1. 8 - 1 © 2000 Prentice-Hall, Inc. Statistics for Business and Economics Inferences Based on a Single Sample: Tests of Hypothesis Chapter 8

2. 8 - 2 © 2000 Prentice-Hall, Inc. Learning Objectives 1.Distinguish Types of Hypotheses 2.Describe Hypothesis Testing Process 3.Explain p-Value Concept 4.Solve Hypothesis Testing Problems Based on a Single Sample 5.Explain Power of a Test

3. 8 - 3 © 2000 Prentice-Hall, Inc. Statistical Methods

4. 8 - 4 © 2000 Prentice-Hall, Inc. Hypothesis Testing Concepts

5. 8 - 5 © 2000 Prentice-Hall, Inc. Hypothesis Testing

6. 8 - 6 © 2000 Prentice-Hall, Inc. Hypothesis Testing Population

7. 8 - 7 © 2000 Prentice-Hall, Inc. Hypothesis Testing Population I believe the population mean age is 50 (hypothesis).

8. 8 - 8 © 2000 Prentice-Hall, Inc. Hypothesis Testing Population I believe the population mean age is 50 (hypothesis). Mean  X = 20 Random sample

9. 8 - 9 © 2000 Prentice-Hall, Inc. Hypothesis Testing Population I believe the population mean age is 50 (hypothesis). Mean  X = 20 Reject hypothesis! Not close. Random sample

10. 8 - 10 © 2000 Prentice-Hall, Inc. What’s a Hypothesis? 1.A Belief about a Population Parameter Parameter Is Population Mean, Proportion, Variance Parameter Is Population Mean, Proportion, Variance Must Be Stated Before Analysis Must Be Stated Before Analysis I believe the mean GPA of this class is 3.5! © 1984-1994 T/Maker Co.

11. 8 - 11 © 2000 Prentice-Hall, Inc. Null Hypothesis 1.What Is Tested 2.Has Serious Outcome If Incorrect Decision Made 3.Always Has Equality Sign: , , or  4.Designated H 0 (Pronounced H-oh) 5.Specified as H 0 :   Some Numeric Value Specified with = Sign Even if , or  Specified with = Sign Even if , or  Example, H 0 :   3 Example, H 0 :   3

12. 8 - 12 © 2000 Prentice-Hall, Inc. Alternative Hypothesis 1.Opposite of Null Hypothesis 2.Always Has Inequality Sign: , , or  3.Designated H a 4.Specified H a :  < Some Value Example, H a :  < 3 Example, H a :  < 3

13. 8 - 13 © 2000 Prentice-Hall, Inc. Identifying Hypotheses Steps 1.Example Problem: Test That the Population Mean Is Not 3 2.Steps State the Question Statistically (   3) State the Question Statistically (   3) State the Opposite Statistically (  = 3) State the Opposite Statistically (  = 3) Must Be Mutually Exclusive & Exhaustive Must Be Mutually Exclusive & Exhaustive Select the Alternative Hypothesis (   3) Select the Alternative Hypothesis (   3) Has the , Sign Has the , Sign State the Null Hypothesis (  = 3) State the Null Hypothesis (  = 3)

14. 8 - 14 © 2000 Prentice-Hall, Inc. State the question statistically:  = 12 State the opposite statistically:   12 Select the alternative hypothesis: H a :   12 State the null hypothesis: H 0 :  = 12 Is the population average amount of TV viewing 12 hours? What Are the Hypotheses?

15. 8 - 15 © 2000 Prentice-Hall, Inc. State the question statistically:   12 State the opposite statistically:  = 12 Select the alternative hypothesis: H a :   12 State the null hypothesis: H 0 :  = 12 Is the population average amount of TV viewing different from 12 hours? What Are the Hypotheses?

16. 8 - 16 © 2000 Prentice-Hall, Inc. State the question statistically:   20 State the opposite statistically:   20 Select the alternative hypothesis: H a :   20 State the null hypothesis: H 0 :   20 Is the average cost per hat less than or equal to $20? What Are the Hypotheses?

17. 8 - 17 © 2000 Prentice-Hall, Inc. State the question statistically:   25 State the opposite statistically:   25 Select the alternative hypothesis: H a :   25 State the null hypothesis: H 0 :   25 Is the average amount spent in the bookstore greater than $25? What Are the Hypotheses?

18. 8 - 18 © 2000 Prentice-Hall, Inc. Basic Idea

19. 8 - 19 © 2000 Prentice-Hall, Inc. Basic Idea H0H0H0H0 H0H0H0H0 Sampling Distribution

20. 8 - 20 © 2000 Prentice-Hall, Inc. Basic Idea Sampling Distribution It is unlikely that we would get a sample mean of this value... 2020 H0H0H0H0 H0H0H0H0

21. 8 - 21 © 2000 Prentice-Hall, Inc. 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 2020 H0H0H0H0 H0H0H0H0

22. 8 - 22 © 2000 Prentice-Hall, Inc. 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... therefore, we reject the hypothesis that  = 50. 2020 H0H0H0H0 H0H0H0H0

23. 8 - 23 © 2000 Prentice-Hall, Inc. Level of Significance 1.Probability 2.Defines Unlikely Values of Sample Statistic if Null Hypothesis Is True Called Rejection Region of Sampling Distribution Called Rejection Region of Sampling Distribution 3.Designated  (alpha) Typical Values Are.01,.05,.10 Typical Values Are.01,.05,.10 4.Selected by Researcher at Start

24. 8 - 24 © 2000 Prentice-Hall, Inc. Rejection Region (One-Tail Test)

25. 8 - 25 © 2000 Prentice-Hall, Inc. Rejection Region (One-Tail Test) Sampling Distribution 1 -  Level of Confidence

26. 8 - 26 © 2000 Prentice-Hall, Inc. Rejection Region (One-Tail Test) Sampling Distribution 1 -  Level of Confidence Observed sample statistic

27. 8 - 27 © 2000 Prentice-Hall, Inc. Rejection Region (One-Tail Test) Sampling Distribution 1 -  Level of Confidence

28. 8 - 28 © 2000 Prentice-Hall, Inc. Rejection Regions (Two-Tailed Test)

29. 8 - 29 © 2000 Prentice-Hall, Inc. Rejection Regions (Two-Tailed Test) Sampling Distribution 1 -  Level of Confidence

30. 8 - 30 © 2000 Prentice-Hall, Inc. Rejection Regions (Two-Tailed Test) Sampling Distribution 1 -  Level of Confidence Observed sample statistic

31. 8 - 31 © 2000 Prentice-Hall, Inc. Rejection Regions (Two-Tailed Test) Sampling Distribution 1 -  Level of Confidence

32. 8 - 32 © 2000 Prentice-Hall, Inc. Rejection Regions (Two-Tailed Test) Sampling Distribution 1 -  Level of Confidence

33. 8 - 33 © 2000 Prentice-Hall, Inc. Decision Making Risks

34. 8 - 34 © 2000 Prentice-Hall, Inc. Errors in Making Decision 1.Type I Error Reject True Null Hypothesis Reject True Null Hypothesis Has Serious Consequences Has Serious Consequences Probability of Type I Error Is  (Alpha) Probability of Type I Error Is  (Alpha) Called Level of Significance Called Level of Significance 2.Type II Error Do Not Reject False Null Hypothesis Do Not Reject False Null Hypothesis Probability of Type II Error Is  (Beta) Probability of Type II Error Is  (Beta)

35. 8 - 35 © 2000 Prentice-Hall, Inc. Decision Results H 0 : Innocent

36. 8 - 36 © 2000 Prentice-Hall, Inc. Decision Results H 0 : Innocent

37. 8 - 37 © 2000 Prentice-Hall, Inc.  &  Have an Inverse Relationship   You can’t reduce both errors simultaneously!

38. 8 - 38 © 2000 Prentice-Hall, Inc. Factors Affecting  1. True Value of Population Parameter Increases When Difference With Hypothesized Parameter Decreases Increases When Difference With Hypothesized Parameter Decreases 2.Significance Level,  Increases When  Decreases Increases When  Decreases 3.Population Standard Deviation,  Increases When  Increases Increases When  Increases 4.Sample Size, n Increases When n Decreases Increases When n Decreases

39. 8 - 39 © 2000 Prentice-Hall, Inc. Hypothesis Testing Steps

40. 8 - 40 © 2000 Prentice-Hall, Inc. H 0 Testing Steps

41. 8 - 41 © 2000 Prentice-Hall, Inc. H 0 Testing Steps n State H 0 n State H a Choose  Choose  n Choose n n Choose test

42. 8 - 42 © 2000 Prentice-Hall, Inc. H 0 Testing Steps n Set up critical values n Collect data n Compute test statistic n Make statistical decision n Express decision n State H 0 n State H a Choose  Choose  n Choose n n Choose test

43. 8 - 43 © 2000 Prentice-Hall, Inc. One Population Tests

44. 8 - 44 © 2000 Prentice-Hall, Inc. Two-Tailed Z Test of Mean (  Known)

45. 8 - 45 © 2000 Prentice-Hall, Inc. One Population Tests

46. 8 - 46 © 2000 Prentice-Hall, Inc. Two-Tailed Z Test for Mean (  Known) 1.Assumptions Population Is Normally Distributed Population Is Normally Distributed If Not Normal, Can Be Approximated by Normal Distribution (n  30) If Not Normal, Can Be Approximated by Normal Distribution (n  30) 2.Alternative Hypothesis Has  Sign

47. 8 - 47 © 2000 Prentice-Hall, Inc. Two-Tailed Z Test for Mean (  Known) 1.Assumptions Population Is Normally Distributed Population Is Normally Distributed If Not Normal, Can Be Approximated by Normal Distribution (n  30) If Not Normal, Can Be Approximated by Normal Distribution (n  30) 2.Alternative Hypothesis Has  Sign 3.Z-Test Statistic

48. 8 - 48 © 2000 Prentice-Hall, Inc. Two-Tailed Z Test Example Does an average box of cereal contain 368 grams of cereal? A random sample of 25 boxes showed  X = 372.5. The company has specified  to be 25 grams. Test at the.05 level. 368 gm.

49. 8 - 49 © 2000 Prentice-Hall, Inc. Two-Tailed Z Test Solution H 0 : H a :   n  Critical Value(s): Test Statistic: Decision:Conclusion:

50. 8 - 50 © 2000 Prentice-Hall, Inc. Two-Tailed Z Test Solution H 0 :  = 368 H a :   368   n  Critical Value(s): Test Statistic: Decision:Conclusion:

51. 8 - 51 © 2000 Prentice-Hall, Inc. Two-Tailed Z Test Solution H 0 :  = 368 H a :   368  .05 n  25 Critical Value(s): Test Statistic: Decision:Conclusion:

52. 8 - 52 © 2000 Prentice-Hall, Inc. Two-Tailed Z Test Solution H 0 :  = 368 H a :   368  .05 n  25 Critical Value(s): Test Statistic: Decision:Conclusion:

53. 8 - 53 © 2000 Prentice-Hall, Inc. Two-Tailed Z Test Solution H 0 :  = 368 H a :   368  .05 n  25 Critical Value(s): Test Statistic: Decision:Conclusion:

54. 8 - 54 © 2000 Prentice-Hall, Inc. Two-Tailed Z Test Solution H 0 :  = 368 H a :   368  .05 n  25 Critical Value(s): Test Statistic: Decision:Conclusion: Do not reject at  =.05

55. 8 - 55 © 2000 Prentice-Hall, Inc. Two-Tailed Z Test Solution H 0 :  = 368 H a :   368  .05 n  25 Critical Value(s): Test Statistic: Decision:Conclusion: Do not reject at  =.05 No evidence average is not 368

56. 8 - 56 © 2000 Prentice-Hall, Inc. 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?

57. 8 - 57 © 2000 Prentice-Hall, Inc. Two-Tailed Z Test Solution* H 0 : H a :  = n = Critical Value(s): Test Statistic: Decision:Conclusion:

58. 8 - 58 © 2000 Prentice-Hall, Inc. Two-Tailed Z Test Solution* H 0 :  = 70 H a :   70  = n = Critical Value(s): Test Statistic: Decision:Conclusion:

59. 8 - 59 © 2000 Prentice-Hall, Inc. Two-Tailed Z Test Solution* H 0 :  = 70 H a :   70  =.05 n = 36 Critical Value(s): Test Statistic: Decision:Conclusion:

60. 8 - 60 © 2000 Prentice-Hall, Inc. Two-Tailed Z Test Solution* H 0 :  = 70 H a :   70  =.05 n = 36 Critical Value(s): Test Statistic: Decision:Conclusion:

61. 8 - 61 © 2000 Prentice-Hall, Inc. Two-Tailed Z Test Solution* H 0 :  = 70 H a :   70  =.05 n = 36 Critical Value(s): Test Statistic: Decision:Conclusion:

62. 8 - 62 © 2000 Prentice-Hall, Inc. Two-Tailed Z Test Solution* H 0 :  = 70 H a :   70  =.05 n = 36 Critical Value(s): Test Statistic: Decision:Conclusion: Do not reject at  =.05

63. 8 - 63 © 2000 Prentice-Hall, Inc. Two-Tailed Z Test Solution* H 0 :  = 70 H a :   70  =.05 n = 36 Critical Value(s): Test Statistic: Decision:Conclusion: Do not reject at  =.05 No evidence average is not 70

64. 8 - 64 © 2000 Prentice-Hall, Inc. One-Tailed Z Test of Mean (  Known)

65. 8 - 65 © 2000 Prentice-Hall, Inc. One-Tailed Z Test for Mean (  Known) 1.Assumptions Population Is Normally Distributed Population Is Normally Distributed If Not Normal, Can Be Approximated by Normal Distribution (n  30) If Not Normal, Can Be Approximated by Normal Distribution (n  30) 2.Alternative Hypothesis Has Sign

66. 8 - 66 © 2000 Prentice-Hall, Inc. One-Tailed Z Test for Mean (  Known) 1.Assumptions Population Is Normally Distributed Population Is Normally Distributed If Not Normal, Can Be Approximated by Normal Distribution (n  30) If Not Normal, Can Be Approximated by Normal Distribution (n  30) 2.Alternative Hypothesis Has  or > Sign 3.Z-test Statistic

67. 8 - 67 © 2000 Prentice-Hall, Inc. One-Tailed Z Test for Mean Hypotheses

68. 8 - 68 © 2000 Prentice-Hall, Inc. One-Tailed Z Test for Mean Hypotheses H 0 :  =  0 H a :  < 0 Must be significantly below 

69. 8 - 69 © 2000 Prentice-Hall, Inc. One-Tailed Z Test for Mean Hypotheses H 0 :  =  0 H a :  < 0 H 0 :  =  0 H a :  > 0 Must be significantly below  Small values satisfy H 0. Don’t reject!

70. 8 - 70 © 2000 Prentice-Hall, Inc. One-Tailed Z Test Finding Critical Z

71. 8 - 71 © 2000 Prentice-Hall, Inc. One-Tailed Z Test Finding Critical Z What Is Z given  =.025?  =.025 

72. 8 - 72 © 2000 Prentice-Hall, Inc. One-Tailed Z Test Finding Critical Z.500 -.025.475 What Is Z given  =.025?  =.025  

73. 8 - 73 © 2000 Prentice-Hall, Inc. One-Tailed Z Test Finding Critical Z.500 -.025.475.06 1.9.4750 Standardized Normal Probability Table (Portion) What Is Z given  =.025?  =.025  

74. 8 - 74 © 2000 Prentice-Hall, Inc. One-Tailed Z Test Finding Critical Z.500 -.025.475.06 1.9.4750 Standardized Normal Probability Table (Portion) What Is Z given  =.025?  =.025   

75. 8 - 75 © 2000 Prentice-Hall, Inc. 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 = 372.5. The company has specified  to be 25 grams. Test at the.05 level. 368 gm.

76. 8 - 76 © 2000 Prentice-Hall, Inc. One-Tailed Z Test Solution H 0 : H a :  = n = Critical Value(s): Test Statistic: Decision:Conclusion:

77. 8 - 77 © 2000 Prentice-Hall, Inc. One-Tailed Z Test Solution H 0 :  = 368 H a :  > 368  = n = Critical Value(s): Test Statistic: Decision:Conclusion:

78. 8 - 78 © 2000 Prentice-Hall, Inc. One-Tailed Z Test Solution H 0 :  = 368 H a :  > 368  =.05 n = 25 Critical Value(s): Test Statistic: Decision:Conclusion:

79. 8 - 79 © 2000 Prentice-Hall, Inc. One-Tailed Z Test Solution H 0 :  = 368 H a :  > 368  =.05 n = 25 Critical Value(s): Test Statistic: Decision:Conclusion:

80. 8 - 80 © 2000 Prentice-Hall, Inc. One-Tailed Z Test Solution H 0 :  = 368 H a :  > 368  =.05 n = 25 Critical Value(s): Test Statistic: Decision:Conclusion:

81. 8 - 81 © 2000 Prentice-Hall, Inc. One-Tailed Z Test Solution H 0 :  = 368 H a :  > 368  =.05 n = 25 Critical Value(s): Test Statistic: Decision:Conclusion: Do not reject at  =.05

82. 8 - 82 © 2000 Prentice-Hall, Inc. One-Tailed Z Test Solution H 0 :  = 368 H a :  > 368  =.05 n = 25 Critical Value(s): Test Statistic: Decision:Conclusion: Do not reject at  =.05 No evidence average is more than 368

83. 8 - 83 © 2000 Prentice-Hall, Inc. 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?

84. 8 - 84 © 2000 Prentice-Hall, Inc. One-Tailed Z Test Solution* H 0 : H a :  = n = Critical Value(s): Test Statistic: Decision:Conclusion: Decision:Conclusion:

85. 8 - 85 © 2000 Prentice-Hall, Inc. One-Tailed Z Test Solution* H 0 :  = 32 H a :  < 32  = n = Critical Value(s): Test Statistic: Decision:Conclusion: Decision:Conclusion:

86. 8 - 86 © 2000 Prentice-Hall, Inc. One-Tailed Z Test Solution* H 0 :  = 32 H a :  < 32  =.01 n = 60 Critical Value(s): Test Statistic: Decision:Conclusion: Decision:Conclusion:

87. 8 - 87 © 2000 Prentice-Hall, Inc. One-Tailed Z Test Solution* H 0 :  = 32 H a :  < 32  =.01 n = 60 Critical Value(s): Test Statistic: Decision:Conclusion: Decision:Conclusion:

88. 8 - 88 © 2000 Prentice-Hall, Inc. One-Tailed Z Test Solution* H 0 :  = 32 H a :  < 32  =.01 n = 60 Critical Value(s): Test Statistic: Decision:Conclusion: Decision:Conclusion:

89. 8 - 89 © 2000 Prentice-Hall, Inc. One-Tailed Z Test Solution* H 0 :  = 32 H a :  < 32  =.01 n = 60 Critical Value(s): Test Statistic: Decision:Conclusion: Decision:Conclusion: Reject at  =.01

90. 8 - 90 © 2000 Prentice-Hall, Inc. One-Tailed Z Test Solution* H 0 :  = 32 H a :  < 32  =.01 n = 60 Critical Value(s): Test Statistic: Decision:Conclusion: Decision:Conclusion: Reject at  =.01 There is evidence average is less than 32

91. 8 - 91 © 2000 Prentice-Hall, Inc. Observed Significance Levels: p-Values

92. 8 - 92 © 2000 Prentice-Hall, Inc. p-Value 1.Probability of Obtaining a Test Statistic More Extreme (  or  than Actual Sample Value Given H 0 Is True 2.Called Observed Level of Significance Smallest Value of  H 0 Can Be Rejected Smallest Value of  H 0 Can Be Rejected 3.Used to Make Rejection Decision If p-Value  , Do Not Reject H 0 If p-Value  , Do Not Reject H 0 If p-Value < , Reject H 0 If p-Value < , Reject H 0

93. 8 - 93 © 2000 Prentice-Hall, Inc. Two-Tailed Z Test p-Value Example Does an average box of cereal contain 368 grams of cereal? A random sample of 25 boxes showed  X = 372.5. The company has specified  to be 25 grams. Find the p-Value. 368 gm.

94. 8 - 94 © 2000 Prentice-Hall, Inc. Two-Tailed Z Test p-Value Solution

95. 8 - 95 © 2000 Prentice-Hall, Inc. Two-Tailed Z Test p-Value Solution Z value of sample statistic (observed) 

96. 8 - 96 © 2000 Prentice-Hall, Inc. Two-Tailed Z Test p-Value Solution Z value of sample statistic (observed)  p-value is P(Z  -1.50 or Z  1.50)

97. 8 - 97 © 2000 Prentice-Hall, Inc. Two-Tailed Z Test p-Value Solution p-value is P(Z  -1.50 or Z  1.50) Z value of sample statistic (observed) 

98. 8 - 98 © 2000 Prentice-Hall, Inc. Two-Tailed Z Test p-Value Solution Z value of sample statistic (observed) From Z table: lookup 1.50.4332  p-value is P(Z  -1.50 or Z  1.50)

99. 8 - 99 © 2000 Prentice-Hall, Inc. Two-Tailed Z Test p-Value Solution Z value of sample statistic (observed) From Z table: lookup 1.50.4332 .5000 -.4332.0668  p-value is P(Z  -1.50 or Z  1.50)

100. 8 - 100 © 2000 Prentice-Hall, Inc. Two-Tailed Z Test p-Value Solution p-value is P(Z  -1.50 or Z  1.50) =.1336 Z value of sample statistic From Z table: lookup 1.50.4332.5000 -.4332.0668  

101. 8 - 101 © 2000 Prentice-Hall, Inc. Two-Tailed Z Test p-Value Solution 1/2 p-Value =.0668 1/2  =.025

102. 8 - 102 © 2000 Prentice-Hall, Inc. Two-Tailed Z Test p-Value Solution (p-Value =.1336)  (  =.05). Do not reject. 1/2 p-Value =.0668 1/2  =.025 Test statistic is in ‘Do not reject’ region

103. 8 - 103 © 2000 Prentice-Hall, Inc. One-Tailed Z Test p-Value Example Does an average box of cereal contain more than 368 grams of cereal? A random sample of 25 boxes showed  X = 372.5. The company has specified  to be 25 grams. Find the p-Value. 368 gm.

104. 8 - 104 © 2000 Prentice-Hall, Inc. One-Tailed Z Test p-Value Solution

105. 8 - 105 © 2000 Prentice-Hall, Inc. One-Tailed Z Test p-Value Solution  Z value of sample statistic

106. 8 - 106 © 2000 Prentice-Hall, Inc. One-Tailed Z Test p-Value Solution Use alternative hypothesis to find direction p-Value is P(Z  1.50) Z value of sample statistic  

107. 8 - 107 © 2000 Prentice-Hall, Inc. One-Tailed Z Test p-Value Solution Use alternative hypothesis to find direction p-Value is P(Z  1.50) Z value of sample statistic From Z table: lookup 1.50.4332   

108. 8 - 108 © 2000 Prentice-Hall, Inc. One-Tailed Z Test p-Value Solution Use alternative hypothesis to find direction p-Value is P(Z  1.50) Z value of sample statistic From Z table: lookup 1.50.4332 .5000 -.4332.0668   

109. 8 - 109 © 2000 Prentice-Hall, Inc. One-Tailed Z Test p-Value Solution Z value of sample statistic From Z table: lookup 1.50.4332 Use alternative hypothesis to find direction.5000 -.4332.0668   p-Value is P(Z  1.50) =.0668  

110. 8 - 110 © 2000 Prentice-Hall, Inc. One-Tailed Z Test p-Value Solution p-Value =.0668  =.05

111. 8 - 111 © 2000 Prentice-Hall, Inc. One-Tailed Z Test p-Value Solution (p-Value =.0668)  (  =.05). Do not reject. p-Value =.0668  =.05 Test statistic is in ‘Do not reject’ region

112. 8 - 112 © 2000 Prentice-Hall, Inc. p-Value 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. What is the value of the observed level of significance (p-Value)?

113. 8 - 113 © 2000 Prentice-Hall, Inc. p-Value Solution* Z value of sample statistic From Z table: lookup 2.65.4960 Use alternative hypothesis to find direction.5000 -.4960.0040    p-Value is P(Z  -2.65) =.004. p-Value < (  =.01). Reject H 0.

114. 8 - 114 © 2000 Prentice-Hall, Inc. Two-Tailed t Test of Mean (  Unknown)

115. 8 - 115 © 2000 Prentice-Hall, Inc. One Population Tests

116. 8 - 116 © 2000 Prentice-Hall, Inc. t Test for Mean (  Unknown) 1.Assumptions Population Is Normally Distributed Population Is Normally Distributed If Not Normal, Only Slightly Skewed & Large Sample (n  30) Taken If Not Normal, Only Slightly Skewed & Large Sample (n  30) Taken 2.Parametric Test Procedure

117. 8 - 117 © 2000 Prentice-Hall, Inc. t Test for Mean (  Unknown) 1.Assumptions Population Is Normally Distributed Population Is Normally Distributed If Not Normal, Only Slightly Skewed & Large Sample (n  30) Taken If Not Normal, Only Slightly Skewed & Large Sample (n  30) Taken 2.Parametric Test Procedure 3.t Test Statistic

118. 8 - 118 © 2000 Prentice-Hall, Inc. Two-Tailed t Test Finding Critical t Values

119. 8 - 119 © 2000 Prentice-Hall, Inc. Two-Tailed t Test Finding Critical t Values Given: n = 3;  =.10

120. 8 - 120 © 2000 Prentice-Hall, Inc. Two-Tailed t Test Finding Critical t Values  /2 =.05  Given: n = 3;  =.10

121. 8 - 121 © 2000 Prentice-Hall, Inc. Two-Tailed t Test Finding Critical t Values  /2 =.05   Given: n = 3;  =.10 df = n - 1 = 2

122. 8 - 122 © 2000 Prentice-Hall, Inc. Two-Tailed t Test Finding Critical t Values Critical Values of t Table (Portion)  /2 =.05    Given: n = 3;  =.10 df = n - 1 = 2

123. 8 - 123 © 2000 Prentice-Hall, Inc. Two-Tailed t Test Finding Critical t Values Critical Values of t Table (Portion)  /2 =.05     Given: n = 3;  =.10 df = n - 1 = 2

124. 8 - 124 © 2000 Prentice-Hall, Inc. 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 372.5 & a standard deviation of 12 grams. Test at the.05 level. 368 gm.

125. 8 - 125 © 2000 Prentice-Hall, Inc. Two-Tailed t Test Solution H 0 : H a :  = df = Critical Value(s): Test Statistic: Decision:Conclusion:

126. 8 - 126 © 2000 Prentice-Hall, Inc. Two-Tailed t Test Solution H 0 :  = 368 H a :   368  = df = Critical Value(s): Test Statistic: Decision:Conclusion:

127. 8 - 127 © 2000 Prentice-Hall, Inc. Two-Tailed t Test Solution H 0 :  = 368 H a :   368  =.05 df = 36 - 1 = 35 Critical Value(s): Test Statistic: Decision:Conclusion:

128. 8 - 128 © 2000 Prentice-Hall, Inc. Two-Tailed t Test Solution H 0 :  = 368 H a :   368  =.05 df = 36 - 1 = 35 Critical Value(s): Test Statistic: Decision:Conclusion:

129. 8 - 129 © 2000 Prentice-Hall, Inc. Two-Tailed t Test Solution H 0 :  = 368 H a :   368  =.05 df = 36 - 1 = 35 Critical Value(s): Test Statistic: Decision:Conclusion:

130. 8 - 130 © 2000 Prentice-Hall, Inc. Two-Tailed t Test Solution H 0 :  = 368 H a :   368  =.05 df = 36 - 1 = 35 Critical Value(s): Test Statistic: Decision:Conclusion: Reject at  =.05

131. 8 - 131 © 2000 Prentice-Hall, Inc. Two-Tailed t Test Solution H 0 :  = 368 H a :   368  =.05 df = 36 - 1 = 35 Critical Value(s): Test Statistic: Decision:Conclusion: Reject at  =.05 There is evidence pop. average is not 368

132. 8 - 132 © 2000 Prentice-Hall, Inc. 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 3.238 lb. with a standard deviation of.117 lb. At the.01 level, is the manufacturer correct? 3.25 lb.

133. 8 - 133 © 2000 Prentice-Hall, Inc. Two-Tailed t Test Solution* H 0 : H a :   df  Critical Value(s): Test Statistic: Decision:Conclusion:

134. 8 - 134 © 2000 Prentice-Hall, Inc. Two-Tailed t Test Solution* H 0 :  = 3.25 H a :   3.25   df  Critical Value(s): Test Statistic: Decision:Conclusion:

135. 8 - 135 © 2000 Prentice-Hall, Inc. Two-Tailed t Test Solution* H 0 :  = 3.25 H a :   3.25  .01 df  64 - 1 = 63 Critical Value(s): Test Statistic: Decision:Conclusion:

136. 8 - 136 © 2000 Prentice-Hall, Inc. Two-Tailed t Test Solution* H 0 :  = 3.25 H a :   3.25  .01 df  64 - 1 = 63 Critical Value(s): Test Statistic: Decision:Conclusion:

137. 8 - 137 © 2000 Prentice-Hall, Inc. Two-Tailed t Test Solution* H 0 :  = 3.25 H a :   3.25  .01 df  64 - 1 = 63 Critical Value(s): Test Statistic: Decision:Conclusion:

138. 8 - 138 © 2000 Prentice-Hall, Inc. Two-Tailed t Test Solution* H 0 :  = 3.25 H a :   3.25  .01 df  64 - 1 = 63 Critical Value(s): Test Statistic: Decision:Conclusion: Do not reject at  =.01

139. 8 - 139 © 2000 Prentice-Hall, Inc. Two-Tailed t Test Solution* H 0 :  = 3.25 H a :   3.25  .01 df  64 - 1 = 63 Critical Value(s): Test Statistic: Decision:Conclusion: Do not reject at  =.01 There is no evidence average is not 3.25

140. 8 - 140 © 2000 Prentice-Hall, Inc. One-Tailed t Test of Mean (  Unknown)

141. 8 - 141 © 2000 Prentice-Hall, Inc. 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 138.47 & a standard deviation of 2.66. Assume a normal distribution. Test at the.05 level.

142. 8 - 142 © 2000 Prentice-Hall, Inc. One-Tailed t Test Solution H 0 : H a :  = df = Critical Value(s): Test Statistic: Decision:Conclusion:

143. 8 - 143 © 2000 Prentice-Hall, Inc. One-Tailed t Test Solution H 0 :  = 140 H a :  < 140  = df = Critical Value(s): Test Statistic: Decision:Conclusion:

144. 8 - 144 © 2000 Prentice-Hall, Inc. One-Tailed t Test Solution H 0 :  = 140 H a :  < 140  =.05 df = 20 - 1 = 19 Critical Value(s): Test Statistic: Decision:Conclusion:

145. 8 - 145 © 2000 Prentice-Hall, Inc. One-Tailed t Test Solution H 0 :  = 140 H a :  < 140  =.05 df = 20 - 1 = 19 Critical Value(s): Test Statistic: Decision:Conclusion:

146. 8 - 146 © 2000 Prentice-Hall, Inc. One-Tailed t Test Solution H 0 :  = 140 H a :  < 140  =.05 df = 20 - 1 = 19 Critical Value(s): Test Statistic: Decision:Conclusion:

147. 8 - 147 © 2000 Prentice-Hall, Inc. One-Tailed t Test Solution H 0 :  = 140 H a :  < 140  =.05 df = 20 - 1 = 19 Critical Value(s): Test Statistic: Decision:Conclusion: Reject at  =.05

148. 8 - 148 © 2000 Prentice-Hall, Inc. One-Tailed t Test Solution H 0 :  = 140 H a :  < 140  =.05 df = 20 - 1 = 19 Critical Value(s): Test Statistic: Decision:Conclusion: Reject at  =.05 There is evidence pop. average is less than 140

149. 8 - 149 © 2000 Prentice-Hall, Inc. 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: 8 11 0 4 7 8 10 5 8 3. At the.05 level, is there evidence that the average bear sales per store is more than 5 ($ 00)?

150. 8 - 150 © 2000 Prentice-Hall, Inc. One-Tailed t Test Solution* H 0 : H a :  = df = Critical Value(s): Test Statistic: Decision:Conclusion:

151. 8 - 151 © 2000 Prentice-Hall, Inc. One-Tailed t Test Solution* H 0 :  = 5 H a :  > 5  = df = Critical Value(s): Test Statistic: Decision:Conclusion:

152. 8 - 152 © 2000 Prentice-Hall, Inc. One-Tailed t Test Solution* H 0 :  = 5 H a :  > 5  =.05 df = 10 - 1 = 9 Critical Value(s): Test Statistic: Decision:Conclusion:

153. 8 - 153 © 2000 Prentice-Hall, Inc. One-Tailed t Test Solution* H 0 :  = 5 H a :  > 5  =.05 df = 10 - 1 = 9 Critical Value(s): Test Statistic: Decision:Conclusion:

154. 8 - 154 © 2000 Prentice-Hall, Inc. One-Tailed t Test Solution* H 0 :  = 5 H a :  > 5  =.05 df = 10 - 1 = 9 Critical Value(s): Test Statistic: Decision:Conclusion:

155. 8 - 155 © 2000 Prentice-Hall, Inc. One-Tailed t Test Solution* H 0 :  = 5 H a :  > 5  =.05 df = 10 - 1 = 9 Critical Value(s): Test Statistic: Decision:Conclusion: Do not reject at  =.05

156. 8 - 156 © 2000 Prentice-Hall, Inc. One-Tailed t Test Solution* H 0 :  = 5 H a :  > 5  =.05 df = 10 - 1 = 9 Critical Value(s): Test Statistic: Decision:Conclusion: Do not reject at  =.05 There is no evidence average is more than 5

157. 8 - 157 © 2000 Prentice-Hall, Inc. Z Test of Proportion

158. 8 - 158 © 2000 Prentice-Hall, Inc. Data Types

159. 8 - 159 © 2000 Prentice-Hall, Inc. Qualitative Data 1.Qualitative Random Variables Yield Responses That Classify e.g., Gender (Male, Female) e.g., Gender (Male, Female) 2.Measurement Reflects # in Category 3.Nominal or Ordinal Scale 4.Examples Do You Own Savings Bonds? Do You Own Savings Bonds? Do You Live On-Campus or Off-Campus? Do You Live On-Campus or Off-Campus?

160. 8 - 160 © 2000 Prentice-Hall, Inc. Proportions 1.Involve Qualitative Variables 2.Fraction or % of Population in a Category 3.If Two Qualitative Outcomes, Binomial Distribution Possess or Don’t Possess Characteristic Possess or Don’t Possess Characteristic

161. 8 - 161 © 2000 Prentice-Hall, Inc. Proportions 1.Involve Qualitative Variables 2.Fraction or % of Population in a Category 3.If Two Qualitative Outcomes, Binomial Distribution Possess or Don’t Possess Characteristic Possess or Don’t Possess Characteristic 4.Sample Proportion (p) ^

162. 8 - 162 © 2000 Prentice-Hall, Inc. p 1.Approximated by Normal Distribution Excludes 0 or n Excludes 0 or n 2.Mean 3.Standard Error Sampling Distribution of Proportion Sampling Distribution where p 0 = Population Proportion  p ^ p n  1.0.1.2.3.0.2.4.6.81.0 P ^ P(P ^ ) 00

163. 8 - 163 © 2000 Prentice-Hall, Inc.  Z = 0  z  = 1 Z Standardizing Sampling Distribution of Proportion Sampling Distribution Standardized Normal Distribution P ^  P  P Z p pp pp n ^ p p ^ ^        ()1 ^ ^ ^ 0 0 0

164. 8 - 164 © 2000 Prentice-Hall, Inc. One Population Tests

165. 8 - 165 © 2000 Prentice-Hall, Inc. One-Sample Z Test for Proportion

166. 8 - 166 © 2000 Prentice-Hall, Inc. One-Sample Z Test for Proportion 1.Assumptions Two Categorical Outcomes Two Categorical Outcomes Population Follows Binomial Distribution Population Follows Binomial Distribution Normal Approximation Can Be Used Normal Approximation Can Be Used Does Not Contain 0 or n Does Not Contain 0 or n

167. 8 - 167 © 2000 Prentice-Hall, Inc. One-Sample Z Test for Proportion 1.Assumptions Two Categorical Outcomes Two Categorical Outcomes Population Follows Binomial Distribution Population Follows Binomial Distribution Normal Approximation Can Be Used Normal Approximation Can Be Used Does Not Contain 0 or n Does Not Contain 0 or n 2.Z-test statistic for proportion Hypothesized population proportion

168. 8 - 168 © 2000 Prentice-Hall, Inc. One-Proportion Z Test Example The present packaging system produces 10% defective cereal boxes. Using a new system, a random sample of 200 boxes had  11 defects. Does the new system produce fewer defects? Test at the.05 level.

169. 8 - 169 © 2000 Prentice-Hall, Inc. One-Proportion Z Test Solution H 0 : H a :  = n = Critical Value(s): Test Statistic: Decision:Conclusion:

170. 8 - 170 © 2000 Prentice-Hall, Inc. One-Proportion Z Test Solution H 0 : p =.10 H a : p <.10  = n = Critical Value(s): Test Statistic: Decision:Conclusion:

171. 8 - 171 © 2000 Prentice-Hall, Inc. One-Proportion Z Test Solution H 0 : p = .10 H a : p <.10  =.05 n = 200 Critical Value(s): Test Statistic: Decision:Conclusion:

172. 8 - 172 © 2000 Prentice-Hall, Inc. One-Proportion Z Test Solution H 0 : p =.10 H a : p <.10  =.05 n = 200 Critical Value(s): Test Statistic: Decision:Conclusion:

173. 8 - 173 © 2000 Prentice-Hall, Inc. One-Proportion Z Test Solution H 0 : p =.10 H a : p <.10  =.05 n = 200 Critical Value(s): Test Statistic: Decision:Conclusion:

174. 8 - 174 © 2000 Prentice-Hall, Inc. One-Proportion Z Test Solution H 0 : p =.10 H a : p <.10  =.05 n = 200 Critical Value(s): Test Statistic: Decision:Conclusion: Reject at  =.05

175. 8 - 175 © 2000 Prentice-Hall, Inc. One-Proportion Z Test Solution H 0 : p =.10 H a : p <.10  =.05 n = 200 Critical Value(s): Test Statistic: Decision:Conclusion: Reject at  =.05 There is evidence new system < 10% defective

176. 8 - 176 © 2000 Prentice-Hall, Inc. One-Proportion Z Test Thinking Challenge You’re an accounting manager. A year-end audit showed 4% of transactions had errors. You implement new procedures. A random sample of 500 transactions had 25 errors. Has the proportion of incorrect transactions changed at the.05 level?

177. 8 - 177 © 2000 Prentice-Hall, Inc. One-Proportion Z Test Solution* H 0 : H a :  = n = Critical Value(s): Test Statistic: Decision:Conclusion:

178. 8 - 178 © 2000 Prentice-Hall, Inc. One-Proportion Z Test Solution* H 0 : p =.04 H a : p .04  = n = Critical Value(s): Test Statistic: Decision:Conclusion:

179. 8 - 179 © 2000 Prentice-Hall, Inc. One-Proportion Z Test Solution* H 0 : p =.04 H a : p .04  =.05 n = 500 Critical Value(s): Test Statistic: Decision:Conclusion:

180. 8 - 180 © 2000 Prentice-Hall, Inc. One-Proportion Z Test Solution* H 0 : p =.04 H a : p .04  =.05 n = 500 Critical Value(s): Test Statistic: Decision:Conclusion:

181. 8 - 181 © 2000 Prentice-Hall, Inc. One-Proportion Z Test Solution* H 0 : p =.04 H a : p .04  =.05 n = 500 Critical Value(s): Test Statistic: Decision:Conclusion:

182. 8 - 182 © 2000 Prentice-Hall, Inc. One-Proportion Z Test Solution* H 0 : p =.04 H a : p .04  =.05 n = 500 Critical Value(s): Test Statistic: Decision:Conclusion: Do not reject at  =.05

183. 8 - 183 © 2000 Prentice-Hall, Inc. One-Proportion Z Test Solution* H 0 : p =.04 H a : p .04  =.05 n = 500 Critical Value(s): Test Statistic: Decision:Conclusion: Do not reject at  =.05 There is evidence proportion is still 4%

184. 8 - 184 © 2000 Prentice-Hall, Inc. Chi-Square (  2 ) Test of Variance

185. 8 - 185 © 2000 Prentice-Hall, Inc. One Population Tests

186. 8 - 186 © 2000 Prentice-Hall, Inc. Chi-Square (  2 ) Test for Variance 1.Tests One Population Variance or Standard Deviation 2.Assumes Population Is Approximately Normally Distributed 3.Null Hypothesis Is H 0 :  2 =  0 2

187. 8 - 187 © 2000 Prentice-Hall, Inc. Chi-Square (  2 ) Test for Variance 1.Tests One Population Variance or Standard Deviation 2.Assumes Population Is Approximately Normally Distributed 3.Null Hypothesis Is H 0 :  2 =  0 2 4.Test Statistic Hypothesized Pop. Variance Sample Variance   2 2 2 1)  (nS 0

188. 8 - 188 © 2000 Prentice-Hall, Inc. Chi-Square (  2 ) Distribution

189. 8 - 189 © 2000 Prentice-Hall, Inc. Finding Critical Value Example What is the critical  2 value given: H a :  2 > 0.7 n = 3  =.05?

190. 8 - 190 © 2000 Prentice-Hall, Inc. Finding Critical Value Example  2 Table (Portion) What is the critical  2 value given: H a :  2 > 0.7 n = 3  =.05?

191. 8 - 191 © 2000 Prentice-Hall, Inc. Finding Critical Value Example  =.05  2 Table (Portion) What is the critical  2 value given: H a :  2 > 0.7 n = 3  =.05?

192. 8 - 192 © 2000 Prentice-Hall, Inc. Finding Critical Value Example  =.05  2 Table (Portion) What is the critical  2 value given: H a :  2 > 0.7 n = 3  =.05?

193. 8 - 193 © 2000 Prentice-Hall, Inc. Finding Critical Value Example  =.05  2 Table (Portion) What is the critical  2 value given: H a :  2 > 0.7 n = 3  =.05?

194. 8 - 194 © 2000 Prentice-Hall, Inc. Finding Critical Value Example  =.05  2 Table (Portion) What is the critical  2 value given: H a :  2 > 0.7 n = 3  =.05?

195. 8 - 195 © 2000 Prentice-Hall, Inc. Finding Critical Value Example  =.05  2 Table (Portion) df= n - 1 = 2 What is the critical  2 value given: H a :  2 > 0.7 n = 3  =.05?

196. 8 - 196 © 2000 Prentice-Hall, Inc. Finding Critical Value Example  =.05  2 Table (Portion) df= n - 1 = 2 What is the critical  2 value given: H a :  2 > 0.7 n = 3  =.05?

197. 8 - 197 © 2000 Prentice-Hall, Inc. Finding Critical Value Example  =.05  2 Table (Portion) df= n - 1 = 2 What is the critical  2 value given: H a :  2 > 0.7 n = 3  =.05?

198. 8 - 198 © 2000 Prentice-Hall, Inc. Finding Critical Value Example  =.05  2 Table (Portion) df= n - 1 = 2 What is the critical  2 value given: H a :  2 > 0.7 n = 3  =.05?

199. 8 - 199 © 2000 Prentice-Hall, Inc. Finding Critical Value Example  =.05  2 Table (Portion) df= n - 1 = 2 What is the critical  2 value given: H a :  2 > 0.7 n = 3  =.05?

200. 8 - 200 © 2000 Prentice-Hall, Inc. Finding Critical Value Example What Do You Do If the Rejection Region Is on the Left? What is the critical  2 value given: H a :  2 < 0.7 n = 3  =.05?

201. 8 - 201 © 2000 Prentice-Hall, Inc. Finding Critical Value Example  2 Table (Portion) What is the critical  2 value given: H a :  2 < 0.7 n = 3  =.05?

202. 8 - 202 © 2000 Prentice-Hall, Inc. What is the critical  2 value given: H a :  2 < 0.7 n = 3  =.05? Finding Critical Value Example  =.05  2 Table (Portion) Reject

203. 8 - 203 © 2000 Prentice-Hall, Inc. What is the critical  2 value given: H a :  2 < 0.7 n = 3  =.05? Finding Critical Value Example  =.05  2 Table (Portion) Reject Upper Tail Area for Lower Critical Value = 1-.05 =.95

204. 8 - 204 © 2000 Prentice-Hall, Inc. What is the critical  2 value given: H a :  2 < 0.7 n = 3  =.05? Finding Critical Value Example  =.05  2 Table (Portion) Reject Upper Tail Area for Lower Critical Value = 1-.05 =.95

205. 8 - 205 © 2000 Prentice-Hall, Inc. What is the critical  2 value given: H a :  2 < 0.7 n = 3  =.05? Finding Critical Value Example  =.05  2 Table (Portion) Reject Upper Tail Area for Lower Critical Value = 1-.05 =.95 df= n - 1 = 2

206. 8 - 206 © 2000 Prentice-Hall, Inc. What is the critical  2 value given: H a :  2 < 0.7 n = 3  =.05? Finding Critical Value Example  =.05  2 Table (Portion) Reject Upper Tail Area for Lower Critical Value = 1-.05 =.95 df= n - 1 = 2

207. 8 - 207 © 2000 Prentice-Hall, Inc. What is the critical  2 value given: H a :  2 < 0.7 n = 3  =.05? Finding Critical Value Example  =.05  2 Table (Portion) df= n - 1 = 2 Upper Tail Area for Lower Critical Value = 1-.05 =.95 Reject

208. 8 - 208 © 2000 Prentice-Hall, Inc. Chi-Square (  2 ) Test Example Is the variation in boxes of cereal, measured by the variance, equal to 15 grams? A random sample of 25 boxes had a standard deviation of 17.7 grams. Test at the.05 level.

209. 8 - 209 © 2000 Prentice-Hall, Inc.  2 0 Chi-Square (  2 ) Test Solution H 0 : H a :  = df = Critical Value(s): Test Statistic: Decision:Conclusion:

210. 8 - 210 © 2000 Prentice-Hall, Inc.  2 0 Chi-Square (  2 ) Test Solution H 0 :  2 = 15 H a :  2  15  = df = Critical Value(s): Test Statistic: Decision:Conclusion:

211. 8 - 211 © 2000 Prentice-Hall, Inc.  2 0 Chi-Square (  2 ) Test Solution H 0 :  2 = 15 H a :  2  15  =.05 df = 25 - 1 = 24 Critical Value(s): Test Statistic: Decision:Conclusion:

212. 8 - 212 © 2000 Prentice-Hall, Inc.  2 0 Chi-Square (  2 ) Test Solution H 0 :  2 = 15 H a :  2  15  =.05 df = 25 - 1 = 24 Critical Value(s): Test Statistic: Decision:Conclusion:  /2 =.025

213. 8 - 213 © 2000 Prentice-Hall, Inc.  2 039.36412.401 Chi-Square (  2 ) Test Solution H 0 :  2 = 15 H a :  2  15  =.05 df = 25 - 1 = 24 Critical Value(s): Test Statistic: Decision:Conclusion:  /2 =.025

214. 8 - 214 © 2000 Prentice-Hall, Inc.  2 039.36412.401 Chi-Square (  2 ) Test Solution H 0 :  2 = 15 H a :  2  15  =.05 df = 25 - 1 = 24 Critical Value(s): Test Statistic: Decision:Conclusion:  /2 =.025   2 2 2 2 2 1) (25 - 1)177 15 3342      (nS 0..

215. 8 - 215 © 2000 Prentice-Hall, Inc.  2 039.36412.401 Chi-Square (  2 ) Test Solution H 0 :  2 = 15 H a :  2  15  =.05 df = 25 - 1 = 24 Critical Value(s): Test Statistic: Decision:Conclusion: Do Not Reject at  =.05  /2 =.025   2 2 2 2 2 1) (25 - 1)177 15 3342      (nS 0..

216. 8 - 216 © 2000 Prentice-Hall, Inc.  2 039.36412.401 Chi-Square (  2 ) Test Solution H 0 :  2 = 15 H a :  2  15  =.05 df = 25 - 1 = 24 Critical Value(s): Test Statistic: Decision:Conclusion: Do Not Reject at  =.05 There Is No Evidence  2 Is Not 15  /2 =.025   2 2 2 2 2 1) (25 - 1)177 15 3342      (nS 0..

217. 8 - 217 © 2000 Prentice-Hall, Inc. Calculating Type II Error Probabilities

218. 8 - 218 © 2000 Prentice-Hall, Inc. Power of Test 1.Probability of Rejecting False H 0 Correct Decision Correct Decision 2.Designated 1 -  3.Used in Determining Test Adequacy 4.Affected by True Value of Population Parameter True Value of Population Parameter Significance Level  Significance Level  Standard Deviation & Sample Size n Standard Deviation & Sample Size n

219. 8 - 219 © 2000 Prentice-Hall, Inc. X  0 = 368 = 368 Reject Do Not Reject Finding Power Step 1 Hypothesis: H 0 :  0  368 H 1 :  0 < 368  =.05  n = 15/  25  Draw

220. 8 - 220 © 2000 Prentice-Hall, Inc. X  1 = 360 = 360 X  0 = 368 = 368 Reject Do Not Reject Finding Power Steps 2 & 3 Hypothesis: H 0 :  0  368 H 1 :  0 < 368 ‘True’ Situation:  1 = 360  =.05  n = 15/  25    Draw Draw Specify  1- 

221. 8 - 221 © 2000 Prentice-Hall, Inc. X  1 = 360 = 360363.065 X  0 = 368 = 368 Reject Do Not Reject Finding Power Step 4 Hypothesis: H 0 :  0  368 H 1 :  0 < 368 ‘True’ Situation:  1 = 360  =.05  n = 15/  25     Draw Draw Specify

222. 8 - 222 © 2000 Prentice-Hall, Inc. X  1 = 360 = 360363.065 X  0 = 368 = 368 Reject Do Not Reject Finding Power Step 5 Hypothesis: H 0 :  0  368 H 1 :  0 < 368 ‘True’ Situation:  1 = 360  =.05  n = 15/  25  =.154 1-  =.846     Draw Draw Specify  Z Table

223. 8 - 223 © 2000 Prentice-Hall, Inc. Power Curves PowerPower Power Possible True Values for  1 H 0 :   0 H 0 :   0 H 0 :  =  0   = 368 in Example

224. 8 - 224 © 2000 Prentice-Hall, Inc. 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

225. End of Chapter Any blank slides that follow are blank intentionally.