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Chapter Nine
Statistical Inferences Based
on Two Samples
McGraw-Hill/Irwin
Copyright © 2003 by The McGraw-Hill Companies, Inc. All rights reserved.
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21
Statistical Inferences Based on Two
Samples
9.1 Comparing Two Population Means Using Large
Independent Samples
9.2 Comparing Two Population Means Using Small
Independent Samples
9.3 Comparing Two Population Variances Using
Independent Samples
9.4 Paired Difference Experiments
9.5 Comparing Two Population Proportions Using
Large Independent Samples
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9.1 Sampling Distribution of
x1  x 2
If independent random samples are taken from two
populationsthen the sampling distribution of the sample
difference in means x1  x 2 is
Normal, if each of the sampled populations is normal and
approximately normal if the sample sizes n1 and n2 are
large
Has mean:
 x -x = 1   2
1
2
Has standard deviation:
 x -x =
1
2
 12
n1

 22
n2
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Sampling Distribution of
(Continued)
x1  x 2
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Large Sample Confidence Interval,
Difference in Mean
If two independent samples are from populations that are
normal or each of the sample sizes is large, 100(1 - a)%
confidence interval for 1 - 2 is
(x1  x 2 )  za/2
 12
n1

 22
n2
If 1 and 2 are unknown and the each of the sample sizes is
large (n1, n2  30), estimate the sample standard deviations by
s1 and s2 and a 100(1 - a)% confidence interval for 1 - 2 is
(x1  x 2 )  za/2
s12 s22

n1 n2
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Large Sample Tests about Differences in
Means
If sampled populations are normal or both samples are large, we can
reject H0: 1 - 2 = D0 at the a level of significance if and only if the
appropriate rejection point condition holds or, equivalently, if the
corresponding p-value is less than a.
Alternative
H a : 1   2  D0
Reject H0 if:
z  za
p-Value
H a : 1   2  D0
z   za
Area under std normal curve left of z
H a : 1   2  D0
z  za / 2 , that is
Twice area under std normal curve right of z
Area under std normal curve right of z
z  za / 2 or z   za / 2
Test Statistic
z
(x1  x 2 )  D 0

2
1
n1


2
2
n2
If population variance unknown and
the sample sizes are large, substitute
sample variances.
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Example: Large Sample Interval and
Test
Bank Waiting Times, Current System versus New System
n1  100, x1  8.79, s12  4.7089,
n 2  100, x2  5.14, s22  1.9044
95% Confidence Interval for 1 - 2
(x1  x 2 )  za/2
s12 s22
4.7089 1.9044

 (8.79  5.14)  1.96

n1 n2
100
100
 3.65  0.5040  [3.15, 4.15]
Test H0: 1 - 2  3 versus Ha: 1 - 2 > 3, a = 0.05
z
(x1  x 2 )  D 0
s12 s22

n1 n2
(8.79  5.14)  3

 2.53  1.645  z.05
4.7089 1.9044

100
100
p  value  P(z  2.53)  (0.5 - 0.4943)  0.0057
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9.2 Small Sample Confidence Interval,
Difference in Mean When Variances are
Equal
If two independent samples are from populations that are
normal with equal variances, 100(1 - a)% confidence interval
for 1 - 2 is
(x1  x 2 )  ta/2
1 1
s   
 n1 n2 
2
p
Where sp2 is the pooled variance
2
2
(
n

1
)
s

(
n

1
)
s
1
2
2
s 2p  1
(n1  n2  2)
And ta/2 is based on (n1 – n2 – 2) degrees of freedom.
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Small Sample Tests about Differences in
Means When Variances are Equal
If sampled populations are both normal with equal variances, we can
reject H0: 1 - 2 = D0 at the a level of significance if and only if the
appropriate rejection point condition holds or, equivalently, if the pvalue is less than a.
Alternative
H a : 1   2  D0
H a : 1   2  D0
H a : 1   2  D0
Test Statistic
(x1  x 2 )  D 0
t
1
2 1
s p   
 n1 n2 
Reject H0 if:
p-Value
Area under t distributi on right of t
t  ta
Area under t distributi on left of t
t  ta
Twice area under t distributi on right of t
t  ta / 2 , that is
t  ta / 2 or t  ta / 2
Pooled Variance
(n1  1) s12  (n2  1) s22
s 
(n1  n2  2)
2
p
ta, ta/2 and p-values are based on (n1 – n2 – 2) df
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Example: Small Sample Difference in
Mean Test
Catalyst Case, Difference in Mean Hourly Yields?
n1  5, x1  811.0, s12  386,
n 2  5, x2  750.2, s22  484.2
Test H0: 1 - 2 = 0 versus Ha: 1 - 2  0, a = 0.01
(n1  1) s12  (n2  1) s22 (5  1)386  (5  1)484.2
s 

 435.1
(n1  n2  2)
(5  5  2)
2
p
t
(x1  x 2 )  D 0

(811  750.2)  0
1 1
1 1
435
.
1
  
s   
5 5
 n1 n2 
p  value  P(t  4.6087)  0.0017
 4.6087  3.355  t.005
2
p
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Small Sample Intervals and Tests about
Differences in Means When Variances
are Not Equal
If sampled populations are both normal, but sample sizes and variances
differ substantially, small-sample estimation and testing can be based on
the following “unequal variance” procedure.
Confidence Interval
(x1  x 2 )  za/2
Test Statistic
s12 s22

n1 n2
t
(x1  x 2 )  D 0
s12 s22

n1 n2
For both the interval and test, the degrees of freedom are equal to
(s12 / n1  s 22 / n 2 ) 2
df  2
(s1 / n1 ) 2 (s 22 / n 2 ) 2

n1  1
n 2 1
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9.3 Comparing Two Population
Variances Using Independent Samples
(One-Tailed)
If both sampled populations are normal
Test of H0: 12 = 22 vs
Ha :  1 2 >  2 2
Test of H0: 12 = 22 vs
Ha :  1 2 <  2 2
s12
Test Statistic F  2
s2
s22
Test Statistic F  2
s1
Reject H0 in favor of Ha if:
F > Fa or if
p-value < a
Fa is based on
(n1 – 1) and (n2 – 1) df
Reject H0 in favor of Ha if:
F > Fa or if
p-value < a
Fa is based on
(n2 – 1) and (n1 – 1) df
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Comparing Two Population Variances
Using Independent Samples (TwoTailed)
If both sampled populations are normal
Test of H0: 12 = 22 vs Ha: 12  22
Test Statistic
larger of s12 and s22
F
smaller of s12and s22
df1 = {size of sample with larger variance} – 1
df2 = {size of sample with smaller variance} – 1
Reject H0 in favor of Ha if:
F > Fa/2 or if
p-value < a
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9.4 Paired Difference Interval for
Difference in Mean
If the sampled population of differences is normally
distributed with mean d, then a (1a)100% confidence
interval for d 1  2 is
d  t a/2
sd
n
t a/2 is based on n – 1 degrees of freedom.
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Paired Difference Test for Difference in
Mean
If the population of differences is normal, we can reject H0: d =
D0 at the a level of significance (probability of Type I error equal
to a) if and only if the appropriate rejection point condition
holds or, equivalently, if the corresponding p-value is less than a.
Alternative
H a :  d  D0
Reject H0 if:
t  ta
H a :  d  D0
t  ta
Area under t distributi on left of t
H a :  d  D0
t  ta / 2 , that is
Twice area under t distributi on right of t
Test Statistic
d-D 0
t=
sd / n
p-Value
Area under t distributi on right of t
t  ta / 2 or t  ta / 2
ta, ta/2 and p-values are based on n – 1 degrees of
freedom.
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Example: Paired Difference Interval
and Test
Car
Car 1
Car 2
Car 3
Car 4
Car 5
Car 6
Car 7
Garage 1 Garage 2 Difference
$ 7.10 $
7.90
-0.8
9.00
10.10
-1.1
11.00
12.20
-1.2
8.90
8.80
0.1
9.90
10.40
-0.5
9.10
9.80
-0.7
10.30
11.70
-1.4
Table 9.3
Mean
Std Dev
Excel Test Output
-0.8
0.5033
95% Confidence Interval
d  ta / 2
sd
 0.5033 
 0.8  2.447
  0.8  0.4654  [1.2654,0.3346]
n
7 

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9.5 Large Sample Interval for the
Difference in Proportions
If two independent samples are both large, a
100(1 - a)% confidence interval for p1 - p2 is
(p̂1  p̂ 2 )  za / 2
p̂1 (1  p̂1 ) p̂ 2 (1  p̂ 2 )

n1  1
n 2 1
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Large Sample Test for Difference in
Proportions
If two sampled populations are both large, we can reject H0: p1 - p2 =
D0 at the a level of significance if and only if the appropriate rejection
point condition holds or, equivalently, if the corresponding p-value is less
than a.
H a : p1  p2  D0
Reject H0 if:
z  za
H a : p1  p2  D0
z   za
Area under std normal curve left of z
H a : p1  p2  D0
z  za / 2 , that is
Twice area under std normal curve right of z
Alternative
p-Value
Area under std normal curve right of z
z  za / 2 or z   za / 2
Test Statistics
z
D0  0
( pˆ 1  pˆ 2 )  D 0
1 1
pˆ (1  pˆ )  
 n1 n2 
D0  0
z
( pˆ 1  pˆ 2 )  D 0
pˆ 1 (1  pˆ 1 ) pˆ 2 (1  pˆ 2 )

n1  1
n2  1
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Example: Paired Difference Interval
and Test
631
798
631  798
Example 9.11
p̂1 
, p̂ 2 
, p̂ 
Advertising Media
1000
1000
1000  1000
95% Confidence Interval for p1 - p2
.631(1  .369) .798(1  .202)
(.631  .798)  1.96


1000 - 1
1000 - 1
 0.167  0.0389  [0.2059,  0.1281]
Test H0: p1 - p2 = 0 versus Ha: p1 - p2  0
z
( pˆ 1  pˆ 2 )  D 0

(0.631  0.798)  0
1 
1 1
 1
0
.
7145
(
1

0
.
7145
)



pˆ (1  pˆ )  
1000
1000


 n1 n2 
 8.2673  3.29   z0.0005, p  value  2  P( z  8.2673)  0.001
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Statistical Inferences Based on Two
Samples
9.1 Comparing Two Population Means Using Large
Summary:
Independent Samples
9.2 Comparing Two Population Means Using Small
Independent Samples
9.3 Comparing Two Population Variances Using
Independent Samples
9.4 Paired Difference Experiments
9.5 Comparing Two Population Proportions Using
Large Independent Samples
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