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QMDS 202 Data Analysis and Modeling

Chapter 14 Analysis Of Variance

Definition: ANOVA is the name given to the approach that allows us to use sample

data to see if the values of two or more unknown population means are

likely to be equal.

Assumptions: . !he populations under study are normally distributed.". !he samples are drawn randomly# and each sample is independent of 

the other samples.

$. !he populations from which the sample values are obtained all have

the same unknown population variance %σ"&. !hat is# === "$"

"

"

  σ  σ  σ   '"

k σ  =  where k ( no. of populations under study.

)ypothesis !esting *rocedure

. )+: µ ( µ" ( ' ( µk 

): Not all population means are equal.

". ,et the level of significance %α&.

$.  F -distribution as the testing distribution.

. /ind the critical value with df  ( k  0 and df " ( n 0 k  and state the re1ection rule.

n ( the total number of items in all samples ( n 2 n" 2 ' 2 nk n j ( sample si3e of the j

th sample

4. 5ompute the test statistic.

6. 7ake the statistical decision.

8ey !erms in ANOVA

SS(Total) ( !otal ,um of ,quares ( =−∑∑= =

"

&%   X  X k 

 j

n

i

ij

 j

n

 X 

 X 

k 

 j

n

i

ijk 

 j

n

i

ij

 j

 j

"

"   

  

 

−∑∑

∑∑   = == =

==

∑∑= =

n

 X 

 X 

k 

 j

n

i

ij

 j

 9rand mean

SST ( ,um of ,quares etween 9roups ( ,um of ,quares of !reatment groups

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(

n

 X 

n

 X 

 X  X n

ij

n

i

k 

 j

 j

ij

n

ik 

 j

 j j

k 

 j

 j j"

"

"

&%

   

  

 

−   

  

 

=−ΣΣΣ

ΣΣ  ===

==

SSE  ( ,um of ,quares ;ithin 9roups ( ,um of ,quares of

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rand 5 $." $.4 $.4 !$ ( +."

rand D $.4 $.@ $.@ ! ( .

! ( $."

SST  (( )

"

".$

$

.

$

".+

$

".+

$

A."""""

−

+++

  ( 46.+6 0 44.4" ( +.4

A.46&@.$%&@.$%...&.%&6.$% """"" =++++=ΣΣ   ij x

SS(Total) ( 46. 0 44.4" ( .@

SSE  ( .@ 0 +.4 ( +.6

!he One-way ANOVA ,ummary !able of this problem:

Source df SS MS F  

Treatments 3 0.54 0.! "."5

 Error ! 0.#4 0.0!

Total .!

)+:  µ  (  µ " (  µ $ (  µ ): Not all population means are equal

α ( +.+4

Bndependent *opulations. Assume that all the populations are normally

distributed with an equal variance σ" ⇒  F$distribution will be used in the

ANOVA test.

df  ( k  0 ( 0 ( $

df " ( nT  0 k  ( " 0 ( @

Ce1ect )+ if TS   .+

"4."+@.+

@.+==TS 

TS  ( "."4 is not great than .+ ⇒ Do not re1ect )+

5onclusion: !he mean cholesterol contents of the four diet foods are notsignificantly different.

OR 

 x " &%   x x   − " x """ &%   x x   −   $ x "$$ &%   x x   − . x " &%   x x   −$.6 %-+.$&" $. %-+.$&" $." %-+."&" $.4 %-+."&"

. %+."&" $." %-+."&" $.4 %+.&" $.@ %+.&"

.+ %+.&" $.? %+.4&" $.4 %+.&" $.@ %+.&"

. +. +." +.$@ +." +.+6 . +.+6

$

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?.$$

A.

  ==

Σ=

n

 x x .$

$

".+

"

""   ==

Σ=

n

 x x

.$$

".+

$

$$   ==

Σ=

n

 x x A.$

$

.

  ==

Σ=

n

 x x

 n ( $ 2 $ 2 $ 2 $ ( " k  (

6.$"

.".+".+A.=

+++= x

&%&%&%&%"

.."

$$"

"""

−−+−+−+−

=k 

 x xn x xn x xn x xn MST 

@.+$

4.+

&6.$A.$%$&6.$.$%$&6.$.$%$&6.$?.$%$ """"==

−

−+−+−+−=

k n

 x x x x x x x x MSE 

−

−Σ+−Σ+−Σ+−Σ=

"..

"$$

"""

" &%&%&%&%

  +@.+@

6.+

"

+6.++6.+$@.+.+==

−

+++=

!he One-;ay ANOVA !able

,ource of

Variation

Degrees of

/reedom %df&

,um of ,quares

%,,&

7ean ,quare

%7,&

/-,tatistic

%!,&

!reatments k  0 ,,! 7,! / ( 7,! = 7,<

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,,%!otal& ((k 

T  xij

"" −ΣΣ

=iS   total of the observations in the ith block 

,, ( ,um of ,quares for locks ( [ ] (k T 

S S S k    (

"""

"

"

...

−+++7, ( 7ean ,quare for locks ( ,, = % 0 &

,,< ( ,,%!otal& 0 ,,! 0 ,,

ample " Bn the previous problem# suppose now we learn something that we did not

know earlier - the measurements of the cholesterol contents were

 performed in different laboratories. !he first value of each sample# welearn# came from one laboratory# the second value came from another 

laboratory# and the third value came from a third laboratory. ;e might

 picture the original data as follows:

Eab. Eab. " Eab. $

rand A $.6 . .+

rand $. $." $.?rand 5 $." $.4 $.4

rand D $.4 $.@ $.@

!est whether there is a difference in the mean cholesterol contents among

the four diet foods as well as whether there is a difference in the mean

results given by the three laboratories. Fse α ( +.+4.

,olution: Eab. Eab. " Eab. $

rand A $.6 . .+ .

rand $. $." $.? +."

rand 5 $." $.4 $.4 +."rand D $.4 $.@ $.@ .

$. .6 4." $."

,,%!otal& ( .@ ,,! ( +.4

,, ( [ ] ".+4".44?.444".44".46..$

""" =−=−++

,,< ( .@ 0 +.4 0 +." ( +.""

Source df SS MS F  

Treatments 3 0.54 0.! 4.*0

 'locks " 0.4" 0." 5.+"

 Error # 0."" 0.03#+ 

Total .!

/or the treatments:

)+: µ ( µ" ( µ$ ( µ

4

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  ): Not all population means are equal

 α ( +.+4

, ( $ ," ( 6 5.V. ( .6

 !, ( .?+ .6 ⇒ Ce1ect )+

∴  !he mean cholesterol contents of the four diet foods are significantly

different.

/or the blocks:

 )+: µ ( µ" ( µ$ 

): Not all population means are equal

 α ( +.+4

, ( " ," ( 6 5.V. ( 4.

 !, ( 4." 4. ⇒ Ce1ect )+

∴ !he mean results given by the three laboratories are significantly different.

ample $ !he sample data in the following table are the marks in a statistics test

obtained by nine students from $ ma1ors who were taught by $ different

instructors:

Bnstructor 

A

Bnstructor 

Bnstructor 

5

7arketing ma1or @@

/inance ma1or @@ ? @

Accounting ma1or @4 ?4 "

At 4G significance level# test whether the mean scores of the ma1ors arethe same by using the instructors as blocks.

,olution: Bnstructor A

Bnstructor 

Bnstructor 5

7arketing ma1or @@

/inance ma1or @@ ? @

Accounting ma1or @4 ?4 "

k  (  (

n (

,,! ( 

,, (

"

ij xΣΣ (

6

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,,%total& (

,,< (

Source df SS MS F  Treatments

 'locks

 Error 

Total 

/or the treatments:)+:

):

α ( +.+4, ( ," ( 5.V. (

Decision:

 5onclusion:

/or the blocks:

 )+:

):

 α ( +.+4

, ( ," ( 5.V. (

Decision:

 5onclusion:

Assumptions in this problem:

Ceview *roblems: .# .$# .4# .46# .4@.