1. Statistics for Business and Economics
Inferences Based on Two Samples: Confidence Intervals & Tests of Hypotheses Chapter 9

2. Learning Objectives
1. Solve Hypothesis Testing Problems for Two Populations
Mean
Proportion
Variance
2. Distinguish Independent & Related Populations
3. Explain the F Distribution
As a result of this class, you will be able to ...

3. Thinking Challenge How Would You Try to Answer These Questions? D O S
Who Gets Higher Grades: Males or Females?
Which Programs Are Faster to Learn: Windows or DOS?
D O S

4. Two Population Tests

5. Independent Sampling & Paired Difference Experiments
Testing Two Means
Independent Sampling & Paired Difference Experiments 9 26

6. Two Population Tests

7. Independent & Related Populations
Matching
Match according to some characteristic of interest.
Repeated Measures
Assumes the same individual behaves similarly under both treatments except for treatment effect. Any difference will be due to treatment effect.

8. Independent & Related Populations
1. Different Data Sources
Unrelated
Independent
Matching
Match according to some characteristic of interest.
Repeated Measures
Assumes the same individual behaves similarly under both treatments except for treatment effect. Any difference will be due to treatment effect.

9. Independent & Related Populations
1. Different Data Sources
Unrelated
Independent
1. Same Data Source
Paired or Matched
Repeated Measures (Before/After)
Matching
Match according to some characteristic of interest.
Repeated Measures
Assumes the same individual behaves similarly under both treatments except for treatment effect. Any difference will be due to treatment effect.

10. Independent & Related Populations
1. Different Data Sources
Unrelated
Independent
2. Use Difference Between the 2 Sample Means
X1 -X2
1. Same Data Source
Paired or Matched
Repeated Measures (Before/After)
Matching
Match according to some characteristic of interest.
Repeated Measures
Assumes the same individual behaves similarly under both treatments except for treatment effect. Any difference will be due to treatment effect.

11. Independent & Related Populations
1. Different Data Sources
Unrelated
Independent
2. Use Difference Between the 2 Sample Means
X1 -X2
1. Same Data Source
Paired or Matched
Repeated Measures (Before/After)
2. Use Difference Between Each Pair of Observations
Di = X1i - X2i
Matching
Match according to some characteristic of interest.
Repeated Measures
Assumes the same individual behaves similarly under both treatments except for treatment effect. Any difference will be due to treatment effect.

12. Two Independent Populations Examples
1. An economist wishes to determine whether there is a difference in mean family income for households in 2 socioeconomic groups.
2. An admissions officer of a small liberal arts college wants to compare the mean SAT scores of applicants educated in rural high schools & in urban high schools.
These are comparative studies.
The general purpose of comparative studies is to establish similarities or to detect and measure differences between populations.
The populations can be (1) existing populations or (2) hypothetical populations.

13. Two Related Populations Examples
1. Nike wants to see if there is a difference in durability of 2 sole materials. One type is placed on one shoe, the other type on the other shoe of the same pair.
2. An analyst for Educational Testing Service wants to compare the mean GMAT scores of students before & after taking a GMAT review course.

14. Thinking Challenge Are They Independent or Paired?
1. Miles per gallon ratings of cars before & after mounting radial tires
2. The life expectancy of light bulbs made in 2 different factories
3. Difference in hardness between 2 metals: one contains an alloy, one doesn’t
4. Tread life of two different motorcycle tires: one on the front, the other on the back
1. Related. Same car should be used since car weights vary.
2. Independent. Can’t match light bulbs.
3. Independent. No relationship between two alloys. Only differ in alloy.
4. Related. Tires are matched to same motorcycle. Should alternate tire types between front and back and motorcycle.

15. Testing 2 Independent Means9 26

16. Two Population Tests

17. Two Independent Populations Hypotheses for Means9

18. Two Independent Populations Hypotheses for Means30

19. Two Independent Populations Hypotheses for Means30

20. Sampling Distribution
In this diagram, do the populations have equal or unequal variances? Unequal. 31

21. Sampling Distribution
Population 1 1
 1
In this diagram, do the populations have equal or unequal variances? Unequal. 32

22. Sampling Distribution 
Population 1 2
Population 1 2
   2 1
In this diagram, do the populations have equal or unequal variances? Unequal. 33

23. Sampling Distribution 
Population 1 2
Population 1 2
   2 1
In this diagram, do the populations have equal or unequal variances? Unequal.
Select simple random
sample, size n . 1
Compute  X 1 34

24. Sampling Distribution 
Population 1 2
Population 1 2
   2 1
In this diagram, do the populations have equal or unequal variances? Unequal.
Select simple random
Select simple random
sample, size n .
sample, size n . 1 2
Compute  X
Compute  X 1 2 35

25. Sampling Distribution 
Population 1 2
Population 1 2
   2 1
In this diagram, do the populations have equal or unequal variances? Unequal.
Select simple random
Compute  X -  X
Select simple random 1 2
sample, size n .
for every pair
sample, size n . 1 2
Compute  X
of samples
Compute  X 1 2 36

26. Sampling Distribution 
Population 1 2
Population 1 2
   2 1
In this diagram, do the populations have equal or unequal variances? Unequal.
Select simple random
Compute  X -  X
Select simple random 1 2
sample, size n .
for every pair
sample, size n . 1 2
Compute  X
of samples
Compute  X 1 2
Astronomical number of  X -  X
values 1 2 37

27. Sampling Distribution
In this diagram, do the populations have equal or unequal variances? Unequal. 38

28. Large-Sample Z Test for 2 Independent Means9 26

29. Two Population Tests

30. Large-Sample Z Test for 2 Independent Means

31. Large-Sample Z Test for 2 Independent Means
1. Assumptions
Independent, Random Samples
Populations Are Normally Distributed
If Not Normal, Can Be Approximated by Normal Distribution (n1  30 & n2  30 )

32. Large-Sample Z Test for 2 Independent Means
1. Assumptions
Independent, Random Samples
Populations Are Normally Distributed
If Not Normal, Can Be Approximated by Normal Distribution (n1  30 & n2  30 )
2. Two Independent Sample Z-Test Statistic  X n    1 2   X n    1 2 s Z

33. Large-Sample Z Test Example
You’re a financial analyst for Charles Schwab. You want to find out if there is a difference in dividend yield between stocks listed on NYSE & NASDAQ. You collect the following data:
NYSE NASDAQ Number
Mean
Std Dev
Is there a difference in average yield ( = .05)?
© T/Maker Co.

34. Large-Sample Z Test Solution

35. Large-Sample Z Test Solution
H0:
Ha:
 
n1 = , n2 =
Critical Value(s):
Test Statistic:
Decision:
Conclusion:

36. Large-Sample Z Test Solution
H0: 1 - 2 = 0 (1 = 2)
Ha: 1 - 2  0 (1  2)
 
n1 = , n2 =
Critical Value(s):
Test Statistic:
Decision:
Conclusion:

37. Large-Sample Z Test Solution
H0: 1 - 2 = 0 (1 = 2)
Ha: 1 - 2  0 (1  2)
  .05
n1 = 121, n2 = 125
Critical Value(s):
Test Statistic:
Decision:
Conclusion:

38. Large-Sample Z Test Solution
H0: 1 - 2 = 0 (1 = 2)
Ha: 1 - 2  0 (1  2)
  .05
n1 = 121, n2 = 125
Critical Value(s):
Test Statistic:
Decision:
Conclusion:
Reject H
Reject H
.025 z
-1.96
1.96

39. Large-Sample Z Test Solution
H0: 1 - 2 = 0 (1 = 2)
Ha: 1 - 2  0 (1  2)
  .05
n1 = 121, n2 = 125
Critical Value(s):
Test Statistic:
Decision:
Conclusion: 3 . 27  2 . 53 z    4 . 69 1 .
698 1 .
353 
121
125
Reject H
Reject H
.025 z
-1.96
1.96

40. Large-Sample Z Test Solution
H0: 1 - 2 = 0 (1 = 2)
Ha: 1 - 2  0 (1  2)
  .05
n1 = 121, n2 = 125
Critical Value(s):
Test Statistic:
Decision:
Conclusion: 3 . 27  2 . 53 z    4 . 69 1 .
698 1 .
353 
121
125
Reject H
Reject H
Reject at  = .05
.025
.025 z
-1.96
1.96

41. Large-Sample Z Test Solution
H0: 1 - 2 = 0 (1 = 2)
Ha: 1 - 2  0 (1  2)
  .05
n1 = 121, n2 = 125
Critical Value(s):
Test Statistic:
Decision:
Conclusion: 3 . 27  2 . 53 z    4 . 69 1 .
698 1 .
353 
121
125
Reject H
Reject H
Reject at  = .05
.025
.025
There is Evidence of a Difference in Means z
-1.96
1.96

42. Large-Sample Z Test Thinking Challenge
You’re an economist for the Department of Education. You want to find out if there is a difference in spending per pupil between urban & rural high schools. You collect the following:
Urban Rural Number 35 35
Mean $ 6,012 $ 5,832
Std Dev $ 602 $ 497
Is there any difference in population means ( = .10)?
Allow students about 15 minutes to solve.

43. Large-Sample Z Test Solution*
H0: 1 - 2 = 0 (1 = 2)
Ha: 1 - 2  0 (1  2)
  .10
n1 = 35, n2 = 35
Critical Value(s):
Test Statistic:
Decision:
Conclusion:
6012 
5832  z    1 . 36 2 2
602
497  35 35
Reject H
Reject H
Do Not Reject at  = .10
.05
.05
There is No Evidence of a Difference in Means z
-1.645
1.645

44. Small-Sample t Test for 2 Independent Means9 26

45. Two Population Tests

46. Small-Sample t Test for 2 Independent Means
1. Tests Means of 2 Independent Populations Having Equal Variances
2. Assumptions
Independent, Random Samples
Both Populations Are Normally Distributed
Population Variances Are Unknown But Assumed Equal

47. Small-Sample t Test Test Statistic
Hypothesized difference 39

48. Small-Sample t Test Example
You’re a financial analyst for Charles Schwab. Is there a difference in dividend yield between stocks listed on the NYSE & NASDAQ? You collect the following data:
NYSE NASDAQ Number
Mean
Std Dev
Assuming normal populations, is there a difference in average yield ( = .05)?
© T/Maker Co.

49. Small-Sample t Test Solution41

50. Small-Sample t Test Solution
H0:
Ha:
 
df 
Critical Value(s):
Test Statistic:
Decision:
Conclusion: 41

51. Small-Sample t Test Solution
H0: 1 - 2 = 0 (1 = 2)
Ha: 1 - 2  0 (1  2)
 
df 
Critical Value(s):
Test Statistic:
Decision:
Conclusion: 41

52. Small-Sample t Test Solution
H0: 1 - 2 = 0 (1 = 2)
Ha: 1 - 2  0 (1  2)
  .05
df  = 44
Critical Value(s):
Test Statistic:
Decision:
Conclusion: 41

53. Small-Sample t Test Solution
H0: 1 - 2 = 0 (1 = 2)
Ha: 1 - 2  0 (1  2)
  .05
df  = 44
Critical Value(s):
Test Statistic:
Decision:
Conclusion: 41

54. Small-Sample t Test Solution42

55. Small-Sample t Test Solution
H0: 1 - 2 = 0 (1 = 2)
Ha: 1 - 2  0 (1  2)
  .05
df  = 44
Critical Value(s):
Test Statistic:
Decision:
Conclusion: 41

56. Small-Sample t Test Solution
H0: 1 - 2 = 0 (1 = 2)
Ha: 1 - 2  0 (1  2)
  .05
df  = 44
Critical Value(s):
Test Statistic:
Decision:
Conclusion:
Reject at  = .05 41

57. Small-Sample t Test Solution
H0: 1 - 2 = 0 (1 = 2)
Ha: 1 - 2  0 (1  2)
  .05
df  = 44
Critical Value(s):
Test Statistic:
Decision:
Conclusion:
Reject at  = .05
There is evidence of a difference in means 41

58. Small-Sample t Test Thinking Challenge
You’re a research analyst for General Motors. Assuming equal variances, is there a difference in the average miles per gallon (mpg) of two car models ( = .05)?
You collect the following:
Sedan Van Number 15 11
Mean
Std Dev
Allow students about 15 minutes to solve. 43

59. Small-Sample t Test Solution*
H0: 1 - 2 = 0 (1 = 2)
Ha: 1 - 2  0 (1  2)
  .05
df  = 24
Critical Value(s):
Test Statistic:
Decision:
Conclusion:
Do not reject at  = .05
There is no evidence of a difference in means 45

60. Small-Sample t Test Solution*46

61. Paired Difference Experiments
Paired-Sample t Test
Paired Difference Experiments 9 26

62. Two Population Tests

63. Paired-Sample t Test for Mean Difference
1. Tests Means of 2 Related Populations
Paired or Matched
Repeated Measures (Before/After)
2. Eliminates Variation Among Subjects
3. Assumptions
Both Population Are Normally Distributed
If Not Normal, Can Be Approximated by Normal Distribution (n1  30 & n2  30 )

64. Paired-Sample t Test Hypotheses
Research Questions
No Difference
Pop 1 
Pop 2
Pop 1 
Pop 2
Hypothesis
Any Difference
Pop 1 < Pop 2
Pop 1 > Pop 2 H   
= 0   D D D H    
< 0
> 0 1 D D D
Note: Di = X1i - X2i for ith observation

65. Paired-Sample t Test Data Collection Table
Observation
Group 1
Group 2
Difference 1 x x D
= x -x 11 21 1 11 21 2 x x D
= x -x 12 22 2 12 22     i x x D
= x
- x 1i 2i i 1i 2i     n x x D
= x
- x 1n 2n n 1n 2n

66. Paired-Sample t Test Test StatisticxD  D t  df  n  1 D S D n
Just take the mean and standard deviation of the difference.
SD is simply the standard deviation. The formula is the computational formula. D
Sample Mean
Sample Standard Deviation n n   D
(Di - xD)2 i x  i  1 S  i  1 D D n n  1 D D

67. Paired-Sample t Test Example
You work in Human Resources. You want to see if a training program is effective. You collect the following test score data:
Name Before (1) After (2)
Sam 85 94
Tamika 94 87
Brian 78 79
Mike 87 88
At the .10 level, was the training effective?

68. Computation Table Observation Before After Difference Sam 85 94 -9
Tamika 94 87 7
Brian 78 79 -1
Mike 87 88 -1
Total
- 4

69. Null Hypothesis Solution
1. Was the training effective?
2. Effective means ‘After’ > ‘Before’.
3. Statistically, this means A > B.
4. Rearranging terms gives 0 B - A.
5. Defining D = B - A & substituting into (4) gives 0 D or D .
6. The alternative hypothesis is Ha: D 0.

70. Paired-Sample t Test Solution

71. Paired-Sample t Test Solution
H0:
Ha:
 =
df =
Critical Value(s):
Test Statistic:
Decision:
Conclusion:

72. Paired-Sample t Test Solution
H0: D = 0 (D = B - A)
Ha: D < 0
 =
df =
Critical Value(s):
Test Statistic:
Decision:
Conclusion:

73. Paired-Sample t Test Solution
H0: D = 0 (D = B - A)
Ha: D < 0
 = .10
df = = 3
Critical Value(s):
Test Statistic:
Decision:
Conclusion:

74. Paired-Sample t Test Solution
H0: D = 0 (D = B - A)
Ha: D < 0
 = .10
df = = 3
Critical Value(s):
Test Statistic:
Decision:
Conclusion:
Reject
.10 t

75. Paired-Sample t Test Solution
H0: D = 0 (D = B - A)
Ha: D < 0
 = .10
df = = 3
Critical Value(s):
Test Statistic:
Decision:
Conclusion: x  D  1  D t     .
306 S 6 . 53 D n 4 D
Reject
.10 t

76. Paired-Sample t Test Solution
H0: D = 0 (D = B - A)
Ha: D < 0
 = .10
df = = 3
Critical Value(s):
Test Statistic:
Decision:
Conclusion: x  D  1  D t     .
306 S 6 . 53 D n 4 D
Reject
Do Not Reject at  = .10
.10 t

77. Paired-Sample t Test Solution
H0: D = 0 (D = B - A)
Ha: D < 0
 = .10
df = = 3
Critical Value(s):
Test Statistic:
Decision:
Conclusion: x  D  1  D t     .
306 S 6 . 53 D n 4 D
Reject
Do Not Reject at  = .10
.10
There Is No Evidence Training Was Effective t

78. Paired-Sample t Test Thinking Challenge
(1) (2) Store Client Competitor
1 $ 10 $ 11
You’re a marketing research analyst. You want to compare a client’s calculator to a competitor’s. You sample 8 retail stores.
At the .01 level, does your client’s calculator sell for less than their competitor’s
Why related populations? Control for differences in store price. Some stores might be higher priced in terms of all goods.
Allow students about 15 minutes to solve this.

79. Paired-Sample t Test Solution*
H0: D = 0 (D = 1 - 2)
Ha: D < 0
 = .01
df = = 7
Critical Value(s):
Test Statistic:
Decision:
Conclusion: x  D  2 . 25  D t     5 .
486 S 1 . 16 D n 8 D
Reject
Reject at  = .01
.01
There Is Evidence Client’s Brand (1) Sells for Less
-2.998 t

80. Z Test for Differences in Two Proportions9 26

81. Two Population Tests

82. Z Test for Difference in Two Proportions

83. Z Test for Difference in Two Proportions
1. Assumptions
Populations Are Independent
Populations Follow Binomial Distribution
Normal Approximation Can Be Used
Does Not Contain 0 or n

84. Z Test for Difference in Two Proportions
1. Assumptions
Populations Are Independent
Populations Follow Binomial Distribution
Normal Approximation Can Be Used
Does Not Contain 0 or n
2. Z-Test Statistic for Two Proportions

85. Hypotheses for Two Proportions9

86. Hypotheses for Two Proportions9

87. Hypotheses for Two Proportions9

88. Hypotheses for Two Proportions9

89. Hypotheses for Two Proportions9

90. Hypotheses for Two Proportions9

91. Z Test for Two Proportions Example
As personnel director, you want to test the perception of fairness of two methods of performance evaluation. 63 of 78 employees rated Method 1 as fair. 49 of 82 rated Method 2 as fair. At the .01 level, is there a difference in perceptions?
To check assumptions, use sample proportions as estimators of population proportion:
n1·p = 78·63/78 = 63
n1·(1-p) = 78·(1-63/78) = 15

92. Z Test for Two Proportions Solution11

93. Z Test for Two Proportions Solution
H0:
Ha:
 =
n1 = n1 =
Critical Value(s):
Test Statistic:
Decision:
Conclusion: 11

94. Z Test for Two Proportions Solution
H0: p1 - p2 = 0
Ha: p1 - p2  0
 =
n1 = n1 =
Critical Value(s):
Test Statistic:
Decision:
Conclusion: 11

95. Z Test for Two Proportions Solution
H0: p1 - p2 = 0
Ha: p1 - p2  0
 = .01
n1 = 78 n1 = 82
Critical Value(s):
Test Statistic:
Decision:
Conclusion: 11

96. Z Test for Two Proportions Solution
H0: p1 - p2 = 0
Ha: p1 - p2  0
 = .01
n1 = 78 n1 = 82
Critical Value(s):
Test Statistic:
Decision:
Conclusion: 11

97. Z Test for Two Proportions Solution12

98. Z Test for Two Proportions Solution
H0: p1 - p2 = 0
Ha: p1 - p2  0
 = .01
n1 = 78 n1 = 82
Critical Value(s):
Test Statistic:
Decision:
Conclusion: 11

99. Z Test for Two Proportions Solution
H0: p1 - p2 = 0
Ha: p1 - p2  0
 = .01
n1 = 78 n1 = 82
Critical Value(s):
Test Statistic:
Decision:
Conclusion:
Reject at  = .01 11

100. Z Test for Two Proportions Solution
H0: p1 - p2 = 0
Ha: p1 - p2  0
 = .01
n1 = 78 n1 = 82
Critical Value(s):
Test Statistic:
Decision:
Conclusion:
Reject at  = .01
There is evidence of a difference in proportions 11

101. Z Test for Two Proportions Thinking Challenge
You’re an economist for the Department of Labor. You’re studying unemployment rates. In MA, 74 of 1500 people surveyed were unemployed. In CA, 129 of 1500 were unemployed. At the .05 level, does MA have a lower unemployment rate? MA CA

102. Z Test for Two Proportions Solution*15

103. Z Test for Two Proportions Solution*
H0:
Ha:
 =
nMA = nCA =
Critical Value(s):
Test Statistic:
Decision:
Conclusion: 15

104. Z Test for Two Proportions Solution*
H0: pMA - pCA = 0
Ha: pMA - pCA < 0
 =
nMA = nCA =
Critical Value(s):
Test Statistic:
Decision:
Conclusion: 15

105. Z Test for Two Proportions Solution*
H0: pMA - pCA = 0
Ha: pMA - pCA < 0
 = .05
nMA = nCA = 1500
Critical Value(s):
Test Statistic:
Decision:
Conclusion: 15

106. Z Test for Two Proportions Solution*
H0: pMA - pCA = 0
Ha: pMA - pCA < 0
 = .05
nMA = nCA = 1500
Critical Value(s):
Test Statistic:
Decision:
Conclusion: 15

107. Z Test for Two Proportions Solution*16

108. Z Test for Two Proportions Solution*
H0: pMA - pCA = 0
Ha: pMA - pCA < 0
 = .05
nMA = nCA = 1500
Critical Value(s):
Test Statistic:
Decision:
Conclusion:
Z = -4.00 15

109. Z Test for Two Proportions Solution*
H0: pMA - pCA = 0
Ha: pMA - pCA < 0
 = .05
nMA = nCA = 1500
Critical Value(s):
Test Statistic:
Decision:
Conclusion:
Z = -4.00
Reject at  = .05 15

110. Z Test for Two Proportions Solution*
H0: pMA - pCA = 0
Ha: pMA - pCA < 0
 = .05
nMA = nCA = 1500
Critical Value(s):
Test Statistic:
Decision:
Conclusion:
Z = -4.00
Reject at  = .05
There is evidence MA is less than CA 15

111. F-Test for Two Population Variances9 26

112. Two Population Tests

113. F-Test for Two Variances
1. Tests for Differences in 2 Population Variances
2. Assumptions
Both Populations Are Normally Distributed
Test Is Not Robust to Violations
Independent, Random Samples
Key words that show a test on variances, not on means or proportions: variance, standard deviation, variability, variation, scatter, dispersion.

114. F-Test for Variances Hypotheses
H0: 12 = 22 OR H0: 12  22 (or )
Ha: 12   Ha: 12 22 (or >)
2. Test Statistic
F = s12 /s22
Two Sets of Degrees of Freedom
1 = n1 - 1; 2 = n2 - 1
Follows F Distribution
F ratio is a statistic defined as the ratio of 2 independent estimates of a normally distributed population’s variance.
Note: degrees of freedom refer to numerator and denominator

115. F Distribution
The sampling distribution is a function of the sample sizes upon which the sample variances are based.
Hint: Recall the formula for variance!
s2 = (x -x)2/(n-1) 58

116. F Distribution   Population 1 1 1
The sampling distribution is a function of the sample sizes upon which the sample variances are based.
Hint: Recall the formula for variance!
s2 = (x -x)2/(n-1) 59

117. F Distribution     Population Population 1 2 1 2 1 2
The sampling distribution is a function of the sample sizes upon which the sample variances are based.
Hint: Recall the formula for variance!
s2 = (x -x)2/(n-1) 60

118. F Distribution     Population Population 1 2
The sampling distribution is a function of the sample sizes upon which the sample variances are based.
Hint: Recall the formula for variance!
s2 = (x -x)2/(n-1)
Select simple random
sample, size n 1
Compute S 2 1 61

119. F Distribution     Population Population 1 2
The sampling distribution is a function of the sample sizes upon which the sample variances are based.
Hint: Recall the formula for variance!
s2 = (x -x)2/(n-1)
Select simple random
Select simple random
sample, size n
sample, size n 1 2
Compute S 2
Compute S 2 1 2 62

120. F Distribution     Population Population 1 2
The sampling distribution is a function of the sample sizes upon which the sample variances are based.
Hint: Recall the formula for variance!
s2 = (x -x)2/(n-1)
Select simple random
Compute F =  S 2 /S 2
Select simple random 1 2
sample, size n
for every pair of
sample, size n 1 2
Compute S 2 n
& n
size samples
Compute S 2 1 1 2 2 63

121. F Distribution     Population Population 1 2
The sampling distribution is a function of the sample sizes upon which the sample variances are based.
Hint: Recall the formula for variance!
s2 = (x -x)2/(n-1)
Select simple random
Compute F =  S 2 /S 2
Select simple random 1 2
sample, size n
for every pair of
sample, size n 1 2
Compute S 2 n
& n
size samples
Compute S 2 1 1 2 2
Astronomical number of S 2 /S 2
values 1 2 64

122. F Distribution
The sampling distribution is a function of the sample sizes upon which the sample variances are based.
Hint: Recall the formula for variance!
s2 = (x -x)2/(n-1) 65

123. F-Test for 2 Variances Critical Values
The lower F is not the reciprocal of the upper F.
What do you do if equal sample sizes?
/2
/2
Note! 66

124. F-Test for Variances Example
You’re a financial analyst for Charles Schwab. You want to compare dividend yields between stocks listed on the NYSE & NASDAQ. You collect the following data:
NYSE NASDAQ Number
Mean
Std Dev
Is there a difference in variances between the NYSE & NASDAQ at the .05 level?
© T/Maker Co.

125. F-Test for 2 Variances Solution68

126. F-Test for 2 Variances Solution
H0:
Ha:
 
1  2 
Critical Value(s):
Test Statistic:
Decision:
Conclusion: 68

127. F-Test for 2 Variances Solution
H0: 12 = 22
Ha: 12  22
 
1  2 
Critical Value(s):
Test Statistic:
Decision:
Conclusion: 68

128. F-Test for 2 Variances Solution
H0: 12 = 22
Ha: 12  22
  .05
1  2  24
Critical Value(s):
Test Statistic:
Decision:
Conclusion: 68

129. F-Test for 2 Variances Solution
H0: 12 = 22
Ha: 12  22
  .05
1  2  24
Critical Value(s):
Test Statistic:
Decision:
Conclusion: 68

130. F-Test for 2 Variances Solution
/2 = .025
/2 = .025 69

131. F-Test for 2 Variances Solution
H0: 12 = 22
Ha: 12  22
  .05
1  2  24
Critical Value(s):
Test Statistic:
Decision:
Conclusion: 68

132. F-Test for 2 Variances Solution
H0: 12 = 22
Ha: 12  22
  .05
1  2  24
Critical Value(s):
Test Statistic:
Decision:
Conclusion:
Do not reject at  = .05 68

133. F-Test for 2 Variances Solution
H0: 12 = 22
Ha: 12  22
  .05
1  2  24
Critical Value(s):
Test Statistic:
Decision:
Conclusion:
Do not reject at  = .05
There is no evidence of a difference in variances 68

134. F-Test for Variances Thinking Challenge
You’re an analyst for the Light & Power Company. You want to compare the electricity consumption of single-family homes in 2 towns. You compute the following from a sample of homes:
Town 1 Town 2 Number
Mean $ 85 $ 68
Std Dev $ 30 $ 18
At the .05 level, is there evidence of a difference in variances between the two towns?
Allow students about 15 minutes to solve this.

135. F-Test for 2 Variances Solution*
H0: 12 = 22
Ha: 12  22
  .05
1  2  20
Critical Value(s):
Test Statistic:
Decision:
Conclusion:
Reject at  = .05
There is evidence of a difference in variances 72

136. Critical Values Solution*
/2 = .025
/2 = .025 73

137. Conclusion 1. Solved Hypothesis Testing Problems for Two Populations
Mean
Proportion
Variance
2. Distinguished Independent & Related Populations
3. Explained the F Distribution
As a result of this class, you will be able to ...

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End of Chapter
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