yly l yk c` zloc`ikùe gqay~ra`n` ay elrg`c#
ae~leiay# ya zqana `zrlvkg`r ω ` y%
Namkekikùe8 Yk ac zrlgankl V ya qy` ilgl ay~kg`ikùe na ±# ae~leiay# zqana
~aearya qe` ilemk`ex` nac 0::&0 ’ ΰ+ zlr ikae~l na pqa ac arrlr s V ’ ±s el yarà
g`{lr pqa qe` i`e~kn`n ayzaiçmki` A iq`enl ac ~`g`ðl na c` gqay~r` ya`
e 7 &xΰ,? ω , A+?% Ay~` mqeikùe zqana yar qy`n` z`r` na~argke`r ac ~`g`ðl na
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 772/846
Nr% Fëi~lr Pqa}anl Qrç`y
0?'>
gqay~r` eaiay`rkl# z`r` zrlnqikr oqaely rayqc~`nly ` qe br`nl na ilemk`ex`
naya`nl { g`rbae na arrlr% El loy~`e~a# ay~` mùrgqc` rapqkara na cly }`clray naω l
na ω?% Ay~ly }`clray ya zqanae ileliar na ay~qnkly zra}kly l zqanae yar
r`xle`ocagae~a# ay~kg`nly na ay~qnkly `e~arklray l ay~qnkly zkcl~ly%
Ahagzcly kcqy~r`enl c` na~argke`ikùe nac ~`g`ðl na gqay~r` gày `zrlzk`nl
z`r` ac zrlgankl V
Ahagzcl )0% Qe ileyqc~lr ay~`nçy~kil ke~ae~` qy`r ac zrlgankl na qe` gqay~r`
`ca`~lrk` na ~`g`ðl e 7 01:# z`r` ay~kg`r c` `z~k~qn gaiàeki` zrlgankl &zrlgankl
gank`e~a ikar~` zrqao`+ na lorarly na c` cçea` na gle~`ha na qe` kenqy~rk`% Yk ile o`ya ae c` avzarkaeik`# ac ay~`nçy~kil zqana yqzlear pqa ω 7 =%?# ae~leiay# z`r`
ay~ly n`~ly# ¼pqë zqana `mkrg`r ay~a ileyqc~lr# ile zrlo`okckn`n na :%;;# `iari` na
c` nkgaeykùe gàvkg` nac arrlr A<
Ylcqikùe8
Z`r` ay~kg`r A qy`gly e 7 01:# ω 7 =%?# xΰ,? 7 x:%:0,? 7 ?%121%
Qy`enl c` mùrgqc` &0?'1+ { yqy~k~q{aenl n`8
A 7 xΰ,? &ω,∘e+
7 ?%121&=%?,∘01:+
7 0%9:
Ile ay~a rayqc~`nl# ac ay~`nçy~kil zqana `mkrg`r# ile qe ek}ac na ilemk`ex` na ;;!
&l ile qe` zrlo`okckn`n na :%;;+# pqa yq arrlr yarà iq`enl gày na 0%9:%
Ahagzcl )?% _amkrkëenlya `c zrlocag` `e~arklr# yqzleb`gly `flr` pqa ac ileyqc~lr
ay~`nçy~kil naya` qe ek}ac na ilemk`ex` nac ;1!# ykaenl `yç# ¼iqàc yarç` c` g`bek~qn
nac arrlr# A<
Ylcqikùe8
Qy`enl# eqa}`gae~a# c` mùrgqc` &0?'=+# ile xΰ,? 7 x:%:1,? 7 x%:?1 7 0%;=
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 773/846
Nr% Fëi~lr Pqa}anl Qrç`y
0?'1
A 7 0%;=&=%?,∘01:+ 7 :%;;?
@pqç# eù~aya pqa naoknl ` pqa pqaragly gaely zraikykùe &qy`enl ac ek}ac na
ilemk`ex` na ;1!+ ac arrlr ay gày zapqaðl pqa yk qy`gly ac ek}ac na ilemk`ex` na
;;!% ^`gokëe ay na el~`rya pqa# ` gankn` pqa ac ~`g`ðl na e ya f`ia gày br`ena#
ac arrlr A nkygkeq{a%
Ahagzcl )9% Ae qe ay~qnkl na pqçgki`# ae qe `r~çiqcl zqocki`nl ae ac Hlqre`c lm
Fa`~ ^r`eymar# ya nayirkoa qe eqa}l gë~lnl z`r` gankr c` ilenqi~k}kn`n ~ërgki`
nac fkarrl @rgil% Yqzùeb`ya pqa ya naya` pqa ac arrlr zrlgankl ae c`
ilenqi~k}kn`n ~ërgki` nac fkarrl @rgil ya` gaelr pqa :%:1 O~q,fr'm~'
l
M# ile qeek}ac na ilemk`ex` nac ;1!% Ae~leiay# yk na ay~qnkly zra}kly ya y`oa pqa c`
nay}k`ikùe ay~àen`r ay na ω 7 :%0:# ay~kg`r ac ~`g`ðl na gqay~r` rapqarknl%
Ylcqikùe8
@pqç# xΰ,? 7 x:%:1,? 7 x:%:?1 7 °0%;=# ω 7 :%0:# A :%:1%
Qy`enl c` aiq`ikùe &0?'>+8 e 7 &xΰ,? ω, A+?
{ yqy~k~q{aenl ay~ly }`clray ely n`8
e 7 T&0%;=+&:%0:+ , :%:1+W?
7 01%92 ≆ 0=
El~` 0% Ykagzra pqaragly ranlena`r ac ~`g`ðl na c` gqay~r` na g`ear` pqa# ac
eøgarl rapqarknl ae c` gqay~r` ya` iq`enl gaely `naiq`nl# ae cqb`r na qe zlil
`naiq`nl% Ay~l ay qe ile}aeikle`ckygl%
Ahagzcl )>% Ae qe ay~qnkl na railcaiikùe na o`yqr` nayaif`n` zlr ac yai~lr
nlgëy~kil# ay naikr# nac y`c}`gae~l na o`yqr` raikic`oca# pqaragly ay~kg`r ac
zrlgankl nac zcày~kil nayaif`nl zlr c`y i`y`y% ¼Pqë ~`g`ðl na gqay~r` na i`y`y
naoa yar yacaiikle`nl# `ca`~lrk`gae~a# yk pqaragly ay~`r yabqrly# ae ;;!# pqa ac
zrlgankl gqay~r`c ay~ë nae~rl na :%?1: dkclbr`gly nac }arn`narl zrlgankl
zloc`ikle`c ±< @yqgkr pqa ay~qnkly zkcl~ly n`e qe` nay}k`ikùe ay~àen`r ilelikn`
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 774/846
Nr% Fëi~lr Pqa}anl Qrç`y
0?'=
na ω 7 0%0:: dkclbr`gly%
Ylcqikùe8
Pqaragly qe ~`g`ðl na gqay~r` e# n`nl pqa ΰ 7 :%:0 &;;! na ek}ac na ilemk`ex`+
na g`ear` pqa# xΰ,? 7 x:%:0,? 7 ?%121 &}`clr iley~`e~a na c` ~`oc` na c` nky~rkoqikùe
elrg`c ile ;;! ek}ac na ilemk`ex`+% @nagày# A 7 :%?1:# ω 7 0%0::% @yç# qy`enl c`
mùrgqc` &0?'1+ ely n`8
e 7 &xΰ,? ω , A+?
7 T&?%121+&0%0::+ , &:%?1:+W?
7 0?6%92≆ 0?;
Ae ileicqykùe# naoagly na lo~aear qe` gqay~r`# na iq`enl gaely 0?; i`y`y
nlgëy~ki`y yacaiikle`n`y `ca`~lrk`gae~a &pqa ay~àe nayi`r~`enl ac zcày~kil+% Ile
yagah`e~a gqay~r`# ay~`ragly ilemk`nly ae qe ;;! na pqa ac zrlgankl gqay~r`c
V ay~`rà nae~rl na :%?1: dkcly na ±%
Ahagzcl )1% _amkrkëenlya `c ahagzcl `e~arklr# yk pqkykër`gly ~aear rayqc~`nly
gaely zraikyly qy`enl qe g`rbae na arrlr na :%1:: dkcly# i`ciqc`r ac ~`g`ðl na c`
gqay~r` e `yqgkaenl c`y gkyg`y ilenkikleay `e~arklray%
Ylcqikùe8
Qy`enl c` mùrgqc` &0?'>+ lo~aeagly8
e 7 T&?%121+&0%0::+ , &:%1::+W?
7 9?%:;
≆ 99
Ya loyar}`e cly ykbqkae~ay zqe~ly ae c` rac`ikùe baear`c ae~ra ac ~`g`ðl na c`
gqay~r`# c` clebk~qn naya`n` nac ke~ar}`cl ?A# ac ek}ac na ilemk`ex` 0::&0 ’ ΰ+ zlr
ikae~l { ω8
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 775/846
Nr% Fëi~lr Pqa}anl Qrç`y
0?'2
0% Ilemlrga nkygkeq{a c` clebk~qn nac ke~ar}`cl ?A# ac ~`g`ðl rapqarknl na c`
gqay~r` e `qgae~` z`r` qe }`clr mkhl na ω { z`r` ac ek}ac na ilemk`ex`
ayzaikmki`nl%
?% @ gankn` pqa ω `qgae~`# ac ~`g`ðl rapqarknl na c` gqay~r` e `qgae~`# z`r` qe`
clebk~qn naya`n` ?A mkh` { qe ek}ac na ilemk`ex` ayzaikmki`nl%
9% Ilemlrga `qgae~` ac ek}ac na ilemk`ex`# ac ~`g`ðl rapqarknl na c` gqay~r` e
`qgae~` z`r` qe` clebk~qn mkh` naya`n` ?A { qe` nay}k`ikùe ay~àen`r ω%
Yacaiikùe nac ~`g`ðl na c` gqay~r` z`r` nly zloc`ikleay
^`gokëe ya zqana yacaiikle`r ac ~`g`ðl na c` gqay~r`# gày `zrlzk`nl# z`r` c`nkmaraeik` na nly zrlgankly% Zlr ahagzcl# yk ya ileliae c`y nay}k`ikleay ay~àen`r
na c`y gqay~r`y qel { nly# ay naikr# ω0 { ω?# { cly ~`g`ðly na c`y nly gqay~r`y yle
kbq`cay# ay naikr# e0 7 e? 7 e# ae~leiay# zqana na~argke`rya ac ~`g`ðl gày
`zrlzk`nl na c` gqay~r`% Ay~l ya f`ia na glnl pqa ya ~aeb` qe` ilemk`ex` na
0::&0 ’ ΰ+ zlr ikae~l ae pqa ac arrlr A ae c` ay~kg`ikùe na c` nkmaraeik` na ±0 ’ ±?#
zlr cly zrlgankly na c`y gqay~r`y V 0 ’ V ? ya` gaelr pqa A%
C` aiq`ikùe qy`n` z`r` i`ciqc`r ac ~`g`ðl na c` gqay~r` gày `zrlzk`nl z`r`
c` nkmaraeik` na nly zloc`ikleay ay8
e 7 &xΰ,? , A+?
&ω?0 $ ω?
?+ &0?'2+
El~` 0% _aiqërnaya pqa ay eaiay`rkl ranlena`r e# yk ay~a }`clr el ay qe ae~arl% Ile
ay~l# ya `yabqr` pqa ac ek}ac na ilemk`ex` el ya` gaelr pqa 0::&0 ’ ΰ+ zlr ikae~l%
Ahagzcl )=% Ya zrqao`e nly mùrgqc`y nkmarae~ay na b`ylcke` lvkbae`n` z`r`
ranqikr c`y agkykleay na gleùvknl na i`rolel &IL+ agk~kn`y zlr cly gl~lray na
ilgoqy~kùe ke~are`% Ya y`oa na `e~ag`el pqa c` }`rk`ex` z`r` c` zrkgar` mùrgqc`
ay na ω?0 7 0%1# gkae~r`y pqa c` }`rk`ex` z`r` c` yabqen` mùrgqc` ay na ω?
? 7 0%?%
¼Pqë ~`g`ðl na gqay~r` naoa qy`rya z`r` i`n` zloc`ikùe gqay~ra`n`# yk ya naya`
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 776/846
Nr% Fëi~lr Pqa}anl Qrç`y
0?'6
~aear qe` ilemk`ex` nac ;1! na pqa ac arrlr# `c ay~kg`r c` nkmaraeik` ae~ra cly
zrlgankly na c`y nly mùrgqc`y nkmarae~ay# ya` gaelr pqa 0<
Ylcqikùe8
@pqç# qy`gly c` mùrgqc` &0?'2+ z`r` i`ciqc`r ac ~`g`ðl na c` gqay~r` na nly
zloc`ikleay# ay naikr#
e 7 &xΰ,? , A+?
&ω?0 $ ω
??+
Nlena8
xΰ,? 7 x:%:1,? 7 x%:?1 7 0%;2# A 7 0# ω?0 7 0%1# ω?
? 7 0%?
Yqy~k~q{aenl ay~ly }`clray ae c` mùrgqc` na `rrko` n`8e 7 &0%;1 , 0+
?&0%1 $ 0%?+
7 0:%?2 ≆ 00
Zlr cl ~`e~l# ac ~`g`ðl na c` gqay~r` z`r` c`y zloc`ikleay ±0 { ±? ay8
e 7 e0 7 e? 7 00
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 777/846
Nr% Fëi~lr Pqa}anl Qrç`y
0?';
Aharikikly I`zç~qcl 0?
0?%0% Ya y`oa pqa c` nqr`ikùe# ae flr`y# na qe mlil na 21 `~~y ~kaea qe`
nky~rkoqikùe# `zrlvkg`n`gae~a elrg`c# ile qe` nay}k`ikùe ay~àen`r na ?1 flr`y%
Yqzùeb`ya pqa ya naya` qe` ilemk`ex` nac ;1! ae pqa ac arrlr ae c` ay~kg`ikùe na
c` nqr`ikùe zrlgankl ya` gaelr pqa 1 flr`y% ¼Pqë ~`g`ðl na gqay~r` naoa
qy`rya< &≆ ;2+
0?%?% F`iar ac gkygl zrlocag` pqa ac `e~arklr# zarl `flr` qy`enl qe` ilemk`ex`
na ;;! { qe arrlr A 7 0 { ilgz`r`r cly rayqc~`nly%
0?%9% Qe kebaekarl `q~lgl~rkx naya` na~argke`r ac ~kagzl zrlgankl pqa ~`rn`rç`
qe gaiàekil ae bkr`r c`y cc`e~`y na qe `q~l% Ay~a kebaekarl pqkara ay~kg`r# ile qe`
ilemk`ex` na ;1!# pqa ac zrlgankl na yq gqay~r` ay kgzraiky` ae iq`enl gày :%1:
gkeq~ly% Yk y`oa na ay~qnkly zkcl~ly `e~arklray pqa c` nay}k`ikùe ay~àen`r ay naω 7
0%= gkeq~ly# ¼pqë ~`e br`ena naoarà yar c` gqay~r` pqa naoa na yacaiikle`r#
`ca`~lrk`gae~a< Yqbaraeik`8 Qy`r c` mùrgqc` e 7 &xΰ,?
ω,A+?
&9;%9 ≆ >: gaiàekily+
0?%>% Ac nkrai~lr na ikar~` qek}arykn`n naya` qy`r ac zrlgankl na qe` gqay~r`
`ca`~lrk` z`r` ay~kg`r ac gle~l zrlgankl na ~kagzl pqa ya cay cca}` ` cly
ay~qnk`e~ay z`r` kr na qe y`cùe ` l~rl { ~lg`r yqy ic`yay yke ccab`r ~`rna% Z`r` ay~l
naya` `mkrg`r ile ;;! na ilemk`ex` pqa ac arrlr ay iq`enl gày na :%?1 gkeq~ly%
Avzarkaeik`y `e~arklray ay~kg`e qe` nay}k`ikùe ay~àen`r na ω 7 0%>: gkeq~ly%
Ykaenl `yç# ¼pqë ~`e br`ena naoarà yar c` gqay~r` pqa ya nao` ~lg`r<
0?%1% C` Ae}krlegae~`c Zrl~ai~kle @baei{ &AZ@+ na cly Ay~`nly Qeknly naya`ilenqikr qe` zrqao` na gkcc`ha na ikar~l glnacl na qe `q~l kgzlr~`nl% Ac
kebaekarl ay~`nçy~kil na c` AZ@ naya` ay~kg r ac zrlgankl ±# na gkcc`y zlr b`cùe
na ilgoqy~koca qy`nl zlr ay~a glnacl# ile ;1! na ek}ac na ilemk`ex`% @yqgkaenl
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 778/846
Nr% Fëi~lr Pqa}anl Qrç`y
0?'0:
pqa ω 7 ?%1 gkcc`y zlr b`cùe# ¼pqë ~`g`ðl na gqay~r` &eøgarl na `q~ly na ay~a
glnacl+ naoarà ~lg`r z`r` ilenqikr ay~` zrqao`< &e 7 ?1+
0?%=% Ileyknar`r ac ahagzcl 0?%0 na c` b`ylcke` lvkbae`n`# z`r` c` ay~kg`ikùe nac
~`g`ðl na c`y gqay~r`y z`r` c`y zloc`ikleay qel { nly% Ykaenl `yç# ay~kg`r cly
~`g`ðly na c`y gqay~r`y `zrlzk`nly# yk pqaragly qe` ilemk`ex` na ;;! { ac arrlr
na c` ay~kg`ikùe na c`y nkmaraeik`y ae~ra cly zrlgankly ya` gaelr pqa >%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 779/846
Nr% Fëi~lr Pqa}anl Qrç`y
0?'0
I@ZK^QCL 0?
Yacaiikùe nac ~`g`ðl na c` gqay~r` Nark}`ikùe na c` mùrgqc` z`r` ay~kg`r ac ~`g`ðl gày `zrlzk`nl na c`
gqay~r` z`r` ac zrlgankl%' Yacaiikùe nac ~`g`ðl na c` gqay~r` z`r` nly
zloc`ikleay%'
Ae ay~qnkly na nkyaðly avzarkgae~`cay ay~`nçy~kily ay eaiay`rkl ay~kg`r ac ~`g`ðl
na c` gqay~r` gày `zrlzk`nl z`r` c` ay~kg`ikùe na zrlgankly# zrlzlrikleay# a~i%
C` yacaiikùe gày `zrlzk`n` nac ~`g`ðl na c` gqay~r` ay kgzlr~`e~a# zlrpqa el
pqaragly y`i`r qe ~`g`ðl na gqay~r` aviayk}`gae~a br`ena# pqa }` ` yar gq{
ily~lyl% Zlr c` gkyg` r`xùe# ~`gzlil pqaragly y`i`r qe ~`g`ðl na gqay~r`
zapqaðl# pqa ely keickea ` `iaz~`r fkzù~ayky eqc`y# ay naikr# na ilga~ar ac arrlr KK%
Na ay~` g`ear`# ac ~`g`ðl `zrlzk`nl na c` gqay~r` ay kgzlr~`e~a# zlrpqa ~`g`ðly
na gqay~r`y keeaiay`rk`gae~a br`enay yle ily~lyly { nayzarnkik`e nkearl { ~kagzl
{# ~`gokëe# zlrpqa ~`g`ðly na gqay~r`y zapqaðly n`e rayqc~`nly zloray%
Avky~ae }`rk`y mqeikleay ay~`nçy~ki`y z`r` na~argke`r ac ~`g`ðl gày
`zrlzk`nl na c` gqay~r` ay~`nçy~ki`# ay naikr# z`r` ay~kg`r ac zrlgankl zloc`ikle`c
±# c` }`rk`ex` ω?# c` nay}k`ikùe ay~àen`r ω# c` zrlzlrikùe π# a~i% Z`r` ay~kg`r ay~ly
z`ràga~rly qy`gly c` nky~rkoqikùe elrg`c# zarl ay eaiay`rkl y`oar yk c` zloc`ikùe
gqay~ra`n` ay elrg`c l `zrlvkg`n`gae~a elrg`c% Ay~l ya f`ia z`r` c`y zrqao`y na
fkzù~ayky qy`enl cly ek}acay na ykbekmki`eik` na :%:1 { :%:0# pqa n`e cly
ilamkikae~ay irç~kily na 0%;= { ?%16# ay naikr# ilrrayzlenkae~ay ` cly ek}acay na
ilemk`ex` na ;1! { ;;!%
Ae yk~q`ikleay nlena zqana ile~rlc`rya ac ~`g`ðl na c` gqay~r` ay zlykoca
acabkr qe ~`g`ðl na gqay~r` e# na glnl pqa ya ~aeb` qe` ilemk`ex` nac 0::&0 ’ ΰ+
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 780/846
Nr% Fëi~lr Pqa}anl Qrç`y
0?'?
zlr ikae~l na pqa ac arrlr# `c ay~kg`r# nkb`gly ±# ya` gaelr pqa ac arrlr
ayzaikmki`nl A# ay~l ay# cl pqa pqaragly `rrkayb`r%
Ae c` na~argke`ikùe nac ~`g`ðl na c` gqay~r` ae qe avzarkgae~l ay~`nçy~kil
~aeagly pqa y`oar nly ily`y8
0% Pqë ~`e iari` naya`gly pqa eqay~r` ay~kg`ikùe ay~ë nac }arn`narl }`clr nac
z`ràga~rl zloc`ikle`c%
?% Pqë ~`e~` iar~ax` naya`gly pqa eqay~r` ay~kg`ikùe ay~ë nae~rl nac eøgarl na
qekn`nay yacaiikle`n`y nac }`clr nac z`ràga~rl%
Nark}`ikùe na c` mùrgqc` z`r` ay~kg`r ac ~`g`ðl gày `zrlzk`nl na c`gqay~r` z`r` ac zrlgankl
Z`r` nark}`r c` mùrgqc` z`r` ay~kg`r ac ~`g`ðl ùz~kgl na c` gqay~r`# qy`gly c`
nky~rkoqikùe na c` ay~`nçy~ki` nac zrlgankl V % Zlr ahagzcl# y`oagly pqë# na c`
nky~rkoqikùe nac zrlgankl V gly~r`n` `o`hl# ac ke~ar}`cl ± ° ?ω V ile~kaea#
`zrlvkg`n`gae~a# ac ;1! na cly }`clray na c` ay~`nçy~ki` nac zrlgankl V %
Mkbqr` 0?%:% Bràmki` gly~r`enl c` nky~rkoqikùe na c` ay~`nçy~ki` nac zrlgankl%
&Ac`olr`ikùe zrlzk`+
@ilrnagae~a# yk naya`gly ay~`r# ` el gày na A qekn`nay na ± ile eqay~rl
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 781/846
Nr% Fëi~lr Pqa}anl Qrç`y
0?'9
ay~kg`nlr ay~`nçy~kil nac zrlgankl V # ae~leiay# nah`gly pqa A 7 ?ω V ay~l ay#
A 7 ? ω , ∘e &0?'0+
@flr`# raylc}kaenl zlr e n`8
e 7 >ω?, A
?&0?'?+
Ay~` mqeikùe &0?'?+ ~kaea qe ilamkikae~a na ilemk`ex` na &0 ’ ΰ+ 7 :%;1>>% Yk
pqaragly qe ilamkikae~a na ilemk`ex` na &0 ’ ΰ+# ae~leiay# ya nah` pqa8
xΰ,? ω V 7 A l okae xΰ,? ω,∘e 7 A &0?'9+
Pqa rayqc~` ae c` mùrgqc`8
e 7 xΰ,? ω?
,A
?
&0?'>+7 &xΰ,? ω,A+
?&0?'1+
Nlena8
xΰ,? 7 }`clr na c` nky~rkoqikùe elrg`c ay~àen`r na ~`c g`ear` pqa# Z&X≡ xΰ,?+ 7 ΰ,?%
@pqç# qyq`cgae~a# cly }`clray irç~kily na
xΰ,? yle na 0%;2 { ?%16#ω 7 nay}k`ikùe ay~àen`r zloc`ikle`c%
A 7 arrlr gàvkgl na c` ay~kg`ikùe
Na `iqarnl ` c` aiq`ikùe `e~arklr# ac arrlr A ay n`nl zlr8
A 7 xΰ,?&ω∘e+ &0?'=+
Z`r` zlnar qy`r c` mùrgqc` &0?'>+ eaiayk~`gly ileliar &0 ’ ΰ+# A { ω% Yk ac
~`g`ðl na c` gqay~r` ay e ≡ 9: i`yly l yk c` zloc`ikùe gqay~ra`n` ay elrg`c#
ae~leiay# ya zqana `zrlvkg`r ω ` y%
Namkekikùe8 Yk ac zrlgankl V ya qy` ilgl ay~kg`ikùe na ±# ae~leiay# zqana
~aearya qe` ilemk`ex` nac 0::&0 ’ ΰ+ zlr ikae~l na pqa ac arrlr s V ’ ±s el yarà
g`{lr pqa qe` i`e~kn`n ayzaiçmki` A iq`enl ac ~`g`ðl na c` gqay~r` ya`
e 7 &xΰ,? ω , A+?% Ay~` mqeikùe zqana yar qy`n` z`r` na~argke`r ac ~`g`ðl na
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 782/846
Nr% Fëi~lr Pqa}anl Qrç`y
0?'>
gqay~r` eaiay`rkl# z`r` zrlnqikr oqaely rayqc~`nly ` qe br`nl na ilemk`ex`
naya`nl { g`rbae na arrlr% El loy~`e~a# ay~` mùrgqc` rapqkara na cly }`clray naω l
na ω?% Ay~ly }`clray ya zqanae ileliar na ay~qnkly zra}kly l zqanae yar
r`xle`ocagae~a# ay~kg`nly na ay~qnkly `e~arklray l ay~qnkly zkcl~ly%
Ahagzcly kcqy~r`enl c` na~argke`ikùe nac ~`g`ðl na gqay~r` gày `zrlzk`nl
z`r` ac zrlgankl V
Ahagzcl )0% Qe ileyqc~lr ay~`nçy~kil ke~ae~` qy`r ac zrlgankl na qe` gqay~r`
`ca`~lrk` na ~`g`ðl e 7 01:# z`r` ay~kg`r c` `z~k~qn gaiàeki` zrlgankl &zrlgankl
gank`e~a ikar~` zrqao`+ na lorarly na c` cçea` na gle~`ha na qe` kenqy~rk`% Yk ile o`ya ae c` avzarkaeik`# ac ay~`nçy~kil zqana yqzlear pqa ω 7 =%?# ae~leiay# z`r`
ay~ly n`~ly# ¼pqë zqana `mkrg`r ay~a ileyqc~lr# ile zrlo`okckn`n na :%;;# `iari` na
c` nkgaeykùe gàvkg` nac arrlr A<
Ylcqikùe8
Z`r` ay~kg`r A qy`gly e 7 01:# ω 7 =%?# xΰ,? 7 x:%:0,? 7 ?%121%
Qy`enl c` mùrgqc` &0?'1+ { yqy~k~q{aenl n`8
A 7 xΰ,? &ω,∘e+
7 ?%121&=%?,∘01:+
7 0%9:
Ile ay~a rayqc~`nl# ac ay~`nçy~kil zqana `mkrg`r# ile qe ek}ac na ilemk`ex` na ;;!
&l ile qe` zrlo`okckn`n na :%;;+# pqa yq arrlr yarà iq`enl gày na 0%9:%
Ahagzcl )?% _amkrkëenlya `c zrlocag` `e~arklr# yqzleb`gly `flr` pqa ac ileyqc~lr
ay~`nçy~kil naya` qe ek}ac na ilemk`ex` nac ;1!# ykaenl `yç# ¼iqàc yarç` c` g`bek~qn
nac arrlr# A<
Ylcqikùe8
Qy`enl# eqa}`gae~a# c` mùrgqc` &0?'=+# ile xΰ,? 7 x:%:1,? 7 x%:?1 7 0%;=
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 783/846
Nr% Fëi~lr Pqa}anl Qrç`y
0?'1
A 7 0%;=&=%?,∘01:+ 7 :%;;?
@pqç# eù~aya pqa naoknl ` pqa pqaragly gaely zraikykùe &qy`enl ac ek}ac na
ilemk`ex` na ;1!+ ac arrlr ay gày zapqaðl pqa yk qy`gly ac ek}ac na ilemk`ex` na
;;!% ^`gokëe ay na el~`rya pqa# ` gankn` pqa ac ~`g`ðl na e ya f`ia gày br`ena#
ac arrlr A nkygkeq{a%
Ahagzcl )9% Ae qe ay~qnkl na pqçgki`# ae qe `r~çiqcl zqocki`nl ae ac Hlqre`c lm
Fa`~ ^r`eymar# ya nayirkoa qe eqa}l gë~lnl z`r` gankr c` ilenqi~k}kn`n ~ërgki`
nac fkarrl @rgil% Yqzùeb`ya pqa ya naya` pqa ac arrlr zrlgankl ae c`
ilenqi~k}kn`n ~ërgki` nac fkarrl @rgil ya` gaelr pqa :%:1 O~q,fr'm~'
l
M# ile qeek}ac na ilemk`ex` nac ;1!% Ae~leiay# yk na ay~qnkly zra}kly ya y`oa pqa c`
nay}k`ikùe ay~àen`r ay na ω 7 :%0:# ay~kg`r ac ~`g`ðl na gqay~r` rapqarknl%
Ylcqikùe8
@pqç# xΰ,? 7 x:%:1,? 7 x:%:?1 7 °0%;=# ω 7 :%0:# A :%:1%
Qy`enl c` aiq`ikùe &0?'>+8 e 7 &xΰ,? ω, A+?
{ yqy~k~q{aenl ay~ly }`clray ely n`8
e 7 T&0%;=+&:%0:+ , :%:1+W?
7 01%92 ≆ 0=
El~` 0% Ykagzra pqaragly ranlena`r ac ~`g`ðl na c` gqay~r` na g`ear` pqa# ac
eøgarl rapqarknl ae c` gqay~r` ya` iq`enl gaely `naiq`nl# ae cqb`r na qe zlil
`naiq`nl% Ay~l ay qe ile}aeikle`ckygl%
Ahagzcl )>% Ae qe ay~qnkl na railcaiikùe na o`yqr` nayaif`n` zlr ac yai~lr
nlgëy~kil# ay naikr# nac y`c}`gae~l na o`yqr` raikic`oca# pqaragly ay~kg`r ac
zrlgankl nac zcày~kil nayaif`nl zlr c`y i`y`y% ¼Pqë ~`g`ðl na gqay~r` na i`y`y
naoa yar yacaiikle`nl# `ca`~lrk`gae~a# yk pqaragly ay~`r yabqrly# ae ;;!# pqa ac
zrlgankl gqay~r`c ay~ë nae~rl na :%?1: dkclbr`gly nac }arn`narl zrlgankl
zloc`ikle`c ±< @yqgkr pqa ay~qnkly zkcl~ly n`e qe` nay}k`ikùe ay~àen`r ilelikn`
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 784/846
Nr% Fëi~lr Pqa}anl Qrç`y
0?'=
na ω 7 0%0:: dkclbr`gly%
Ylcqikùe8
Pqaragly qe ~`g`ðl na gqay~r` e# n`nl pqa ΰ 7 :%:0 &;;! na ek}ac na ilemk`ex`+
na g`ear` pqa# xΰ,? 7 x:%:0,? 7 ?%121 &}`clr iley~`e~a na c` ~`oc` na c` nky~rkoqikùe
elrg`c ile ;;! ek}ac na ilemk`ex`+% @nagày# A 7 :%?1:# ω 7 0%0::% @yç# qy`enl c`
mùrgqc` &0?'1+ ely n`8
e 7 &xΰ,? ω , A+?
7 T&?%121+&0%0::+ , &:%?1:+W?
7 0?6%92≆ 0?;
Ae ileicqykùe# naoagly na lo~aear qe` gqay~r`# na iq`enl gaely 0?; i`y`y
nlgëy~ki`y yacaiikle`n`y `ca`~lrk`gae~a &pqa ay~àe nayi`r~`enl ac zcày~kil+% Ile
yagah`e~a gqay~r`# ay~`ragly ilemk`nly ae qe ;;! na pqa ac zrlgankl gqay~r`c
V ay~`rà nae~rl na :%?1: dkcly na ±%
Ahagzcl )1% _amkrkëenlya `c ahagzcl `e~arklr# yk pqkykër`gly ~aear rayqc~`nly
gaely zraikyly qy`enl qe g`rbae na arrlr na :%1:: dkcly# i`ciqc`r ac ~`g`ðl na c`
gqay~r` e `yqgkaenl c`y gkyg`y ilenkikleay `e~arklray%
Ylcqikùe8
Qy`enl c` mùrgqc` &0?'>+ lo~aeagly8
e 7 T&?%121+&0%0::+ , &:%1::+W?
7 9?%:;
≆ 99
Ya loyar}`e cly ykbqkae~ay zqe~ly ae c` rac`ikùe baear`c ae~ra ac ~`g`ðl na c`
gqay~r`# c` clebk~qn naya`n` nac ke~ar}`cl ?A# ac ek}ac na ilemk`ex` 0::&0 ’ ΰ+ zlr
ikae~l { ω8
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 785/846
Nr% Fëi~lr Pqa}anl Qrç`y
0?'2
0% Ilemlrga nkygkeq{a c` clebk~qn nac ke~ar}`cl ?A# ac ~`g`ðl rapqarknl na c`
gqay~r` e `qgae~` z`r` qe }`clr mkhl na ω { z`r` ac ek}ac na ilemk`ex`
ayzaikmki`nl%
?% @ gankn` pqa ω `qgae~`# ac ~`g`ðl rapqarknl na c` gqay~r` e `qgae~`# z`r` qe`
clebk~qn naya`n` ?A mkh` { qe ek}ac na ilemk`ex` ayzaikmki`nl%
9% Ilemlrga `qgae~` ac ek}ac na ilemk`ex`# ac ~`g`ðl rapqarknl na c` gqay~r` e
`qgae~` z`r` qe` clebk~qn mkh` naya`n` ?A { qe` nay}k`ikùe ay~àen`r ω%
Yacaiikùe nac ~`g`ðl na c` gqay~r` z`r` nly zloc`ikleay
^`gokëe ya zqana yacaiikle`r ac ~`g`ðl na c` gqay~r`# gày `zrlzk`nl# z`r` c`nkmaraeik` na nly zrlgankly% Zlr ahagzcl# yk ya ileliae c`y nay}k`ikleay ay~àen`r
na c`y gqay~r`y qel { nly# ay naikr# ω0 { ω?# { cly ~`g`ðly na c`y nly gqay~r`y yle
kbq`cay# ay naikr# e0 7 e? 7 e# ae~leiay# zqana na~argke`rya ac ~`g`ðl gày
`zrlzk`nl na c` gqay~r`% Ay~l ya f`ia na glnl pqa ya ~aeb` qe` ilemk`ex` na
0::&0 ’ ΰ+ zlr ikae~l ae pqa ac arrlr A ae c` ay~kg`ikùe na c` nkmaraeik` na ±0 ’ ±?#
zlr cly zrlgankly na c`y gqay~r`y V 0 ’ V ? ya` gaelr pqa A%
C` aiq`ikùe qy`n` z`r` i`ciqc`r ac ~`g`ðl na c` gqay~r` gày `zrlzk`nl z`r`
c` nkmaraeik` na nly zloc`ikleay ay8
e 7 &xΰ,? , A+?
&ω?0 $ ω?
?+ &0?'2+
El~` 0% _aiqërnaya pqa ay eaiay`rkl ranlena`r e# yk ay~a }`clr el ay qe ae~arl% Ile
ay~l# ya `yabqr` pqa ac ek}ac na ilemk`ex` el ya` gaelr pqa 0::&0 ’ ΰ+ zlr ikae~l%
Ahagzcl )=% Ya zrqao`e nly mùrgqc`y nkmarae~ay na b`ylcke` lvkbae`n` z`r`
ranqikr c`y agkykleay na gleùvknl na i`rolel &IL+ agk~kn`y zlr cly gl~lray na
ilgoqy~kùe ke~are`% Ya y`oa na `e~ag`el pqa c` }`rk`ex` z`r` c` zrkgar` mùrgqc`
ay na ω?0 7 0%1# gkae~r`y pqa c` }`rk`ex` z`r` c` yabqen` mùrgqc` ay na ω?
? 7 0%?%
¼Pqë ~`g`ðl na gqay~r` naoa qy`rya z`r` i`n` zloc`ikùe gqay~ra`n`# yk ya naya`
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 786/846
Nr% Fëi~lr Pqa}anl Qrç`y
0?'6
~aear qe` ilemk`ex` nac ;1! na pqa ac arrlr# `c ay~kg`r c` nkmaraeik` ae~ra cly
zrlgankly na c`y nly mùrgqc`y nkmarae~ay# ya` gaelr pqa 0<
Ylcqikùe8
@pqç# qy`gly c` mùrgqc` &0?'2+ z`r` i`ciqc`r ac ~`g`ðl na c` gqay~r` na nly
zloc`ikleay# ay naikr#
e 7 &xΰ,? , A+?
&ω?0 $ ω
??+
Nlena8
xΰ,? 7 x:%:1,? 7 x%:?1 7 0%;2# A 7 0# ω?0 7 0%1# ω?
? 7 0%?
Yqy~k~q{aenl ay~ly }`clray ae c` mùrgqc` na `rrko` n`8e 7 &0%;1 , 0+
?&0%1 $ 0%?+
7 0:%?2 ≆ 00
Zlr cl ~`e~l# ac ~`g`ðl na c` gqay~r` z`r` c`y zloc`ikleay ±0 { ±? ay8
e 7 e0 7 e? 7 00
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 787/846
Nr% Fëi~lr Pqa}anl Qrç`y
0?';
Aharikikly I`zç~qcl 0?
0?%0% Ya y`oa pqa c` nqr`ikùe# ae flr`y# na qe mlil na 21 `~~y ~kaea qe`
nky~rkoqikùe# `zrlvkg`n`gae~a elrg`c# ile qe` nay}k`ikùe ay~àen`r na ?1 flr`y%
Yqzùeb`ya pqa ya naya` qe` ilemk`ex` nac ;1! ae pqa ac arrlr ae c` ay~kg`ikùe na
c` nqr`ikùe zrlgankl ya` gaelr pqa 1 flr`y% ¼Pqë ~`g`ðl na gqay~r` naoa
qy`rya< &≆ ;2+
0?%?% F`iar ac gkygl zrlocag` pqa ac `e~arklr# zarl `flr` qy`enl qe` ilemk`ex`
na ;;! { qe arrlr A 7 0 { ilgz`r`r cly rayqc~`nly%
0?%9% Qe kebaekarl `q~lgl~rkx naya` na~argke`r ac ~kagzl zrlgankl pqa ~`rn`rç`
qe gaiàekil ae bkr`r c`y cc`e~`y na qe `q~l% Ay~a kebaekarl pqkara ay~kg`r# ile qe`
ilemk`ex` na ;1!# pqa ac zrlgankl na yq gqay~r` ay kgzraiky` ae iq`enl gày :%1:
gkeq~ly% Yk y`oa na ay~qnkly zkcl~ly `e~arklray pqa c` nay}k`ikùe ay~àen`r ay naω 7
0%= gkeq~ly# ¼pqë ~`e br`ena naoarà yar c` gqay~r` pqa naoa na yacaiikle`r#
`ca`~lrk`gae~a< Yqbaraeik`8 Qy`r c` mùrgqc` e 7 &xΰ,?
ω,A+?
&9;%9 ≆ >: gaiàekily+
0?%>% Ac nkrai~lr na ikar~` qek}arykn`n naya` qy`r ac zrlgankl na qe` gqay~r`
`ca`~lrk` z`r` ay~kg`r ac gle~l zrlgankl na ~kagzl pqa ya cay cca}` ` cly
ay~qnk`e~ay z`r` kr na qe y`cùe ` l~rl { ~lg`r yqy ic`yay yke ccab`r ~`rna% Z`r` ay~l
naya` `mkrg`r ile ;;! na ilemk`ex` pqa ac arrlr ay iq`enl gày na :%?1 gkeq~ly%
Avzarkaeik`y `e~arklray ay~kg`e qe` nay}k`ikùe ay~àen`r na ω 7 0%>: gkeq~ly%
Ykaenl `yç# ¼pqë ~`e br`ena naoarà yar c` gqay~r` pqa ya nao` ~lg`r<
0?%1% C` Ae}krlegae~`c Zrl~ai~kle @baei{ &AZ@+ na cly Ay~`nly Qeknly naya`ilenqikr qe` zrqao` na gkcc`ha na ikar~l glnacl na qe `q~l kgzlr~`nl% Ac
kebaekarl ay~`nçy~kil na c` AZ@ naya` ay~kg r ac zrlgankl ±# na gkcc`y zlr b`cùe
na ilgoqy~koca qy`nl zlr ay~a glnacl# ile ;1! na ek}ac na ilemk`ex`% @yqgkaenl
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 788/846
Nr% Fëi~lr Pqa}anl Qrç`y
0?'0:
pqa ω 7 ?%1 gkcc`y zlr b`cùe# ¼pqë ~`g`ðl na gqay~r` &eøgarl na `q~ly na ay~a
glnacl+ naoarà ~lg`r z`r` ilenqikr ay~` zrqao`< &e 7 ?1+
0?%=% Ileyknar`r ac ahagzcl 0?%0 na c` b`ylcke` lvkbae`n`# z`r` c` ay~kg`ikùe nac
~`g`ðl na c`y gqay~r`y z`r` c`y zloc`ikleay qel { nly% Ykaenl `yç# ay~kg`r cly
~`g`ðly na c`y gqay~r`y `zrlzk`nly# yk pqaragly qe` ilemk`ex` na ;;! { ac arrlr
na c` ay~kg`ikùe na c`y nkmaraeik`y ae~ra cly zrlgankly ya` gaelr pqa >%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 789/846
@ZAENKIA @% CKY^@ NA ^@OC@Y
^@OC@ 0% Zrlo`okckn`nay okelgk`cay `iqgqc`n`y
^@OC@ ?% Zrlo`okckn`nay na Zlkyyle `iqgqc`n`y
^@OC@ 9% Àra`y o`hl c` iqr}` elrg`c Z&x ≮ xl+
^@OC@ >% Zqe~ly zlriae~q`cay na ~Tδ4 ηW na c` nky~rkoqikùe na ~ na Ay~qnk`e~a
^@OC@ 1% Zqe~ly zlriae~q`cay na ς ?& δ4 η+ na c` nky~rkoqikùe na HK iq`nr`n`
^@OC@ =% Mqeikùe na b`gg` keilgzca~a
^@OC@ 2% ]`clray irç~kily z`r` c` nky~rkoqikùe na M Z&M ≮ Ml+
^@OC@ 6% Zqe~ly zlriae~q`cay na c` nky~rkoqikùe na r 0:
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 790/846
^@OC@ 0% Zrlo`okckn`nay okelgk`cay `iqgqc`n`y
Mqae~a8 N`ekac R% R% { H`gay ^arracc% Oqykeayy Y~`~ky~kiy% Flqbf~le GkmmckeIlgz`e{ &0;6;+%
Zlr ahagzcl# yk M&V+ 7 Z&V ≮ v+# { yk z 7 :%?:# v 7 ?# e 7 2# ae~leiay#M&?+ 7 Z&V ≮ ?+ 7 :%61?:
@zëenkia @'0
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 791/846
^@OC@ 0% Zrlo`okckn`nay okelgk`cay `iqgqc`n`y &Ile~keq`ikùe+
@zëenkia @'?
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 792/846
^@OC@ 0% Zrlo`okckn`nay okelgk`cay `iqgqc`n`y &Ile~keq`ikùe+
@zëenkia @'9
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 793/846
^@OC@ 0% Zrlo`okckn`nay okelgk`cay `iqgqc`n`y &Ile~keq`ikùe+
@zëenkia @'>
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 794/846
^@OC@ 0% Zrlo`okckn`nay okelgk`cay `iqgqc`n`y &Ile~keq`ikùe+
@zëenkia @'1
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 795/846
^@OC@ 0% Zrlo`okckn`nay okelgk`cay `iqgqc`n`y &Ile~keq`ikùe+
@zëenkia @'=
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 796/846
^`oc` 0% Zrlo`okckn`nay okelgk`cay `iqgqc`n`y &Ile~keq`ikùe+
@zëenkia @'2
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 797/846
^@OC@ 0% Zrlo`okckn`nay okelgk`cay `iqgqc`n`y &Ile~keq`ikùe+
@zëenkia @'6
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 798/846
^@OC@ 0% Zrlo`okckn`nay okelgk`cay `iqgqc`n`y &Ile~keq`ikùe+
@zëenkia @';
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 799/846
^@OC@ 0% Zrlo`okckn`nay okelgk`cay `iqgqc`n`y
@zëenkia @'0:
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 800/846
^@OC@ 0% Zrlo`okckn`nay okelgk`cay `iqgqc`n`y &Ile~keq`ikùe+
@zëenkia @'00
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 801/846
^@OC@ 0% Zrlo`okckn`nay okelgk`cay `iqgqc`n`y &Ile~keq`ikùe+
@zëenkia @'0?
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 802/846
^@OC@ 0% Zrlo`okckn`nay okelgk`cay `iqgqc`n`y &Ile~keq`ikùe+
@zëenkia @'09
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 803/846
^@OC@ 0% Zrlo`okckn`nay okelgk`cay `iqgqc`n`y &Ile~keq`ikùe+
@zëenkia @'0>
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 804/846
^@OC@ 0% Zrlo`okckn`nay okelgk`cay `iqgqc`n`y &Ile~keq`ikùe+
@zëenkia @'01
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 805/846
^@OC@ 0% Zrlo`okckn`nay okelgk`cay `iqgqc`n`y &Ile~keq`ikùe+
@zëenkia @'0=
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 806/846
^@OC@ 0% Zrlo`okckn`nay okelgk`cay `iqgqc`n`y &Ile~keq`ikùe+
@zëenkia @'02
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 807/846
^@OC@ 0% Zrlo`okckn`nay okelgk`cay `iqgqc`n`y &Ile~keq`ikùe+
@zëenkia @'06
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 808/846
^@OC@ 0% Zrlo`okckn`nay okelgk`cay `iqgqc`n`y &Ile~keq`ikùe+
@zëenkia @'0;
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 809/846
^@OC@ 0% Zrlo`okckn`nay okelgk`cay `iqgqc`n`y &Ile~keq`ikùe+
@zëenkia @'?:
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 810/846
^@OC@ 0% Zrlo`okckn`nay okelgk`cay `iqgqc`n`y &Ile~keq`ikùe+
@zëenkia @'?0
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 811/846
^@OC@ 0% Zrlo`okckn`nay okelgk`cay `iqgqc`n`y &Ile~keq`ikùe+
@zëenkia @'??
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 812/846
^@OC@ 0% Zrlo`okckn`nay okelgk`cay `iqgqc`n`y &Ile~keq`ikùe+
@zëenkia @'?9
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 813/846
^@OC@ 0% Zrlo`okckn`nay okelgk`cay `iqgqc`n`y &Ile~keq`ikùe+
@zëenkia @'?>
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 814/846
^@OC@ 0% Zrlo`okckn`nay okelgk`cay `iqgqc`n`y &Ile~keq`ikùe+
@zëenkia @'?1
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 815/846
^@OC@ ?% Zrlo`okckn`nay `iqgqc`n`y na Zlkyyle%i
M&i+ 7 Z&V ≮ i+ 7 Ω ±v a'±, v!
v7:
Mqae~a8 Glrrky F`goqrb% Y~`~ky~ki`c @e`c{yky mlr Naikykle G`dkeb % F`rilqr~ Or`iaH`}`el}kif# Kei% &0;;0+%
Ahagzcl8 Yk ± 7 0%:: { v 7 i 7 ?# ae~leiay# M&?+ 7 Z&V ≮ ?+ 7 :%;?::
@zëenkia @'?=
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 816/846
^@OC@ ?% Zrlo`okckn`nay `iqgqc`n`y na Zlkyyle &Ile~keq`ikùe+
@zëenkia @'?2
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 817/846
^@OC@ ?% Zrlo`okckn`nay `iqgqc`n`y na Zlkyyle &Ile~keq`ikùe+
@zëenkia @'?6
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 818/846
^@OC@ ?% Zrlo`okckn`nay `iqgqc`n`y na Zlkyyle &Ile~keq`ikùe+
@zëenkia @'?;
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 819/846
^@OC@ 9% Àra`y o`hl c` iqr}` elrg`c
Mqae~a8 N`ekac R% R% { H`gay ^arrac% Oqykeayy Y~`~ky~kiy% Flqbf~le GkmmckeIlgz`e{ &0;6;+%
@zëenkia @'9:
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 820/846
^@OC@ 9% Àra`y o`hl c` iqr}` elrg`c &Ile~keq`ikùe+
@zëenkia @'90
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 821/846
^@OC@ 9% Àra`y o`hl c` iqr}` elrg`c &Ile~keq`ikùe+
@zëenkia @'9?
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 822/846
^@OC@ 9% Àra`y o`hl c` iqr}` elrg`c% &Ile~keq`ikùe+
@zëenkia @'99
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 823/846
^@OC@ 9% Àra`y o`hl c` iqr}` elrg`c% &Ile~keq`ikùe+
@zëenkia @'9>
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 824/846
^@OC@ >% Zqe~ly zlriae~q`cay na ~&δ4 η+ na c` nky~rkoqikùe na ~ na Ay~qnk`e~a%
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ Mqae~a8 Nqee# L% H% { ]krbkek` @% Ic`rd% @zzckan Y~`~ky~kiy8 @e`c{yky lm ]`rk`eia `en
_abrayykle% Hlfe Rkca{ `en Yley# Kei%# Ea [lrd &0;2>+%
Ahagzcl na ke~arzlc`ikùe qy`enl η 7 9? br`nly na ckoar~`n ile δ 7 :%;1
91 ’ 0%=:; &9? ’ 9:+,&91 ’ 9:+ 7 V,&0%=;2 ' 0%=;:+# v 7 %::?69? ’ v Aeyabqkn` `brab`r %::?6 ` 0%=;: z`r` n`r 0%=;?9%9: ’ 0%=;2 Zlr cl ~`e~l# ac }`clr na ~T%;149?W 7 0%=;9
@zëenkia @'91
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 825/846
^@OC@ >% Zqe~ly zlriae~q`cay na ~&δ4 η+ na c` nky~rkoqikùe na ~ naAy~qnk`e~a &Ile~keq`ikùe+%
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
@zëenkia @'9=
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 826/846
^@OC@ 1% Nky~rkoqikùe na HK iq`nr`n` &ς ?+%
Mqae~a8 G`rkl M% ^rklc`% Acagae~`r{ Y~`~ky~kiy% @nnkyle'Rayca{ Zqockyfkeb Ilgz`e{&0;;1+%
@zëenkia @'92
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 827/846
^`oc` 2% ]`clray irç~kily z`r` c` nky~rkoqikùe M&Z+M ≮ Ml+
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ Mqae~a8 H% C% Na}lra% Zrlo`okckn`n { Ay~`nçy~ki` z`r` Kebaekarç` { Ikaeik`y%^flgyle Ca`rekeb &?::0+%
@zëenkia @'96
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 828/846
^@OC@ 2% ]`clray irç~kily z`r` c` nky~rkoqikùe M &Z&M ≮ Ml+&Ile~keq`ikùe+%
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
@zëenkia @'9;
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 829/846
^@OC@ 2% ]`clray irç~kily z`r` c` nky~rkoqikùe M &Z&M ≮ Ml+ &Ile~keq`ikùe+%
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
@zëenkia @'>:
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 830/846
^@OC@ 2% ]`clray irç~kily z`r` c` nky~r~koqikùe M &Z&M ≮ Ml+ Ile~keq`ikùe+%
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
@zëenkia @'>0
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 831/846
^@OC@ 2% ]`clray irç~kily z`r` c` nky~rkoqikùe M &Z&M ≮ Ml+ &Ile~keq`ikùe+%
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
@zëenkia @'>?
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 832/846
v
^@OC@ =% Mqeikùe na b`gg` keilgzca~`% M&v4ΰ+ 7 ∤ : 0 , Ι&ΰ+ {
ΰ'0
a
'{
n{
Mqae~a8 H`{ C% Na}lra% Zrlo`okckn`n { ay~`nçy~ki` z`r` kebaekarç` { ikaeik`y% ^flgyle'Ca`rekeb &?::0+%
@zëenkia @'>9
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 833/846
^@OC@ 6% Zqe~ly zlriae~q`cay na c` nky~rkoqikùe na r 0:%
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ Mqae~a8 Nqee# L% H% { ]krbkek` @% Ic`rd% @zzckan Y~`~ky~kiy8 @e`c{yky lm
]`rk`eia `en _abrayykle% Hlfe Rkca{ `en Yley# Kei% Ea [lrd &0;2>+
@zëenkia @'>>
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 834/846
@zëenkia O
Okocklbr`mç`
@enaryle# N% _# Naeeky H% Yaaea{# ^flg`y @% Rkcck`gy% Ay~`nçy~ki` z`r` @ngkeky~r`ikùe { Ailelgç`% ]lc% 0% Yëz~kg` ankikùe% Ylq~f'Ray~areZqockyfkeb &0;;;+%
Oar~flqv# Z% G%# Ckemkacn I% Orle% Y~`~ky~kiy mlr Ae}krlegae~`c Aebkeaary%Caky Zqockyfary &0;;>+%
Orle C% ^falnlra# F% Aqbaea Ca G`{# Hr%# Orqia A% Oqry~ae% Ifagky~r{%
^fa Iae~r`c Yikaeia% Zrae~kia F`cc# Qzzar Y`nnca _k}ar# Ea Harya{% Akbf~Ank~kle# &?:::+%
N`ekac# R% R%# H`gay I% ^arracc% Oqykeayy Y~`~ky~kiy% Mkry~ Ank~kle% Flqbf~leGkmmcke Ilgz`e{ &0;6;+%
Na}lra# H% C% Zrlo`okckn`n { Ay~`nçy~ki` Z`r` Kebaekarç` { Ikaeik`y% Pqke~`ankikùe% ^flgyle Ca`rekeb% &?::0+
Nqee# L% H%# ]krbkek` @% Ic`rd% @zzckan Y~`~ky~kiy8 @e`c{yky lm ]`rk`eia `en _abrayykle% Hlfe Rkca{ `en yley% Ea [lrd Clenle# Y{nea{# ^lrle~l
&0;2>+%
Mraqen# H%A% Y~`~ky~kiy% @ Mkry~ Ilqrya% Yailen Ank~kle% Zrae~kia F`cc# Kei%Aebcalln Ickmmy# Ea Harya{ &0;2=+%
Blcnoar# Yy% Zrlo`okck~{% @e Ke~rlnqi~kle% Zqockyfan o{ Zrae~keia F`cc# Kei%Aebcalln Ickmmy# E% H% &0;=:+%
F`goqrb# G% Y~`~ky~ki`c @e`c{yky mlr Naikykle G`dkeb % Mkm~f Ank~kle%F`rilqr~ Or`ia Hl}`el}kif# Zqockyfary% @i`nagki Zrayy% Y`e Nkabl# Ea[lrd# Ifki`bl# @qy~ke# R`yfkeb~le# N% I% &0;6;+%
Faroar @%# _`{glen _% Ilc~le% Y~`~ky~ki`c Ga~flny% Mlqr~f Ank~kle% O`reay`en Eloca# Kei% Ea [lrd &0;==+%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 835/846
Harlga# I% _% Ck% Y~`~ky~ki`c Kemaraeia% Nky~rkoq~an o{ An`rny Orl~fary# Kei%@ee @rolr# Gkifkb`e% &0;=>+%
Daccar# B%# Ork`e R`rrlid# Faer{ O`r~ac% Y~`~ky~kiy mlr G`e`bagae~ `en
Ailelgkiy8 ` Y{y~ag`~ki @zzrl`if% Yailen Ank~kle% R`rnylr~f ZqockyfkebIlgz`e{# Oacgle~# I`ckmlrek` &0;;:+%
Dq~ear# G% F%# Ifky~lzfar H% E`if~yfakg# Hlfe Ea~ar# Rkccck`g Ck% @zzckan Ckea r Y~`~ky~ki`c Glnacy% Mkm~f ank~kle% GiBr`'Fkcc Ke~are`~kle`c Ank~kle&?::1+%
C`zke# C% C% Y~`~ky~kiy mlr Glnare Oqykeayy Naikykley% F`rilqr~ Or`iaH`}`el}kif# Kei% &0;60+%
G`ec{# O% M% H% Y~`~ky~kiy mlr Ae}krlegae~`c Yikaeia `en G`e`bagae~ %If`zg`e - F`cc,I_I &?::0+%
Gle~blgar{# N%# Balrba I% _qebar% Zrlo`okckn`n { Ay~`nçy~ki` @zcki`n`y `
c` Kebaekarç`% Gibr` Fkcc Ke~ar`garki`e` Ank~lray# Y%@% Na I% ]% &0;;=+%
Gle~blgar{# N% I% Ackx`oa~f @% Zaid# B% Balmmra{ ]kekeb% Ke~rlnqiikùe `c
@eàckyky Ckea c % Brqzl Z`~rk` Iqc~qr`c# Y% @% Na I% ]% &?::?+%
G{ary# R%# _`{glen F% G{ary% Zrlo`okckn`n { Ay~`nçy~ki`% Iq`r~` Ankikùe%Gibr` Fkcc,Ke~ar`garki`e` na Gavkil# Y% Y% Na I% ]% &0;;?+%
Ea~ar# H%# Gkif`ac F% Dq~ear# Ifrky~lzfar H% E`if~yfakg# Rkcck`gR`yyarg`e% @zzckan Ckea`r _abrayykle Glnacy% ^fkrn Ank~kle% Krke &0;;=+%
Y`enary# N% F% Y~`~ky~kiy% @ Mkry~ Ilqrya% Mkm~f Ank~kle%
Y`{ar# E%I%# Zarr{ C% Gii`r~{% Ifagky~r{ mlr Y`ek~`r{ Aebkeaary% YailenAnk~kle% Gibr`'Fkcc &0;=2+%
Ygk~f# B% Y~`~ky~ki`c _a`ylekeb % @cc{e @en O`ile# Kei% Oly~le ClenleY{nea{ ^lrle~l &0;61+%
Yzkabac# G% _% Yif`qg"y Lq~ckea lm ^falr{ `en Zrlocagy lm Y~`~ky~kiy%Yif`qg Zqockyfkeb Ilgz`e{# Ea [lrd &0;=0+%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 836/846
Y~`en`rn Ga~flny mlr ~fa Av`gke`~kle lm R`~ar `en R`y~a`~ar % Zraz`ran@en Zqockyfan Hlke~c{ o{8 @garki`e Zqocki Fa`c~f @yylik`~kle# @garki`eR`~ar Rlrdy @yylik`~kle `en R`~ar Zlccq~kle Ile~rlc Manar`~kle% @garki`eZqocki Fa`c~f @yylik`~kle# 0:01 Akbf~aae~f Y~raa~# E%R%# R`yfkeb~le# N%I%?::9= &0;20+%
GiIc`}a# H% ^%# Balrba Oaeyle% Y~`~ky~kiy mlr Oqykeayy `en Ailelgkiy%Yailen Ank~kle% Naccae Zqockyfkeb Ilgz`e{# Y`e Mr`eikyil `en Y`e~` Ic`r`#I`ckmlrek` &0;6?+%
^rklc`# G% M% Acagae~`r{ Y~`~ky~kiy% Ykv~f Ank~kle% Ilz{rkbf~ 0;;1% @nnkyle'Rayca{ Zqockyfkeb Ilgz`e{# Kei%
R`czlca# A% _%# _`{glen F% G{ary% Zrlo`okck~{ `en Y~`~ky~kiy mlr Aebkeaary
`en Yikae~ky~y% Mkm~f Ank~kle% Zrae~kia F`cc# Kei% &0;;9+%
[`g`ea# ^% Y~`~ky~kiy# `e Ke~rlnqi~lr{ @e`c{yky% F`rzar - _l# Zqockyfary#Keilrzlr`~an# >; A`y~ 99rn Y~raa~# Ea [lrd 0=# E%[% &0;=>+%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 837/846
@zëenkia I
Z`zac na bràmki`
Z`zac na bràmki` yagkclb`rç~gkil na 1 ikicly
Z`zac na bràmki` clb`rç~gkil
Z`zac na bràmki` na zrlo`okckn`n
Z`zac na bràmki` na zrlo`okckn`n okelgk`c z`r` `e`ckx`r n`~ly aeqgar`nly
Z`zac na bràmki` na mraiqaeik` rac`~k}` `iqgqc`n` ae mqeikùe na c` }`rk`oca`ca`~lrk` V
@zëenkia I
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 838/846
Z`zac na ayi`c` yagkclb`rç~gki`
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 839/846
Z`zac na bràmki` na ayi`c` clb`rç~gki` ilgzca~`%
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 840/846
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 841/846
Z`zac na bràmki` clb`rç~gkil na ?v? ikicly
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 842/846
Z`zac na bràmki` na ayi`c` `rk~gë~ki`
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 843/846
Z`zac na br`mki` na mraiqaeik` rac`~k}` `iqgqc`n` ae mqeikùe na c`}`rk`oca `ca`~lrk` V
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 844/846
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 845/846
@zëenkia N
Çenkia
@hqy~`gkae~l na iqr}`y# ;'?;# ;'9:
@eàckyky na }`rk`ex` ae nly yae~knly# 2'?1@eàckyky na }`rk`ex` ae ~ray yae~knly# 2'9= ’ 2'9;@eàckyky na }`rk`ex`# 2'0
`eàckyky na }`rk`ex` na oclpqay ilgzca~`gae~a `ca`~lrkx`nly# 2'02nkyaðly na @EL]@ ilgzca~`gae~a ca`~lrkx`nly# 2'9
@EL]@ ile ~ray m`i~lray qy`enl ac Gkek~`o# 2'1:# 2'19@zcki`ikleay na c` nky~rkoqikùe na Zlkyyle# >'0% ]ar nky~rkoqikùe na Zlkyyle
nae~rl na yqy zrlzkly ~ërgkely { ilgl qe` `zrlvkg`ikùe ` c` nky~rkoqikùe okelgk`c# >'=# >'2@zcki`ikleay na c` nky~rkoqikùe na ~ na Ay~qnk`e~a# ='>@zcki`ikleay na c` nky~rkoqikùe fkzarbalgë~rki` qy`enl ac zrlbr`g` Gkek~`o# 9'>:Àra`y o`hl c` iqr}` elrg`c# 1'2@q~lilrrac`ikùe# 6'9:# 6'16# ;';# ;'>2# ;'>6# ;'>;# ;'1>
@vklg`y { zrlzkan`nay oàyki`y na c` zrlo`okckn`n# ?'=
Ilamkikae~a na ilrrac`ikùe _# 6';# 6'02Ilamkikae~a na na~argke`ikùe _ ?# 6'6# 6'02Ilgoke`ikleay lr~lble`cay# 2'>9Ilgoke`ikleay# ?'9?Ilgzcagae~l# ?'00Ilgzleae~ay na c` zrqao` na fkzù~ayky# 1'>0Iq`r~kcay# 0'9:# 0'90Iqr}`y na mraiqaeik`# ~kzly na# 0'0;
Nay}k`ikùe ay~àen`r# 0'0:Nay}k`ikleay nac zrlgankl# 0'09Nk`br`g`y na àrolc# ?'?>Nk`br`g`y na ~`ccl { flh`# 0'?2Nk`br`g`y na ]aee# ?'06Nkmaraeik`y ae~ra c` nky~rkoqikùe na Zlkyyle { c` nky~rkoqikùe okelgk c# >'?Nky~rkoqikùe okelgk`c# 9'0Nky~rkoqikùe na b`gg`# 1'?6Nky~rkoqikùe na HK iq`nr`n`# ='?>Nky~rkoqikùe na Zlkyyle# >'0Nky~rkoqikùe na ~ na Ay~qnk`e~a# ='0
Nky~rkoqikùe na Rakoqcc# 1'90Nky~rkoqikùe avzleaeik`c# 1'?>Nky~rkoqikùe fkzarbalgë~rki`# 9'0# 9'90Nky~rkoqikùe elrg`c ay~àen`r { nky~rkoqikùe elrg`c el ay~àen`r# 1'0:Nky~rkoqikùe elrg`c# 1'=Nky~rkoqikleay na mraiqaeik`# 0'02# 0'?0Nky~rkoqikleay na zrlo`okckn`n ile~keq`# 1'0Nqroke'R`~yle# zrqao` na `q~l`ilrrac`ikùe# ;'>6' ;'19
7/23/2019 Estadistica Aplicada a La Ingenieria Ambiental
http://slidepdf.com/reader/full/estadistica-aplicada-a-la-ingenieria-ambiental 846/846
Ay~`nçy~ki` kemaraeik`c# 1'9>Arrlr ay~àen`r# 0'0># 1'9=Ay~`nçy~ki`# namkekikùe na# 0'0Ay~`nçy~ki` el z`r`gë~rki`# 0:'0Aey`{l na Oarelqcck# 9'?
Aiq`ikùe na c` cçea` na rabraykùe# 6'?A}ae~ly gq~qly avicq{ae~ay# ?'09A}ae~ly nazaenkae~ay a kenazaenkae~ay# ?'0=Ay~liày~kil# namkekikùe na# ?'06Ay~`nçy~ki` nayirkz~k}`# 0'0# 0'9Ayz`ikl gqay~r`c# ?'6A}`cq`ikleay na cly glnacly na rabraykùe# 6'92# ;'6
Mqeikùe na naeykn`n na zrlo`okckn`n na c` }`rk`oca `ca`~lrk` ile~keq` v# 1'?Mùrgqc` agzçrki` z`r` f`iar ke~arzlc`ikleay z`r` i`ciqc`r ac }`clr na c` zrlo`okckn`n z# 1'19Mù c m n ~ c n c àc c 1 9