1. Statistical Methods 5

2. Hypothesis Testing Concepts9

3. Hypothesis Testing

4. Hypothesis Testing
Population       

5. Hypothesis Testing        Population
I believe the population mean age is 50 (hypothesis).
Population       

6. Hypothesis Testing        Population  
I believe the population mean age is 50 (hypothesis).
Population      
Random sample 
Mean X = 20  

7. Reject hypothesis! Not close.
Hypothesis Testing
I believe the population mean age is 50 (hypothesis).
Reject hypothesis! Not close.
Population      
Random sample 
Mean X = 20  

8. What’s a Hypothesis? 1. A Belief about a Population Parameter
Parameter Is Population Mean, Proportion, Variance
Must Be Stated Before Analysis
I believe the mean GPA of this class is 3.5!
© T/Maker Co.

9. Null Hypothesis 1. What Is Tested
2. Has Serious Outcome If Incorrect Decision Made
3. Designated H0 (Pronounced H-nought)
4. Specified as H0:   Some Numeric Value
Specified with = Sign Even if , or 
Example, H0:   3

10. Alternative Hypothesis
1. Opposite of Null Hypothesis
2. Always Has Inequality Sign: ,, or 
3. Designated Ha
4. Specified Ha:  < Some Value
Example, Ha:  < 3
 will lead to two-sided tests
<, > will lead to one-sided tests

11. Identifying Hypotheses Steps
1. Example Problem: Test That the Population Mean Is Not 3
2. Steps
State the Question Statistically (  3)
State the Opposite Statistically ( = 3)
Must Be Mutually Exclusive & Exhaustive
Select the Alternative Hypothesis (  3)
Has the , <, or > Sign
State the Null Hypothesis ( = 3)

12. What Are the Hypotheses?
Is the population average amount of TV viewing 12 hours?
State the question statistically:  = 12
State the opposite statistically:   12
Select the alternative hypothesis: Ha:   12
State the null hypothesis: H0:  = 12

13. What Are the Hypotheses?
Is the population average amount of TV viewing different from 12 hours?
State the question statistically:   12
State the opposite statistically:  = 12
Select the alternative hypothesis: Ha:   12
State the null hypothesis: H0:  = 12

14. What Are the Hypotheses?
Is the average cost per hat less than or equal to $20?
State the question statistically:   20
State the opposite statistically:   20
Select the alternative hypothesis: Ha:   20
State the null hypothesis: H0:   20

15. What Are the Hypotheses?
Is the average amount spent in the bookstore greater than $25?
State the question statistically:   25
State the opposite statistically:   25
Select the alternative hypothesis: Ha:   25
State the null hypothesis: H0:   25

16. Basic Idea

17. Sampling Distribution
Basic Idea
Sampling Distribution H0

18. Sampling Distribution
Basic Idea
Sampling Distribution
It is unlikely that we would get a sample mean of this value ... 20 H0

19. Sampling Distribution
Basic Idea
Sampling Distribution
It is unlikely that we would get a sample mean of this value ...
... if in fact this were the population mean 20 H0

20. Sampling Distribution
Basic Idea
Sampling Distribution
It is unlikely that we would get a sample mean of this value ...
... therefore, we reject the hypothesis that  = 50.
... if in fact this were the population mean 20 H0

21. Level of Significance 1. Probability
2. Defines Unlikely Values of Sample Statistic if Null Hypothesis Is True
Called Rejection Region of Sampling Distribution
3. Designated (alpha)
Typical Values Are .01, .05, .10
4. Selected by Researcher at Start

22. Rejection Region (One-Tail Test)
Rejection region does NOT include critical value.

23. Rejection Region (One-Tail Test)
Sampling Distribution
Level of Confidence
Rejection region does NOT include critical value.
1 - 

24. Rejection Region (One-Tail Test)
Sampling Distribution
Level of Confidence
Rejection region does NOT include critical value.
1 - 
Observed sample statistic

25. Rejection Region (One-Tail Test)
Sampling Distribution
Level of Confidence
Rejection region does NOT include critical value.
1 - 

26. Rejection Regions (Two-Tailed Test)
Rejection region does NOT include critical value.

27. Rejection Regions (Two-Tailed Test)
Sampling Distribution
Level of Confidence
Rejection region does NOT include critical value.
1 - 

28. Rejection Regions (Two-Tailed Test)
Sampling Distribution
Level of Confidence
Rejection region does NOT include critical value.
1 - 
Observed sample statistic

29. Rejection Regions (Two-Tailed Test)
Sampling Distribution
Level of Confidence
Rejection region does NOT include critical value.
1 - 

30. Rejection Regions (Two-Tailed Test)
Sampling Distribution
Level of Confidence
Rejection region does NOT include critical value.
1 - 

31. Hypothesis Testing Steps9

32. H0 Testing Steps

33. H0 Testing Steps
State H0
State Ha
Choose 
Choose n
Choose test

34. H0 Testing Steps State H0 State Ha Choose  Choose n Choose test
Set up critical values
Collect data
Compute test statistic
Make statistical decision
Express decision

35. One Population Tests

36. Two-Tailed Z Test of Mean ( Known)9

37. One Population Tests

38. Two-Tailed Z Test for Mean ( Known)
1. Assumptions
Population Is Normally Distributed
If Not Normal, Can Be Approximated by Normal Distribution (n  30)
2. Alternative Hypothesis Has  Sign

39. Two-Tailed Z Test for Mean ( Known)
1. Assumptions
Population Is Normally Distributed
If Not Normal, Can Be Approximated by Normal Distribution (n  30)
2. Alternative Hypothesis Has  Sign
3. Z-Test Statistic

40. Two-Tailed Z Test Example
Does an average box of cereal contain 368 grams of cereal? A random sample of 25 boxes showedX = The company has specified  to be 25 grams. Test at the .05 level.
368 gm.

41. Two-Tailed Z Test Solution
H0:
Ha:
 
n 
Critical Value(s):
Test Statistic:
Decision:
Conclusion:

42. Two-Tailed Z Test Solution
H0:  = 368
Ha:   368
 
n 
Critical Value(s):
Test Statistic:
Decision:
Conclusion:

43. Two-Tailed Z Test Solution
H0:  = 368
Ha:   368
  .05
n  25
Critical Value(s):
Test Statistic:
Decision:
Conclusion:

44. Two-Tailed Z Test Solution
H0:  = 368
Ha:   368
  .05
n  25
Critical Value(s):
Test Statistic:
Decision:
Conclusion:

45. Two-Tailed Z Test Solution
H0:  = 368
Ha:   368
  .05
n  25
Critical Value(s):
Test Statistic:
Decision:
Conclusion:

46. Two-Tailed Z Test Solution
H0:  = 368
Ha:   368
  .05
n  25
Critical Value(s):
Test Statistic:
Decision:
Conclusion:
Do not reject at  = .05

47. Two-Tailed Z Test Solution
H0:  = 368
Ha:   368
  .05
n  25
Critical Value(s):
Test Statistic:
Decision:
Conclusion:
Do not reject at  = .05
No evidence average is not 368

48. Two-Tailed Z Test Thinking Challenge
You’re a Q/C inspector. You want to find out if a new machine is making electrical cords to customer specification: average breaking strength of 70 lb. with  = 3.5 lb. You take a sample of 36 cords & compute a sample mean of 69.7 lb. At the .05 level, is there evidence that the machine is not meeting the average breaking strength?

49. Two-Tailed Z Test Solution*
H0:
Ha:
 =
n =
Critical Value(s):
Test Statistic:
Decision:
Conclusion:

50. Two-Tailed Z Test Solution*
H0:  = 70
Ha:   70
 =
n =
Critical Value(s):
Test Statistic:
Decision:
Conclusion:

51. Two-Tailed Z Test Solution*
H0:  = 70
Ha:   70
 = .05
n = 36
Critical Value(s):
Test Statistic:
Decision:
Conclusion:

52. Two-Tailed Z Test Solution*
H0:  = 70
Ha:   70
 = .05
n = 36
Critical Value(s):
Test Statistic:
Decision:
Conclusion:

53. Two-Tailed Z Test Solution*
H0:  = 70
Ha:   70
 = .05
n = 36
Critical Value(s):
Test Statistic:
Decision:
Conclusion:

54. Two-Tailed Z Test Solution*
H0:  = 70
Ha:   70
 = .05
n = 36
Critical Value(s):
Test Statistic:
Decision:
Conclusion:
Do not reject at  = .05

55. Two-Tailed Z Test Solution*
H0:  = 70
Ha:   70
 = .05
n = 36
Critical Value(s):
Test Statistic:
Decision:
Conclusion:
Do not reject at  = .05
No evidence average is not 70

56. One-Tailed Z Test of Mean ( Known)9

57. One-Tailed Z Test for Mean ( Known)
1. Assumptions
Population Is Normally Distributed
If Not Normal, Can Be Approximated by Normal Distribution (n  30)
2. Alternative Hypothesis Has < or > Sign

58. One-Tailed Z Test for Mean ( Known)
1. Assumptions
Population Is Normally Distributed
If Not Normal, Can Be Approximated by Normal Distribution (n  30)
2. Alternative Hypothesis Has or > Sign
3. Z-test Statistic

59. One-Tailed Z Test for Mean Hypotheses

60. One-Tailed Z Test for Mean Hypotheses
H0:=0 Ha: < 0
Must be significantly below 

61. One-Tailed Z Test for Mean Hypotheses
H0:=0 Ha: < 0
H0:=0 Ha: > 0
Must be significantly below 
Small values satisfy H0 . Don’t reject!

62. One-Tailed Z Test Finding Critical Z

63. One-Tailed Z Test Finding Critical Z
What Is Z given  = .025?
 = .025 

64. One-Tailed Z Test Finding Critical Z
What Is Z given  = .025? 
 = .025 

65. One-Tailed Z Test Finding Critical Z
What Is Z given  = .025?
Standardized Normal Probability Table (Portion)  
.06
 = .025 
1.9
.4750

66. One-Tailed Z Test Finding Critical Z
What Is Z given  = .025?
Standardized Normal Probability Table (Portion)  
.06
 = .025  
1.9
.4750

67. One-Tailed Z Test Example
Does an average box of cereal contain more than 368 grams of cereal? A random sample of 25 boxes showedX = The company has specified  to be 25 grams. Test at the .05 level.
368 gm.

68. One-Tailed Z Test Solution
H0:
Ha:
 =
n =
Critical Value(s):
Test Statistic:
Decision:
Conclusion:

69. One-Tailed Z Test Solution
H0:  = 368
Ha:  > 368
 =
n =
Critical Value(s):
Test Statistic:
Decision:
Conclusion:

70. One-Tailed Z Test Solution
H0:  = 368
Ha:  > 368
 = .05
n = 25
Critical Value(s):
Test Statistic:
Decision:
Conclusion:

71. One-Tailed Z Test Solution
H0:  = 368
Ha:  > 368
 = .05
n = 25
Critical Value(s):
Test Statistic:
Decision:
Conclusion:

72. One-Tailed Z Test Solution
H0:  = 368
Ha:  > 368
 = .05
n = 25
Critical Value(s):
Test Statistic:
Decision:
Conclusion:

73. One-Tailed Z Test Solution
H0:  = 368
Ha:  > 368
 = .05
n = 25
Critical Value(s):
Test Statistic:
Decision:
Conclusion:
Do not reject at  = .05

74. One-Tailed Z Test Solution
H0:  = 368
Ha:  > 368
 = .05
n = 25
Critical Value(s):
Test Statistic:
Decision:
Conclusion:
Do not reject at  = .05
No evidence average is more than 368

75. One-Tailed Z Test Thinking Challenge
You’re an analyst for Ford. You want to find out if the average miles per gallon of Escorts is at least 32 mpg. Similar models have a standard deviation of 3.8 mpg. You take a sample of 60 Escorts & compute a sample mean of 30.7 mpg. At the .01 level, is there evidence that the miles per gallon is at least 32?

76. One-Tailed Z Test Solution*
H0:
Ha:
 =
n =
Critical Value(s):
Test Statistic:
Decision:
Conclusion:

77. One-Tailed Z Test Solution*
H0:  = 32
Ha:  < 32
 =
n =
Critical Value(s):
Test Statistic:
Decision:
Conclusion:

78. One-Tailed Z Test Solution*
H0:  = 32
Ha:  < 32
 = .01
n = 60
Critical Value(s):
Test Statistic:
Decision:
Conclusion:

79. One-Tailed Z Test Solution*
H0:  = 32
Ha:  < 32
 = .01
n = 60
Critical Value(s):
Test Statistic:
Decision:
Conclusion:

80. One-Tailed Z Test Solution*
H0:  = 32
Ha:  < 32
 = .01
n = 60
Critical Value(s):
Test Statistic:
Decision:
Conclusion:

81. One-Tailed Z Test Solution*
H0:  = 32
Ha:  < 32
 = .01
n = 60
Critical Value(s):
Test Statistic:
Decision:
Conclusion:
Reject at  = .01

82. One-Tailed Z Test Solution*
H0:  = 32
Ha:  < 32
 = .01
n = 60
Critical Value(s):
Test Statistic:
Decision:
Conclusion:
Reject at  = .01
There is evidence average is less than 32

83. Decision Making Risks 9

84. Errors in Making Decision
1. Type I Error
Reject True Null Hypothesis
Has Serious Consequences
Probability of Type I Error Is (Alpha)
Called Level of Significance
2. Type II Error
Do Not Reject False Null Hypothesis
Probability of Type II Error Is (Beta)

85. Decision Results
H0: Innocent

86. Decision Results
H0: Innocent

87.  &  Have an Inverse Relationship
You can’t reduce both errors simultaneously!  

88. Factors Affecting  1. True Value of Population Parameter
 Increases When Difference With Hypothesized Parameter Decreases
2. Significance Level, 
 Increases When Decreases
3. Population Standard Deviation, 
 Increases When  Increases
4. Sample Size, n
 Increases When n Decreases

89. Two-Tailed t Test of Mean ( Unknown)9

90. One Population Tests

91. t Test for Mean ( Unknown)
1. Assumptions
Population Is Normally Distributed
If Not Normal, Only Slightly Skewed & Large Sample (n  30) Taken
2. Parametric Test Procedure

92. t Test for Mean ( Unknown)
1. Assumptions
Population Is Normally Distributed
If Not Normal, Only Slightly Skewed & Large Sample (n  30) Taken
2. Parametric Test Procedure
3. t Test Statistic

93. Two-Tailed t Test Finding Critical t Values

94. Two-Tailed t Test Finding Critical t Values
Given: n = 3;  = .10

95. Two-Tailed t Test Finding Critical t Values
Given: n = 3;  = .10 
 /2 = .05
 /2 = .05

96. Two-Tailed t Test Finding Critical t Values
Given: n = 3;  = .10 
df = n - 1 = 2 
 /2 = .05
 /2 = .05

97. Two-Tailed t Test Finding Critical t Values
Given: n = 3;  = .10
Critical Values of t Table (Portion)  
df = n - 1 = 2 
 /2 = .05
 /2 = .05

98. Two-Tailed t Test Finding Critical t Values
Given: n = 3;  = .10
Critical Values of t Table (Portion)  
df = n - 1 = 2 
 /2 = .05
 /2 = .05 

99. Two-Tailed t Test Example
Does an average box of cereal contain 368 grams of cereal? A random sample of 36 boxes had a mean of & a standard deviation of 12 grams. Test at the .05 level.
368 gm.

100. Two-Tailed t Test Solution
H0:
Ha:
 =
df =
Critical Value(s):
Test Statistic:
Decision:
Conclusion:

101. Two-Tailed t Test Solution
H0:  = 368
Ha:   368
 =
df =
Critical Value(s):
Test Statistic:
Decision:
Conclusion:

102. Two-Tailed t Test Solution
H0:  = 368
Ha:   368
 = .05
df = = 35
Critical Value(s):
Test Statistic:
Decision:
Conclusion:

103. Two-Tailed t Test Solution
H0:  = 368
Ha:   368
 = .05
df = = 35
Critical Value(s):
Test Statistic:
Decision:
Conclusion:

104. Two-Tailed t Test Solution
H0:  = 368
Ha:   368
 = .05
df = = 35
Critical Value(s):
Test Statistic:
Decision:
Conclusion:

105. Two-Tailed t Test Solution
H0:  = 368
Ha:   368
 = .05
df = = 35
Critical Value(s):
Test Statistic:
Decision:
Conclusion:
Reject at  = .05

106. Two-Tailed t Test Solution
H0:  = 368
Ha:   368
 = .05
df = = 35
Critical Value(s):
Test Statistic:
Decision:
Conclusion:
Reject at  = .05
There is evidence pop. average is not 368

107. Two-Tailed t Test Thinking Challenge
You work for the FTC. A manufacturer of detergent claims that the mean weight of detergent is 3.25 lb. You take a random sample of 64 containers. You calculate the sample average to be lb. with a standard deviation of .117 lb. At the .01 level, is the manufacturer correct?
Allow students about 10 minutes to finish this.
3.25 lb.

108. Two-Tailed t Test Solution*
H0:
Ha:
 
df 
Critical Value(s):
Test Statistic:
Decision:
Conclusion:

109. Two-Tailed t Test Solution*
H0:  = 3.25
Ha:   3.25
 
df 
Critical Value(s):
Test Statistic:
Decision:
Conclusion:

110. Two-Tailed t Test Solution*
H0:  = 3.25
Ha:   3.25
  .01
df  = 63
Critical Value(s):
Test Statistic:
Decision:
Conclusion:

111. Two-Tailed t Test Solution*
H0:  = 3.25
Ha:   3.25
  .01
df  = 63
Critical Value(s):
Test Statistic:
Decision:
Conclusion:

112. Two-Tailed t Test Solution*
H0:  = 3.25
Ha:   3.25
  .01
df  = 63
Critical Value(s):
Test Statistic:
Decision:
Conclusion:

113. Two-Tailed t Test Solution*
H0:  = 3.25
Ha:   3.25
  .01
df  = 63
Critical Value(s):
Test Statistic:
Decision:
Conclusion:
Do not reject at  = .01

114. Two-Tailed t Test Solution*
H0:  = 3.25
Ha:   3.25
  .01
df  = 63
Critical Value(s):
Test Statistic:
Decision:
Conclusion:
Do not reject at  = .01
There is no evidence average is not 3.25

115. One-Tailed t Test of Mean ( Unknown)9

116. One-Tailed t Test Example
Is the average capacity of batteries at least 140 ampere-hours? A random sample of 20 batteries had a mean of & a standard deviation of Assume a normal distribution. Test at the .05 level.

117. One-Tailed t Test Solution
H0:
Ha:
 =
df =
Critical Value(s):
Test Statistic:
Decision:
Conclusion:

118. One-Tailed t Test Solution
H0:  = 140
Ha:  < 140
 =
df =
Critical Value(s):
Test Statistic:
Decision:
Conclusion:

119. One-Tailed t Test Solution
H0:  = 140
Ha:  < 140
 = .05
df = = 19
Critical Value(s):
Test Statistic:
Decision:
Conclusion:

120. One-Tailed t Test Solution
H0:  = 140
Ha:  < 140
 = .05
df = = 19
Critical Value(s):
Test Statistic:
Decision:
Conclusion:

121. One-Tailed t Test Solution
H0:  = 140
Ha:  < 140
 = .05
df = = 19
Critical Value(s):
Test Statistic:
Decision:
Conclusion:

122. One-Tailed t Test Solution
H0:  = 140
Ha:  < 140
 = .05
df = = 19
Critical Value(s):
Test Statistic:
Decision:
Conclusion:
Reject at  = .05

123. One-Tailed t Test Solution
H0:  = 140
Ha:  < 140
 = .05
df = = 19
Critical Value(s):
Test Statistic:
Decision:
Conclusion:
Reject at  = .05
There is evidence pop. average is less than 140

124. One-Tailed t Test Thinking Challenge
You’re a marketing analyst for Wal-Mart. Wal-Mart had teddy bears on sale last week. The weekly sales ($ 00) of bears sold in 10 stores was: At the .05 level, is there evidence that the average bear sales per store is more than 5 ($ 00)?
Assume that the population is normally distributed.
Allow students about 10 minutes to solve this.

125. One-Tailed t Test Solution*
H0:
Ha:
 =
df =
Critical Value(s):
Test Statistic:
Decision:
Conclusion:
Note: More than 5 have been sold (6.4), but not enough to be significant.

126. One-Tailed t Test Solution*
H0:  = 5
Ha:  > 5
 =
df =
Critical Value(s):
Test Statistic:
Decision:
Conclusion:
Note: More than 5 have been sold (6.4), but not enough to be significant.

127. One-Tailed t Test Solution*
H0:  = 5
Ha:  > 5
 = .05
df = = 9
Critical Value(s):
Test Statistic:
Decision:
Conclusion:
Note: More than 5 have been sold (6.4), but not enough to be significant.

128. One-Tailed t Test Solution*
H0:  = 5
Ha:  > 5
 = .05
df = = 9
Critical Value(s):
Test Statistic:
Decision:
Conclusion:
Note: More than 5 have been sold (6.4), but not enough to be significant.

129. One-Tailed t Test Solution*
H0:  = 5
Ha:  > 5
 = .05
df = = 9
Critical Value(s):
Test Statistic:
Decision:
Conclusion:
Note: More than 5 have been sold (6.4), but not enough to be significant.

130. One-Tailed t Test Solution*
H0:  = 5
Ha:  > 5
 = .05
df = = 9
Critical Value(s):
Test Statistic:
Decision:
Conclusion:
Note: More than 5 have been sold (6.4), but not enough to be significant.
Do not reject at  = .05

131. One-Tailed t Test Solution*
H0:  = 5
Ha:  > 5
 = .05
df = = 9
Critical Value(s):
Test Statistic:
Decision:
Conclusion:
Note: More than 5 have been sold (6.4), but not enough to be significant.
Do not reject at  = .05
There is no evidence average is more than 5

132. One Population Tests

133. Confidence Intervals, Hypothesis Tests, and p-values
All Start with Known Sampling Distribution for
Confidence Interval
Pr( > given distance from ) =
Draw an interval of size around actual
1- is the confidence level
P-Value
Assume true mean
Pr( > measured distance) = p
For one-sided value, no absolute value
Hypothesis test
Pick , If p < , reject the null hypothesis

134. Calculating Type II Error Probabilities9

135. Power of Test 1. Probability of Rejecting False H0 2. Designated 1 - 
Correct Decision
2. Designated 1 - 
3. Used in Determining Test Adequacy
4. Affected by
True Value of Population Parameter
Significance Level 
Standard Deviation & Sample Size n

136. Finding Power Step 1   X Reject n = 15/25
Hypothesis: H0: 0  368 H1: 0 < 368
Do Not
Draw
Reject
 = .05 
= 368  X

137. Finding Power Steps 2 & 3     X  X  1- Reject n = 15/25
Hypothesis: H0: 0  368 H1: 0 < 368
Do Not
Draw
Reject
 = .05 
= 368  X 
‘True’ Situation: 1 = 360
Draw  
1-
Specify 
= 360  X 1

138. Finding Power Step 4      X  X Reject n = 15/25
Hypothesis: H0: 0  368 H1: 0 < 368
Do Not
Draw
Reject
 = .05 
= 368  X 
‘True’ Situation: 1 = 360
Draw  
Specify 
= 360  X 1

139. Finding Power Step 5       X  X Reject n = 15/25
Hypothesis: H0: 0  368 H1: 0 < 368
Do Not
Draw
Reject
 = .05 
= 368  X 
‘True’ Situation: 1 = 360
Draw
 = .154   
1- =.846
Specify
Z Table 
= 360  X 1

140. Power Curves H0:  0 H0:  0 H0:  =0 Power Power
Possible True Values for 1
Possible True Values for 1
H0:  =0
Power
 = 368 in Example
Possible True Values for 1

141. Conclusion 1. Distinguished Types of Hypotheses
2. Described Hypothesis Testing Process
3. Explained p-Value Concept
4. Solved Hypothesis Testing Problems Based on a Single Sample
5. Explained Power of a Test
As a result of this class, you will be able to ...

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End of Chapter
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