8/17/2019 2013_Ch.13_Notes-2

1/9

QMDS 202 Data Analysis and Modeling

Chapter 13 Inference About Comparing Two Populations

Dependent Versus Independent Samples

When making comparisons between the means of two populations, we need to pay particular attention to how we intend to collect sample data.

1. If there is a definite reason for pairing (matching corresponding data !alues, thetwo samples are dependent  samples.

". If the two samples were obtained independently and there is no reason for pairing thedata !alues, the resulting samples are independent  samples.

Inference #bout $he Difference %etween $wo &eans' Independent Samples

When the populations are normally distributed or the sies of two independent samples

are large (both n1 and n" are greater than or e)ual to *+, the sample statistic "1  X  X    − is a

normal random !ariable with mean ( )"1   µ  µ   −  and !ariance    

  

 +

"

"

"

1

"

1

nn

σ  σ  

. $he confidence

inter!al of ( µ 1 µ " is found by'

( ) ( ) ( )"

""

1

"1

"-"1"1"

""

1

"1

"-"1nn

 z  x xnn

 z  x x  σ σ  µ  µ 

σ σ 

α α   ++−≤−≤+−−

If σ1 and σ" are unknown but equal, the confidence inter!al of ( µ 1 µ " is found by'

( ) ( ) ( )    

  

 ++−≤−≤  

 

  

 +−−

"1

""-"1"1

"1

""-"1

1111

nn st  x x

nn st  x x  p p   α α    µ  µ 

where "-α t   is a score obtained from the t distribution with v  n1 / n" 0 " and

"

1(1(

"1

"""1

"1"

−+−+−

=nn

n sn s s p pooled sample variance.

If σ1 and σ" are unknown and unequal, the confidence inter!al of ( µ 1 µ " is found by'

( ) ( ) ( )"

""

1

"1

"-"1"1"

""

1

"1

"-"1n

 s

n

 st  x x

n

 s

n

 st  x x   ++−≤−≤+−−   α α    µ  µ 

where "-α t   is a score obtained from the t distribution with v 

( )( ) ( )

  

 

 

 

 

−+

−

+

1

-

1

-

--

"

"

"""

1

"

1"1

"

"""1

"1

n

n s

n

n s

n sn s

1

8/17/2019 2013_Ch.13_Notes-2

2/9

ound the result of the calculation of v to the nearest integer.

23ample 1 # corporation owns two outlets # and %. # random sample of *4 days atoutlet # had a mean of 15+ sales daily. # random sample of *4 days at

outlet % had a mean sales of 146. #ssuming σ#"  *4 and σ%

"  "6, can we

conclude that there are more sales in outlet # at +.+6 le!el of significance7

Solution' 8+' µ1 ≤ µ"  ⇔  µ1 −µ" ≤ + ( D+  the claimed !alue of µ1 −µ" stated in 8+  +81' µ1 > µ"  ⇔  µ1 − µ" > +

α  +.+6

%oth sample sies are large ⇒   "1  X  X    − is a normal random !ariable.Independent samples, σ1  and σ" are known ⇒ distribution will be used as the testing

distribution.

e9ect 8+ if $S : 1.4;6.

( )

"

""

1

"1

+"1

nn

 D x xTS 

σ σ +

−−=   ( )

8/17/2019 2013_Ch.13_Notes-2

3/9

Independent samples, σ1 and σ" are unknown and une)ual⇒ tdistribution will be used as

the testing distribution.

1*?".1"

11"

1"*"

11+

1+4"

1"

*"

1+

4"

11

""

""

""

"

"

"

""

1

"

1

"1

"

"

""

1

"1

==

−

    

  

+−

    

  

+

=

−

    

  

+−

    

  

+

=

n

n s

n

n s

n

 s

n

 s

v

 

e9ect 8+ if $S A ".14 or $S : ".14.

( )

"

""

1

"1

+"1

n

 s

n

 s

 D x xTS 

+

−−=   ( ) 4?".+

1"

*"(

1+

4"(

+6

8/17/2019 2013_Ch.13_Notes-2

4/9

 Cote. We will use the e)ual!ariances test statistic and confidence inter!al estimator unless there is e!idence (based on the sample !ariances to indicate that the population!ariances are une)ual, in which case we will apply the une)ual!ariances test statistic andconfidence inter!al estimator.

Inference #bout $he Difference %etween $wo &eans' &atched @airs 23periment

When the populations are normally distributed and dependent samples (matched data are

obtained, the confidence inter!al of ( µ 1 µ " is found by'

( ) D

 D D

 D

 D D

n

 st  x

n

 st  x "-"1"-   α α    µ  µ    +≤−≤−

23ample ; #n industrial engineer is e!aluating a new techni)ue to assemble air compressors. # sample of < employees is selected at random, and thenumber of compressors they each produce in one week using the e3isting procedure is recorded. $he same < workers are then trained to use the newtechni)ue and their output for one week is then noted. =onduct a test to

determine whether there is a difference between the two techni)ues with α

+.+6.

 Employee A B C D E F G H 

Old Method 

8/17/2019 2013_Ch.13_Notes-2

5/9

2 5;

8/17/2019 2013_Ch.13_Notes-2

6/9

to be 1".

 v1  n1 0 1   v" n" 0 1

 e9ect 8+ if $S : "1 ,, vv F 

α  

""

"1   - s sTS =

Suppose' 8+'""

"1   σ σ    ≥

81'""

"1   σ σ    <

v1  n1 0 1   v"  n" 0 1

e9ect 8+ if $S A "1,,1   vv F  α  −

"""1   - s sTS =

If independent samples are obtained from two normal populations, the confidence

inter!al of """1   -σ σ   can be found by'

4

8/17/2019 2013_Ch.13_Notes-2

7/9

 1""1   ,,"-

""

"1

""

"1

,,"-

""

"1

-1

--

vvvv   F 

 s s

 F 

 s s

α α    σ 

σ ≤≤

In 23ample 6, the ?6B confidence inter!al of """1   -σ σ   is'

 ?4."-1

8/17/2019 2013_Ch.13_Notes-2

8/9

+6.+"++

6.+6.+

"++

;.+4.+I1(II1(II

"

""

1

11II "1

=×+×=−

+−

=−n

 p p

n

 p p p pσ  

$he ?6B confidence inter!al of ( p1 0 p" is'

( )   +6.+?4.11.++6.+?4.11.+ "1   ×+≤−≤×−   p p1?

8/17/2019 2013_Ch.13_Notes-2

9/9

( )

"

""

1

11

"1

I1(II1(I

II

n

 p p

n

 p p

 D p pTS 

−+

−

−−=

e!iew @roblems' 1*.4, 1*.5, 1*.< a to c, 1*.1", 1*.6", 1*.64, 1*.54, 1*.