This document contains slides about inferences for the difference between two population means. It discusses both the cases when the population standard deviations (σ1 and σ2) are known and unknown. When the standard deviations are known, it describes how to construct confidence intervals and conduct hypothesis tests for the difference between two means (μ1 - μ2) using a z-statistic. When the standard deviations are unknown, it replaces the z-statistic with a t-statistic and uses the sample standard deviations in the calculations. Examples are provided to demonstrate both cases.

1. 1Slide
© 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
John Loucks
St. Edward’s
University
...
...
...
..
SLIDES .BY

2. 2Slide
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or duplicated, or posted to a publicly accessible website, in whole or in part.
Chapter 10, Part A
Inference About Means and Proportions
with Two Populations
 Inferences About the Difference Between
Two Population Means: s 1 and s 2 Known
 Inferences About the Difference Between
Two Population Means: Matched Samples
 Inferences About the Difference Between
Two Population Means: s 1 and s 2 Unknown

3. 3Slide
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or duplicated, or posted to a publicly accessible website, in whole or in part.
Inferences About the Difference Between
Two Population Means: s 1 and s 2 Known
 Interval Estimation of m 1 – m 2
 Hypothesis Tests About m 1 – m 2

4. 4Slide
© 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Estimating the Difference Between
Two Population Means
 Let m1 equal the mean of population 1 and m2 equal
the mean of population 2.
 The difference between the two population means is
m1 - m2.
 To estimate m1 - m2, we will select a simple random
sample of size n1 from population 1 and a simple
random sample of size n2 from population 2.
 Let equal the mean of sample 1 and equal the
mean of sample 2.
x1 x2
 The point estimator of the difference between the
means of the populations 1 and 2 is .x x1 2

5. 5Slide
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or duplicated, or posted to a publicly accessible website, in whole or in part.
 Expected Value
Sampling Distribution of x x1 2
E x x( )1 2 1 2  m m
 Standard Deviation (Standard Error)
s
s s
x x
n n1 2
1
2
1
2
2
2
  
where: s1 = standard deviation of population 1
s2 = standard deviation of population 2
n1 = sample size from population 1
n2 = sample size from population 2

6. 6Slide
© 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
 Interval Estimate
Interval Estimation of m1 - m2:
s 1 and s 2 Known
2 2
1 2
1 2 /2
1 2
x x z
n n

s s
  
where:
1 -  is the confidence coefficient

7. 7Slide
© 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Interval Estimation of m1 - m2:
s 1 and s 2 Known
In a test of driving distance using a mechanical
driving device, a sample of Par golf balls was
compared with a sample of golf balls made by Rap,
Ltd., a competitor. The sample statistics appear on
the next slide.
Par, Inc. is a manufacturer of golf equipment and
has developed a new golf ball that has been
designed to provide “extra distance.”
 Example: Par, Inc.

8. 8Slide
© 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
 Example: Par, Inc.
Interval Estimation of m1 - m2:
s 1 and s 2 Known
Sample Size
Sample Mean
Sample #1
Par, Inc.
Sample #2
Rap, Ltd.
120 balls 80 balls
295 yards 278 yards
Based on data from previous driving distance
tests, the two population standard deviations are
known with s 1 = 15 yards and s 2 = 20 yards.

9. 9Slide
© 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Interval Estimation of m1 - m2:
s 1 and s 2 Known
 Example: Par, Inc.
Let us develop a 95% confidence interval estimate
of the difference between the mean driving distances of
the two brands of golf ball.

10. 10Slide
© 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Estimating the Difference Between
Two Population Means
m1 – m2 = difference between
the mean distances
x1 - x2 = Point Estimate of m1 – m2
Population 1
Par, Inc. Golf Balls
m1 = mean driving
distance of Par
golf balls
Population 2
Rap, Ltd. Golf Balls
m2 = mean driving
distance of Rap
golf balls
Simple random sample
of n2 Rap golf balls
x2 = sample mean distance
for the Rap golf balls
Simple random sample
of n1 Par golf balls
x1 = sample mean distance
for the Par golf balls

11. 11Slide
© 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Point Estimate of m1 - m2
Point estimate of m1  m2 = x x1 2
where:
m1 = mean distance for the population
of Par, Inc. golf balls
m2 = mean distance for the population
of Rap, Ltd. golf balls
= 295  278
= 17 yards

12. 12Slide
© 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
x x z
n n
1 2 2
1
2
1
2
2
2
2 2
17 1 96
15
120
20
80
     
s s
/ .
( ) ( )
Interval Estimation of m1 - m2:
s 1 and s 2 Known
We are 95% confident that the difference between
the mean driving distances of Par, Inc. balls and Rap,
Ltd. balls is 11.86 to 22.14 yards.
17 + 5.14 or 11.86 yards to 22.14 yards

13. 13Slide
© 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Hypothesis Tests About m 1  m 2:
s 1 and s 2 Known
 Hypotheses
1 2 0
2 2
1 2
1 2
( )x x D
z
n n
s s
 


m m 1 2 0:aH D
m m 0 1 2 0:H Dm m 0 1 2 0:H D
m m 1 2 0:aH D
m m 0 1 2 0:H D
m m 1 2 0:aH D
Left-tailed Right-tailed Two-tailed
 Test Statistic

14. 14Slide
© 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
 Example: Par, Inc.
Hypothesis Tests About m 1  m 2:
s 1 and s 2 Known
Can we conclude, using  = .01, that the
mean driving distance of Par, Inc. golf balls is
greater than the mean driving distance of Rap, Ltd.
golf balls?

15. 15Slide
© 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
H0: m1 - m2 < 0
Ha: m1 - m2 > 0
where:
m1 = mean distance for the population
of Par, Inc. golf balls
m2 = mean distance for the population
of Rap, Ltd. golf balls
1. Develop the hypotheses.
 p –Value and Critical Value Approaches
Hypothesis Tests About m 1  m 2:
s 1 and s 2 Known
2. Specify the level of significance.  = .01

16. 16Slide
© 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
3. Compute the value of the test statistic.
Hypothesis Tests About m 1  m 2:
s 1 and s 2 Known
 p –Value and Critical Value Approaches
s s
 


1 2 0
2 2
1 2
1 2
( )x x D
z
n n
2 2
(295 278) 0 17
6.49
2.62(15) (20)
120 80
z
 
  


17. 17Slide
© 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
 p –Value Approach
4. Compute the p–value.
For z = 6.49, the p –value < .0001.
Hypothesis Tests About m 1  m 2:
s 1 and s 2 Known
5. Determine whether to reject H0.
Because p–value <  = .01, we reject H0.
At the .01 level of significance, the sample evidence
indicates the mean driving distance of Par, Inc. golf
balls is greater than the mean driving distance of Rap,
Ltd. golf balls.

18. 18Slide
© 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Hypothesis Tests About m 1  m 2:
s 1 and s 2 Known
5. Determine whether to reject H0.
Because z = 6.49 > 2.33, we reject H0.
 Critical Value Approach
For  = .01, z.01 = 2.33
4. Determine the critical value and rejection rule.
Reject H0 if z > 2.33
The sample evidence indicates the mean driving
distance of Par, Inc. golf balls is greater than the mean
driving distance of Rap, Ltd. golf balls.

19. 19Slide
© 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Inferences About the Difference Between
Two Population Means: s 1 and s 2 Unknown
 Interval Estimation of m 1 – m 2
 Hypothesis Tests About m 1 – m 2

20. 20Slide
© 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Interval Estimation of m1 - m2:
s 1 and s 2 Unknown
When s 1 and s 2 are unknown, we will:
• replace z/2 with t/2.
• use the sample standard deviations s1 and s2
as estimates of s 1 and s 2 , and

21. 21Slide
© 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
2 2
1 2
1 2 /2
1 2
s s
x x t
n n
  
Where the degrees of freedom for t/2 are:
Interval Estimation of m1 - m2:
s 1 and s 2 Unknown
 Interval Estimate
2
2 2
1 2
1 2
2 2
2 2
1 2
1 1 2 2
1 1
1 1
s s
n n
df
s s
n n n n
 
 
 
   
   
    

22. 22Slide
© 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
 Example: Specific Motors
Difference Between Two Population Means:
s 1 and s 2 Unknown
Specific Motors of Detroit has developed a new
Automobile known as the M car. 24 M cars and 28 J
cars (from Japan) were road tested to compare miles-
per-gallon (mpg) performance. The sample statistics
are shown on the next slide.

23. 23Slide
© 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Difference Between Two Population Means:
s 1 and s 2 Unknown
 Example: Specific Motors
Sample Size
Sample Mean
Sample Std. Dev.
Sample #1
M Cars
Sample #2
J Cars
24 cars 28 cars
29.8 mpg 27.3 mpg
2.56 mpg 1.81 mpg

24. 24Slide
© 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Difference Between Two Population Means:
s 1 and s 2 Unknown
Let us develop a 90% confidence interval estimate
of the difference between the mpg performances of
the two models of automobile.
 Example: Specific Motors

25. 25Slide
© 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Point estimate of m1  m2 = x x1 2
Point Estimate of m 1  m 2
where:
m1 = mean miles-per-gallon for the
population of M cars
m2 = mean miles-per-gallon for the
population of J cars
= 29.8 - 27.3
= 2.5 mpg

26. 26Slide
© 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Interval Estimation of m 1  m 2:
s 1 and s 2 Unknown
The degrees of freedom for t/2 are:
22 2
2 22 2
(2.56) (1.81)
24 28
24.07 24
1 (2.56) 1 (1.81)
24 1 24 28 1 28
df
 
 
   
   
   
    
With /2 = .05 and df = 24, t/2 = 1.711

27. 27Slide
© 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Interval Estimation of m 1  m 2:
s 1 and s 2 Unknown
2 2 2 2
1 2
1 2 /2
1 2
(2.56) (1.81)
29.8 27.3 1.711
24 28
s s
x x t
n n
      
We are 90% confident that the difference between
the miles-per-gallon performances of M cars and J cars
is 1.431 to 3.569 mpg.
2.5 + 1.069 or 1.431 to 3.569 mpg

28. 28Slide
© 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Hypothesis Tests About m 1  m 2:
s 1 and s 2 Unknown
 Hypotheses
1 2 0
2 2
1 2
1 2
( )x x D
t
s s
n n
 


m m 1 2 0:aH D
m m 0 1 2 0:H Dm m 0 1 2 0:H D
m m 1 2 0:aH D
m m 0 1 2 0:H D
m m 1 2 0:aH D
Left-tailed Right-tailed Two-tailed
 Test Statistic

29. 29Slide
© 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
 Example: Specific Motors
Hypothesis Tests About m 1  m 2:
s 1 and s 2 Unknown
Can we conclude, using a .05 level of significance,
that the miles-per-gallon (mpg) performance of M cars
is greater than the miles-per-gallon performance of J
cars?

30. 30Slide
© 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
H0: m1 - m2 < 0
Ha: m1 - m2 > 0
where:
m1 = mean mpg for the population of M cars
m2 = mean mpg for the population of J cars
1. Develop the hypotheses.
 p –Value and Critical Value Approaches
Hypothesis Tests About m 1  m 2:
s 1 and s 2 Unknown

31. 31Slide
© 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
2. Specify the level of significance.
3. Compute the value of the test statistic.
 = .05
 p –Value and Critical Value Approaches
Hypothesis Tests About m 1  m 2:
s 1 and s 2 Unknown
1 2 0
2 2 2 2
1 2
1 2
( ) (29.8 27.3) 0
4.003
(2.56) (1.81)
24 28
x x D
t
s s
n n
   
  
 

32. 32Slide
© 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Hypothesis Tests About m 1  m 2:
s 1 and s 2 Unknown
 p –Value Approach
4. Compute the p –value.
The degrees of freedom for t are:
Because t = 4.003 > t.05 = 1.683, the p–value < .05.
22 2
2 22 2
(2.56) (1.81)
24 28
40.566 41
1 (2.56) 1 (1.81)
24 1 24 28 1 28
 
 
   
   
   
    
df
In fact, the p–value < .005.

33. 33Slide
© 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
5. Determine whether to reject H0.
We are at least 95% confident that the miles-per-
gallon (mpg) performance of M cars is greater than
the miles-per-gallon performance of J cars?.
 p –Value Approach
Because p–value <  = .05, we reject H0.
Hypothesis Tests About m 1  m 2:
s 1 and s 2 Unknown

34. 34Slide
© 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
4. Determine the critical value and rejection rule.
 Critical Value Approach
Hypothesis Tests About m 1  m 2:
s 1 and s 2 Unknown
For  = .05 and df = 41, t.05 = 1.683
Reject H0 if t > 1.683
5. Determine whether to reject H0.
Because 4.003 > 1.683, we reject H0.
We are at least 95% confident that the miles-per-
gallon (mpg) performance of M cars is greater than
the miles-per-gallon performance of J cars?.

35. 35Slide
© 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
 With a matched-sample design each sampled item
provides a pair of data values.
 This design often leads to a smaller sampling error
than the independent-sample design because
variation between sampled items is eliminated as a
source of sampling error.
Inferences About the Difference Between
Two Population Means: Matched Samples

36. 36Slide
© 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
 Example: Express Deliveries
Inferences About the Difference Between
Two Population Means: Matched Samples
A Chicago-based firm has documents that must
be quickly distributed to district offices throughout
the U.S. The firm must decide between two delivery
services, UPX (United Parcel Express) and INTEX
(International Express), to transport its documents.

37. 37Slide
© 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
 Example: Express Deliveries
Inferences About the Difference Between
Two Population Means: Matched Samples
In testing the delivery times of the two services,
the firm sent two reports to a random sample of its
district offices with one report carried by UPX and
the other report carried by INTEX. Do the data on
the next slide indicate a difference in mean delivery
times for the two services? Use a .05 level of
significance.

38. 38Slide
© 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
32
30
19
16
15
18
14
10
7
16
25
24
15
15
13
15
15
8
9
11
UPX INTEX DifferenceDistrict Office
Seattle
Los Angeles
Boston
Cleveland
New York
Houston
Atlanta
St. Louis
Milwaukee
Denver
Delivery Time (Hours)
7
6
4
1
2
3
-1
2
-2
5
Inferences About the Difference Between
Two Population Means: Matched Samples

39. 39Slide
© 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
H0: md = 0
Ha: md  
Let md = the mean of the difference values for the
two delivery services for the population
of district offices
1. Develop the hypotheses.
Inferences About the Difference Between
Two Population Means: Matched Samples
 p –Value and Critical Value Approaches

40. 40Slide
© 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
2. Specify the level of significance.  = .05
Inferences About the Difference Between
Two Population Means: Matched Samples
 p –Value and Critical Value Approaches
3. Compute the value of the test statistic.
d
d
n
i



  

( ... )
.
7 6 5
10
2 7
s
d d
n
d
i



 
( ) .
.
2
1
76 1
9
2 9
2.7 0
2.94
2.9 10
d
d
d
t
s n
m 
  

41. 41Slide
© 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
5. Determine whether to reject H0.
We are at least 95% confident that there is a
difference in mean delivery times for the two
services?
4. Compute the p –value.
For t = 2.94 and df = 9, the p–value is between
.02 and .01. (This is a two-tailed test, so we double
the upper-tail areas of .01 and .005.)
Because p–value <  = .05, we reject H0.
Inferences About the Difference Between
Two Population Means: Matched Samples
 p –Value Approach

42. 42Slide
© 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
4. Determine the critical value and rejection rule.
Inferences About the Difference Between
Two Population Means: Matched Samples
 Critical Value Approach
For  = .05 and df = 9, t.025 = 2.262.
Reject H0 if t > 2.262
5. Determine whether to reject H0.
Because t = 2.94 > 2.262, we reject H0.
We are at least 95% confident that there is a
difference in mean delivery times for the two
services?

43. 43Slide
© 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
End of Chapter 10
Part A