1. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. Chapter Inferences on Two Samples 11

2. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. Section Inference about Two Population Proportions 11.1

3. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. Objectives 1.Distinguish between independent and dependent sampling 2.Test hypotheses regarding two proportions from independent samples 3.Construct and interpret confidence intervals for the difference between two population proportions 4.Test hypotheses regarding two proportions from dependent samples 5.Determine the sample size necessary for estimating the difference between two population proportions 11-3

4. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. Objective 1 Distinguish between Independent and Dependent Sampling 11-4

5. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. A sampling method is independent when the individuals selected for one sample do not dictate which individuals are to be in a second sample. A sampling method is dependent when the individuals selected to be in one sample are used to determine the individuals to be in the second sample. 11-5

6. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. Dependent samples are often referred to as matched-pairs samples. It is possible for an individual to be matched against him- or herself. 11-6

7. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. For each of the following, determine whether the sampling method is independent or dependent. a)A researcher wants to know whether the price of a one night stay at a Holiday Inn Express is less than the price of a one night stay at a Red Roof Inn. She randomly selects 8 towns where the location of the hotels is close to each other and determines the price of a one night stay. b)A researcher wants to know whether the “state” quarters (introduced in 1999) have a mean weight that is different from “traditional” quarters. He randomly selects 18 “state” quarters and 16 “traditional” quarters and compares their weights. Parallel Example 1: Distinguish between Independent and Dependent Sampling 11-7

8. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. a)The sampling method is dependent since the 8 Holiday Inn Express hotels can be matched with one of the 8 Red Roof Inn hotels by town. b)The sampling method is independent since the “state” quarters which were sampled had no bearing on which “traditional” quarters were sampled. Solution 11-8

9. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. Objective 2 Test Hypotheses Regarding Two Population Proportions from Independent Samples 11-9

10. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. Suppose that a simple random sample of size n 1 is taken from a population where x 1 of the individuals have a specified characteristic, and a simple random sample of size n 2 is independently taken from a different population where x 2 of the individuals have a specified characteristic. The sampling distribution of, where and, is approximately normal, with mean and standard deviation provided that and and each sample size is no more than 5% of the population size. Sampling Distribution of the Difference between Two Proportions (Independent Sample) 11-10

11. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. The standardized version of is then written as which has an approximate standard normal distribution. Sampling Distribution of the Difference between Two Proportions 11-11

12. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. The best point estimate of p is called the pooled estimate of p, denoted, where Test statistic for Comparing Two Population Proportions 11-12

13. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. To test hypotheses regarding two population proportions, p 1 and p 2, we can use the steps that follow, provided that:  the samples are independently obtained using simple random sampling,  and and  n 1 ≤ 0.05N 1 and n 2 ≤ 0.05N 2 (the sample size is no more than 5% of the population size); this requirement ensures the independence necessary for a binomial experiment. Hypothesis Test Regarding the Difference between Two Population Proportions 11-13

14. Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Step 1: Determine the null and alternative hypotheses. The hypotheses can be structured in one of three ways: 11-14

15. Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Step 2: Select a level of significance, α, based on the seriousness of making a Type I error. 11-15

16. Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Step 3: Compute the test statistic where Classical Approach 11-16

17. Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Use Table V to determine the critical value. Classical Approach 11-17

18. Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Classical Approach (critical value) Two-Tailed 11-18

19. Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Classical Approach (critical value) Left-Tailed 11-19

20. Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Classical Approach (critical value) Right-Tailed 11-20

21. Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Step 4: Compare the critical value with the test statistic: Classical Approach 11-21

22. Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Step 4: Use Table V to estimate the P-value.. P-Value Approach 11-22

23. Copyright © 2013, 2010 and 2007 Pearson Education, Inc. P-Value Approach Two-Tailed 11-23

24. Copyright © 2013, 2010 and 2007 Pearson Education, Inc. P-Value Approach Left-Tailed 11-24

25. Copyright © 2013, 2010 and 2007 Pearson Education, Inc. P-Value Approach Right-Tailed 11-25

26. Copyright © 2013, 2010 and 2007 Pearson Education, Inc. By Hand Step 3: Compute the test statistic where P-Value Approach 11-26

27. Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Use Table V to determine the P-Value. P-Value Approach 11-27

28. Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Two-Tailed P-Value Approach 11-28

29. Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Left-Tailed P-Value Approach 11-29

30. Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Right-Tailed P-Value Approach 11-30

31. Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Technology Step 3: Use a statistical spreadsheet or calculator with statistical capabilities to obtain the P-value. The directions for obtaining the P-value using the TI-83/84 Plus graphing calculator, Excel, MINITAB, and StatCrunch are in the Technology Step-by-Step in the text. P-Value Approach 11-31

32. Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Step 4: If P-value < α, reject the null hypothesis. P-Value Approach 11-32

33. Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Step 5: State the conclusion. 11-33

34. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. An economist believes that the percentage of urban households with Internet access is greater than the percentage of rural households with Internet access. He obtains a random sample of 800 urban households and finds that 338 of them have Internet access. He obtains a random sample of 750 rural households and finds that 292 of them have Internet access. Test the economist’s claim at the α = 0.05 level of significance. Parallel Example 1: Testing Hypotheses Regarding Two Population Proportions 11-34

35. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. We must first verify that the requirements are satisfied: 1.The samples are simple random samples that were obtained independently. 2. x 1 =338, n 1 =800, x 2 =292 and n 2 =750, so 3.The sample sizes are less than 5% of the population size. Solution 11-35

36. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. Step 1: We want to determine whether the percentage of urban households with Internet access is greater than the percentage of rural households with Internet access. So, H 0 : p 1 = p 2 versus H 1 : p 1 > p 2 or, equivalently, H 0 : p 1 - p 2 =0 versus H 1 : p 1 - p 2 > 0 Step 2: The level of significance is α = 0.05. Solution 11-36

37. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. Step 3: The pooled estimate of is: The test statistic is: Solution 11-37

38. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. This is a right-tailed test with α = 0.05. The critical value is z 0.05 =1.645. Solution: Classical Approach 11-38

39. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. Step 4: Since the test statistic, z 0 =1.33 is less than the critical value z.05 =1.645, we fail to reject the null hypothesis. Solution: Classical Approach 11-39

40. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. Because this is a right-tailed test, the P-value is the area under the normal to the right of the test statistic z 0 =1.33. That is, P-value = P(Z > 1.33) ≈ 0.09. Solution: P-Value Approach 11-40

41. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. Step 4: Since the P-value is greater than the level of significance α = 0.05, we fail to reject the null hypothesis. Solution: P-Value Approach 11-41

42. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. Step 5: There is insufficient evidence at the α = 0.05 level to conclude that the percentage of urban households with Internet access is greater than the percentage of rural households with Internet access. Solution 11-42

43. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. Objective 3 Construct and Interpret Confidence Intervals for the Difference between Two Population Proportions 11-43

44. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. To construct a (1 – α)100% confidence interval for the difference between two population proportions, the following requirements must be satisfied: 1.the samples are obtained independently using simple random sampling, 2., and 3.n 1 ≤ 0.05N 1 and n 2 ≤ 0.05N 2 (the sample size is no more than 5% of the population size); this requirement ensures the independence necessary for a binomial experiment. Constructing a (1 – α)100% Confidence Interval for the Difference between Two Population Proportions 11-44

45. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. Provided that these requirements are met, a (1 – α)100% confidence interval for p 1 –p 2 is given by Lower bound: Upper bound: Constructing a (1 – α)100% Confidence Interval for the Difference between Two Population Proportions 11-45

46. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. An economist obtains a random sample of 800 urban households and finds that 338 of them have Internet access. He obtains a random sample of 750 rural households and finds that 292 of them have Internet access. Find a 99% confidence interval for the difference between the proportion of urban households that have Internet access and the proportion of rural households that have Internet access. Parallel Example 3: Constructing a Confidence Interval for the Difference between Two Population Proportions 11-46

47. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. We have already verified the requirements for constructing a confidence interval for the difference between two population proportions in the previous example. Recall Solution 11-47

48. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. Thus, Lower bound = Upper bound = Solution 11-48

49. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. We are 99% confident that the difference between the proportion of urban households that have Internet access and the proportion of rural households that have Internet access is between –0.03 and 0.10. Since the confidence interval contains 0, we are unable to conclude that the proportion of urban households with Internet access is greater than the proportion of rural households with Internet access. Solution 11-49

50. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. Objective 4 Test Hypotheses Regarding Two Proportions from Dependent Samples 11-50

51. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. McNemar’s Test is a test that can be used to compare two proportions with matched-pairs data (i.e., dependent samples) 11-51

52. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. Testing a Hypothesis Regarding the Difference of Two Population Proportions: Dependent Samples To test hypotheses regarding two population proportions p 1 and p 2, where the samples are dependent, arrange the data in a contingency table as follows: Treatment A Treatment B SuccessFailure Successf 11 f 12 Failuref 21 f 22 11-52

53. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. Testing a Hypothesis Regarding the Difference of Two Population Proportions: Dependent Samples We can use the steps that follow provided that: 1. the samples are dependent and are obtained randomly and 2.the total number of observations where the outcomes differ must be greater than or equal to 10. That is, f 12 + f 21 ≥ 10. 11-53

54. Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Step 1: Determine the null and alternative hypotheses. H 0 : the proportions between the two populations are equal (p 1 = p 2 ) H 1 : the proportions between the two populations differ (p 1 ≠ p 2 ) 11-54

55. Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Step 2: Select a level of significance, α, based on the seriousness of making a Type I error. 11-55

56. Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Step 3: Compute the test statistic 11-56

57. Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Step 3: Use Table V to determine the critical value. This is a two tailed test. However, z 0 is always positive, so we only need to find the right critical value z α/2. Classical Approach 11-57

58. Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Classical Approach 11-58

59. Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Step 4: If z 0 > z α/2, reject the null hypothesis. Classical Approach 11-59

60. Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Step 3 (continued): Use Table V to determine the P-value. Because z 0 is always positive, we find the area to the right of z 0 and then double this area (since this is a two- tailed test). P-Value Approach 11-60

61. Copyright © 2013, 2010 and 2007 Pearson Education, Inc. P-Value Approach 11-61

62. Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Step 4: If P-value < α, reject the null hypothesis. P-Value Approach 11-62

63. Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Step 5: State the conclusion. 11-63

64. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. A recent General Social Survey asked the following two questions of a random sample of 1483 adult Americans under the hypothetical scenario that the government suspected that a terrorist act was about to happen: Do you believe the authorities should have the right to tap people’s telephone conversations? Do you believe the authorities should have the right to detain people for as long as they want without putting them on trial? Parallel Example 4: Analyzing the Difference of Two Proportions from Matched-Pairs Data 11-64

65. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. The results of the survey are shown below: Do the proportions who agree with each scenario differ significantly? Use the α = 0.05 level of significance. Parallel Example 4: Analyzing the Difference of Two Proportions from Matched-Pairs Data Detain Tap Phone AgreeDisagree Agree572237 Disagree224450 11-65

66. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. The sample proportion of individuals who believe that the authorities should be able to tap phones is. The sample proportion of individuals who believe that the authorities should have the right to detain people is. We want to determine whether the difference in sample proportions is due to sampling error or to the fact that the population proportions differ. Solution 11-66

67. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. The samples are dependent and were obtained randomly. The total number of individuals who agree with one scenario, but disagree with the other is 237 + 224 = 461, which is greater than 10. We can proceed with McNemar’s Test. Step 1: The hypotheses are as follows H 0 : the proportions between the two populations are equal (p T = p D ) H 1 : the proportions between the two populations differ (p T ≠ p D ) Step 2: The level of significance is α = 0.05. Solution 11-67

68. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. Step 3: The test statistic is: Solution 11-68

69. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. The critical value with an α = 0.05 level of significance is z 0.025 = 1.96. Solution: Classical Approach 11-69

70. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. Step 4: Since the test statistic, z 0 = 0.56 is less than the critical value z.025 = 1.96, we fail to reject the null hypothesis. Solution: Classical Approach 11-70

71. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. The P-value is two times the area under the normal curve to the right of the test statistic z 0 =0.56. That is, P-value = 2P(Z > 0.56) ≈ 0.5754. Solution: P-Value Approach 11-71

72. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. Step 4: Since the P-value is greater than the level of significance α = 0.05, we fail to reject the null hypothesis. Solution: P-Value Approach 11-72

73. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. Step 5: There is insufficient evidence at the α = 0.05 level to conclude that there is a difference in the proportion of adult Americans who believe it is okay to phone tap versus detaining people for as long as they want without putting them on trial in the event that the government believed a terrorist plot was about to happen. Solution 11-73

74. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. Objective 5 Determine the Sample Size Necessary for Estimating the Difference between Two Population Proportions within a Specified Margin of Error 11-74

75. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. Sample Size for Estimating p 1 – p 2 The sample size required to obtain a (1 – α)100% confidence interval with a margin of error, E, is given by rounded up to the next integer, if prior estimates of p 1 and p 2,, are available. If prior estimates of p 1 and p 2 are unavailable, the sample size is rounded up to the next integer. 11-75

76. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. A doctor wants to estimate the difference in the proportion of 15-19 year old mothers that received prenatal care and the proportion of 30-34 year old mothers that received prenatal care. What sample size should be obtained if she wished the estimate to be within 2 percentage points with 95% confidence assuming: a)she uses the results of the National Vital Statistics Report results in which 98% of the 15-19 year old mothers received prenatal care and 99.2% of 30-34 year old mothers received prenatal care. b)she does not use any prior estimates. Parallel Example 5: Determining Sample Size 11-76

77. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. We have E = 0.02 and z α/2 = z 0.025 = 1.96. a)Letting, The doctor must sample 265 randomly selected 15-19 year old mothers and 265 randomly selected 30-34 year old mothers. Solution 11-77

78. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. b)Without prior estimates of p 1 and p 2, the sample size is The doctor must sample 4802 randomly selected 15-19 year old mothers and 4802 randomly selected 30-34 year old mothers. Note that having prior estimates of p 1 and p 2 reduces the number of mothers that need to be surveyed. Solution 11-78

79. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. Section Inference about Two Means: Dependent Samples 11.2

80. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. Objectives 1.Test hypotheses regarding matched-pairs data 2.Construct and interpret confidence intervals about the population mean difference of matched-pairs data 11-80

81. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. Objective 1 Test Hypotheses Regarding Matched-Pairs Data 11-81

82. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. “In Other Words” Statistical inference methods on matched-pairs data use the same methods as inference on a single population mean, except that the differences are analyzed. 11-82

83. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. To test hypotheses regarding the mean difference of matched-pairs data, the following must be satisfied: the sample is obtained using simple random sampling the sample data are matched pairs, the differences are normally distributed with no outliers or the sample size, n, is large (n ≥ 30). Testing Hypotheses Regarding the Difference of Two Means Using a Matched-Pairs Design 11-83

84. Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Step 1: Determine the null and alternative hypotheses. The hypotheses can be structured in one of three ways, where μ d is the population mean difference of the matched-pairs data. 11-84

85. Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Step 2: Select a level of significance, α, based on the seriousness of making a Type I error. 11-85

86. Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Step 3: Compute the test statistic which approximately follows Student’s t-distribution with n – 1 degrees of freedom. The values of and s d are the mean and standard deviation of the differenced data. Classical Approach 11-86

87. Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Use Table VI to determine the critical value using n – 1 degrees of freedom. Classical Approach 11-87

88. Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Classical Approach (critical value) Two-Tailed 11-88

89. Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Classical Approach (critical value) Left-Tailed 11-89

90. Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Classical Approach (critical value) Right-Tailed 11-90

91. Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Step 4: Compare the critical value with the test statistic: Classical Approach 11-91

92. Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Step 3: Compute the test statistic which approximately follows Student’s t-distribution with n – 1 degrees of freedom. The values of and s d are the mean and standard deviation of the differenced data. P-Value Approach 11-92

93. Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Use Table VI to determine the P-value using n – 1 degrees of freedom. P-Value Approach 11-93

94. Copyright © 2013, 2010 and 2007 Pearson Education, Inc. P-Value Approach Two-Tailed 11-94

95. Copyright © 2013, 2010 and 2007 Pearson Education, Inc. P-Value Approach Left-Tailed 11-95

96. Copyright © 2013, 2010 and 2007 Pearson Education, Inc. P-Value Approach Right-Tailed 11-96

97. Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Technology Step 3: Use a statistical spreadsheet or calculator with statistical capabilities to obtain the P- value. The directions for obtaining the P- value using the TI-83/84 Plus graphing calculator, MINITAB, Excel and StatCrunch, are in the Technology Step-by-Step on page in the text. P-Value Approach 11-97

98. Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Step 4: If P-value < α, reject the null hypothesis. P-Value Approach 11-98

99. Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Step 5: State the conclusion. 11-99

100. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. These procedures are robust, which means that minor departures from normality will not adversely affect the results. However, if the data have outliers, the procedure should not be used. 11-100

101. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. The following data represent the cost of a one night stay in Hampton Inn Hotels and La Quinta Inn Hotels for a random sample of 10 cities. Test the claim that Hampton Inn Hotels are priced differently than La Quinta Hotels at the α = 0.05 level of significance. Parallel Example 2: Testing a Claim Regarding Matched-Pairs Data 11-101

102. Copyright © 2013, 2010 and 2007 Pearson Education, Inc. CityHampton InnLa Quinta Dallas129105 Tampa Bay14996 St. Louis14949 Seattle189149 San Diego109119 Chicago16089 New Orleans14972 Phoenix12959 Atlanta12990 Orlando11969 11-102

103. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. This is a matched-pairs design since the hotel prices come from the same ten cities. To test the hypothesis, we first compute the differences and then verify that the differences come from a population that is approximately normally distributed with no outliers because the sample size is small. The differences (Hampton - La Quinta) are: 24 53 100 40 –10 71 77 70 39 50 with = 51.4 and s d = 30.8336. Solution 11-103

104. Copyright © 2013, 2010 and 2007 Pearson Education, Inc. No violation of normality assumption. Solution 11-104

105. Copyright © 2013, 2010 and 2007 Pearson Education, Inc. No outliers. Solution 11-105

106. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. Step 1: We want to determine if the prices differ: H 0 : μ d = 0 versus H 1 : μ d ≠ 0 Step 2: The level of significance is α = 0.05. Step 3: The test statistic is Solution 11-106

107. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. This is a two-tailed test so the critical values at the α = 0.05 level of significance with n – 1 = 10 – 1 = 9 degrees of freedom are –t 0.025 = –2.262 and t 0.025 = 2.262. Solution: Classical Approach 11-107

108. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. Step 4: Since the test statistic, t 0 = 5.27 is greater than the critical value t.025 = 2.262, we reject the null hypothesis. Solution: Classical Approach 11-108

109. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. Because this is a two-tailed test, the P-value is two times the area under the t-distribution with n – 1 = 10 – 1 = 9 degrees of freedom to the right of the test statistic t 0 = 5.27. That is, P-value = 2P(t > 5.27) ≈ 2(0.00026) = 0.00052 (using technology). Approximately 5 samples in 10,000 will yield results as extreme as we obtained if the null hypothesis is true. Solution: P-Value Approach 11-109

110. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. Step 4: Since the P-value is less than the level of significance α = 0.05, we reject the null hypothesis. Solution: P-Value Approach 11-110

111. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. Step 5: There is sufficient evidence to conclude that Hampton Inn hotels and La Quinta hotels are priced differently at the α = 0.05 level of significance. Solution 11-111

112. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. Objective 2 Construct and Interpret Confidence Intervals for the Population Mean Difference of Matched-Pairs Data 11-112

113. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. A (1 – α)100% confidence interval for μ d is given by Lower bound: Upper bound: The critical value t α/2 is determined using n – 1 degrees of freedom. The values of and s d are the mean and standard deviation of the differenced data. Confidence Interval for Matched-Pairs Data 11-113

114. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. Note: The interval is exact when the population is normally distributed and approximately correct for nonnormal populations, provided that n is large. Confidence Interval for Matched-Pairs Data 11-114

115. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. Construct a 90% confidence interval for the mean difference in price of Hampton Inn versus La Quinta hotel rooms. Parallel Example 4: Constructing a Confidence Interval for Matched-Pairs Data 11-115

116. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. We have already verified that the differenced data come from a population that is approximately normal with no outliers. Recall = 51.4 and s d = 30.8336. From Table VI with α = 0.10 and 9 degrees of freedom, we find t α/2 = 1.833. Solution 11-116

117. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. Thus, Lower bound = Upper bound = We are 90% confident that the mean difference in hotel room price for Ramada Inn versus La Quinta Inn is between $33.53 and $69.27. Solution 11-117

118. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. Section Inference about Two Means: Independent Samples 11.3

119. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. Objectives 1.Test hypotheses regarding the difference of two independent means 2.Construct and interpret confidence intervals regarding the difference of two independent means 11-119

120. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. Suppose that a simple random sample of size n 1 is taken from a population with unknown mean μ 1 and unknown standard deviation σ 1. In addition, a simple random sample of size n 2 is taken from a population with unknown mean μ 2 and unknown standard deviation σ 2. If the two populations are normally distributed or the sample sizes are sufficiently large (n 1 ≥ 30, n 2 ≥ 30), then Sampling Distribution of the Difference of Two Means: Independent Samples with Population Standard Deviations Unknown (Welch’s t) 11-120

121. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. approximately follows Student’s t-distribution with the smaller of n 1 -1 or n 2 -1 degrees of freedom where is the sample mean and s i is the sample standard deviation from population i. Sampling Distribution of the Difference of Two Means: Independent Samples with Population Standard Deviations Unknown (Welch’s t) 11-121

122. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. Objective 1 Test Hypotheses Regarding the Difference of Two Independent Means 11-122

123. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. To test hypotheses regarding two population means, μ 1 and μ 2, with unknown population standard deviations, we can use the following steps, provided that:  the samples are obtained using simple random sampling;  the samples are independent;  the populations from which the samples are drawn are normally distributed or the sample sizes are large (n 1 ≥ 30, n 2 ≥ 30);  For each sample, the sample size is no more than 5% of the population size. Testing Hypotheses Regarding the Difference of Two Means 11-123

124. Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Step 1: Determine the null and alternative hypotheses. The hypotheses are structured in one of three ways: 11-124

125. Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Step 2: Select a level of significance, α, based on the seriousness of making a Type I error. 11-125

126. Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Step 3: Compute the test statistic which approximately follows Student’s t- distribution. Classical Approach 11-126

127. Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Use Table VI to determine the critical value using the smaller of n 1 – 1 or n 2 – 1 degrees of freedom. Classical Approach 11-127

128. Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Classical Approach (critical value) Two-Tailed 11-128

129. Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Classical Approach (critical value) Left-Tailed 11-129

130. Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Classical Approach (critical value) Right-Tailed 11-130

131. Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Step 4: Compare the critical value with the test statistic: Classical Approach 11-131

132. Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Step 3: Compute the test statistic which approximately follows Student’s t- distribution. P-Value Approach 11-132

133. Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Use Table VI to determine the P-value using the smaller of n 1 – 1 or n 2 – 1 degrees of freedom. P-Value Approach 11-133

134. Copyright © 2013, 2010 and 2007 Pearson Education, Inc. P-Value Approach Two-Tailed 11-134

135. Copyright © 2013, 2010 and 2007 Pearson Education, Inc. P-Value Approach Left-Tailed 11-135

136. Copyright © 2013, 2010 and 2007 Pearson Education, Inc. P-Value Approach Right-Tailed 11-136

137. Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Technology Step 3: Use a statistical spreadsheet or calculator with statistical capabilities to obtain the P-value. The directions for obtaining the P-value using the TI- 83/84 Plus graphing calculator, Excel, MINITAB, and StatCrunch are in the Technology Step-by-Step in the text. P-Value Approach 11-137

138. Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Step 4: If P-value < α, reject the null hypothesis. P-Value Approach 11-138

139. Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Step 5: State the conclusion. 11-139

140. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. These procedures are robust, which means that minor departures from normality will not adversely affect the results. However, if the data have outliers, the procedure should not be used. 11-140

141. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. A researcher wanted to know whether “state” quarters had a weight that is more than “traditional” quarters. He randomly selected 18 “state” quarters and 16 “traditional” quarters, weighed each of them and obtained the following data. Parallel Example 1: Testing Hypotheses Regarding Two Means 11-141

142. Copyright © 2013, 2010 and 2007 Pearson Education, Inc. 11-142

143. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. Test the claim that “state” quarters have a mean weight that is more than “traditional” quarters at the α = 0.05 level of significance. NOTE: A normal probability plot of “state” quarters indicates the population could be normal. A normal probability plot of “traditional” quarters indicates the population could be normal 11-143

144. Copyright © 2013, 2010 and 2007 Pearson Education, Inc. No outliers. 11-144

145. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. Step 1: We want to determine whether state quarters weigh more than traditional quarters: H 0 : μ 1 = μ 2 versus H 1 : μ 1 > μ 2 Step 2: The level of significance is α = 0.05. Step 3: The test statistic is Solution 11-145

146. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. Solution: Classical Approach This is a right-tailed test with α = 0.05. Since n 1 – 1 = 17 and n 2 – 1 = 15, we will use 15 degrees of freedom. The corresponding critical value is t 0.05 =1.753. 11-146

147. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. Step 4: Since the test statistic, t 0 = 2.53 is greater than the critical value t.05 = 1.753, we reject the null hypothesis. Solution: Classical Approach 11-147

148. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. Because this is a right-tailed test, the P-value is the area under the t-distribution to the right of the test statistic t 0 = 2.53. That is, P-value = P(t > 2.53) ≈ 0.01. Solution: P-Value Approach 11-148

149. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. Step 4: Since the P-value is less than the level of significance α = 0.05, we reject the null hypothesis. Solution: P-Value Approach 11-149

150. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. Step 5: There is sufficient evidence at the α = 0.05 level to conclude that the state quarters weigh more than the traditional quarters. Solution 11-150

151. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. NOTE: The degrees of freedom used to determine the critical value in the last example are conservative. Results that are more accurate can be obtained by using the following degrees of freedom: 11-151

152. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. Objective 2 Construct and Interpret Confidence Intervals Regarding the Difference of Two Independent Means 11-152

153. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. A simple random sample of size n 1 is taken from a population with unknown mean μ 1 and unknown standard deviation σ 1. Also, a simple random sample of size n 2 is taken from a population with unknown mean μ 2 and unknown standard deviation σ 2. If the two populations are normally distributed or the sample sizes are sufficiently large (n 1 ≥ 30 and n 2 ≥ 30), a (1 – α)100% confidence interval about μ 1 – μ 2 is given by... Constructing a (1 – α)100% Confidence Interval for the Difference of Two Means 11-153

154. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. Lower bound: and Upper bound: where t α/2 is computed using the smaller of n 1 – 1 or n 2 – 1 degrees of freedom or Formula (2). Constructing a (1 – α)100% Confidence Interval for the Difference of Two Means 11-154

155. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. Construct a 95% confidence interval about the difference between the population mean weight of a “state” quarter versus the population mean weight of a “traditional” quarter. Parallel Example 3: Constructing a Confidence Interval for the Difference of Two Means 11-155

156. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. We have already verified that the populations are approximately normal and that there are no outliers. Recall = 5.702, s 1 = 0.0497, =5.6494 and s 2 = 0.0689. From Table VI with α = 0.05 and 15 degrees of freedom, we find t α/2 = 2.131. Solution 11-156

157. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. Thus, Lower bound = Upper bound = Solution 11-157

158. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. We are 95% confident that the mean weight of the “state” quarters is between 0.0086 and 0.0974 ounces more than the mean weight of the “traditional” quarters. Since the confidence interval does not contain 0, we conclude that the “state” quarters weigh more than the “traditional” quarters. Solution 11-158

159. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. When the population variances are assumed to be equal, the pooled t-statistic can be used to test for a difference in means for two independent samples. The pooled t- statistic is computed by finding a weighted average of the sample variances and using this average in the computation of the test statistic. The advantage to this test statistic is that it exactly follows Student’s t-distribution with n 1 +n 2 -2 degrees of freedom. The disadvantage to this test statistic is that it requires that the population variances be equal. 11-159

160. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. Section Putting It Together: Which Method Do I Use? 11.4

161. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. Objective 1 Determine the Appropriate Hypothesis Test to Perform 11-161

162. Copyright © 2013, 2010 and 2007 Pearson Education, Inc. What parameter is addressed in the hypothesis? Proportion, p σ or σ 2 Mean, μ 11-162

163. Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Proportion, p Is the sampling Dependent or Independent? 11-163

164. Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Proportion, p Dependent samples: Provided the samples are obtained randomly and the total number of observations where the outcomes differ is at least 10, use the normal distribution with 11-164

165. Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Proportion, p Independent samples: Provided for each sample and the sample size is no more than 5% of the population size, use the normal distribution with where 11-165

166. Copyright © 2013, 2010 and 2007 Pearson Education, Inc. σ or σ 2 Provided the data are normally distributed, use the F-distribution with 11-166

167. Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Mean, μ Is the sampling Dependent or Independent? 11-167

168. Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Mean, μ Dependent samples: Provided each sample size is greater than 30 or the differences come from a population that is normally distributed, use Student’s t-distribution with n-1 degrees of freedom with 11-168

169. Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Mean, μ Independent samples: Provided each sample size is greater than 30 or each population is normally distributed, use Student’s t-distribution 11-169

170. Copyright © 2013, 2010 and 2007 Pearson Education, Inc. 11-170