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1 © 2010 Pearson Prentice Hall. All rights reserved Chapter Inferences on Two Samples 11

2 © 2010 Pearson Prentice Hall. All rights reserved Section Inference about Two Means: Dependent Samples 11.1

3 © 2010 Pearson Prentice Hall. All rights reserved Objectives 1.Distinguish between independent and dependent sampling 2.Test hypotheses regarding matched-pairs data 3.Construct and interpret confidence intervals about the population mean difference of matched-pairs data

4 © 2010 Pearson Prentice Hall. All rights reserved Objective 1 Distinguish between Independent and Dependent Sampling

5 © 2010 Pearson Prentice Hall. All rights reserved 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. Dependent samples are often referred to as matched-pairs samples.

6 © 2010 Pearson Prentice Hall. All rights reserved 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

7 © 2010 Pearson Prentice Hall. All rights reserved 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

8 © 2010 Pearson Prentice Hall. All rights reserved Objective 2 Test Hypotheses Regarding Matched-Pairs Data

9 © 2010 Pearson Prentice Hall. All rights reserved “In Other Words” Statistical inference methods on matched-pairs data use the same methods as inference on a single population mean with  unknown, except that the differences are analyzed.

10 © 2010 Pearson Prentice Hall. All rights reserved To test hypotheses regarding the mean difference of matched-pairs data, the following must be satisfied: 1.the sample is obtained using simple random sampling 2.the sample data are matched pairs, 3.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 © 2010 Pearson Prentice Hall. All rights reserved Step 1: Determine the null and alternative hypotheses. The hypotheses can be structured in one of three ways, where  d is the population mean difference of the matched-pairs data.

12 © 2010 Pearson Prentice Hall. All rights reserved Step 2: Select a level of significance, , based on the seriousness of making a Type I error.

13 © 2010 Pearson Prentice Hall. All rights reserved 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.

14 © 2010 Pearson Prentice Hall. All rights reserved Step 4: Use Table VI to determine the critical value using n-1 degrees of freedom. Classical Approach

15 © 2010 Pearson Prentice Hall. All rights reserved Classical Approach (critical value) Two-Tailed

16 © 2010 Pearson Prentice Hall. All rights reserved Classical Approach (critical value) Left-Tailed

17 © 2010 Pearson Prentice Hall. All rights reserved Classical Approach (critical value) Right-Tailed

18 © 2010 Pearson Prentice Hall. All rights reserved Step 5: Compare the critical value with the test statistic: Classical Approach

19 © 2010 Pearson Prentice Hall. All rights reserved Step 4: Use Table VI to determine the P-value using n-1 degrees of freedom. P-Value Approach

20 © 2010 Pearson Prentice Hall. All rights reserved P-Value Approach Two-Tailed

21 © 2010 Pearson Prentice Hall. All rights reserved P-Value Approach Left-Tailed

22 © 2010 Pearson Prentice Hall. All rights reserved P-Value Approach Right-Tailed

23 © 2010 Pearson Prentice Hall. All rights reserved Step 5: If P-value < , reject the null hypothesis. P-Value Approach

24 © 2010 Pearson Prentice Hall. All rights reserved Step 6: State the conclusion.

25 © 2010 Pearson Prentice Hall. All rights reserved 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.

26 © 2010 Pearson Prentice Hall. All rights reserved 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  =0.05 level of significance. Parallel Example 2: Testing a Claim Regarding Matched-Pairs Data

27 © 2010 Pearson Prentice Hall. All rights reserved CityHampton InnLa Quinta Dallas Tampa Bay14996 St. Louis14949 Seattle San Diego Chicago16089 New Orleans14972 Phoenix12959 Atlanta12990 Orlando11969

28 © 2010 Pearson Prentice Hall. All rights reserved 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: with = 51.4 and s d = Solution

29 © 2010 Pearson Prentice Hall. All rights reserved No violation of normality assumption. Solution

30 © 2010 Pearson Prentice Hall. All rights reserved No outliers. Solution

31 © 2010 Pearson Prentice Hall. All rights reserved Step 1: We want to determine if the prices differ: H 0 :  d = 0 versus H 1 :  d  0 Step 2: The level of significance is  =0.05. Step 3: The test statistic is Solution

32 © 2010 Pearson Prentice Hall. All rights reserved Step 4: This is a two-tailed test so the critical values at the  =0.05 level of significance with n-1=10-1=9 degrees of freedom are -t = and t = Solution: Classical Approach

33 © 2010 Pearson Prentice Hall. All rights reserved Step 5: 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

34 © 2010 Pearson Prentice Hall. All rights reserved Step 4: 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( )= (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

35 © 2010 Pearson Prentice Hall. All rights reserved Step 5: Since the P-value is less than the level of significance  =0.05, we reject the null hypothesis. Solution: P-Value Approach

36 © 2010 Pearson Prentice Hall. All rights reserved Step 6: There is sufficient evidence to conclude that Hampton Inn hotels and La Quinta hotels are priced differently at the  =0.05 level of significance. Solution

37 © 2010 Pearson Prentice Hall. All rights reserved Objective 3 Construct and Interpret Confidence Intervals for the Population Mean Difference of Matched-Pairs Data

38 © 2010 Pearson Prentice Hall. All rights reserved A (1-  )  100% confidence interval for  d is given by Lower bound: Upper bound: The critical value t  /2 is determined using n-1 degrees of freedom. Confidence Interval for Matched-Pairs Data

39 © 2010 Pearson Prentice Hall. All rights reserved 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

40 © 2010 Pearson Prentice Hall. All rights reserved 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

41 © 2010 Pearson Prentice Hall. All rights reserved 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 = From Table VI with  = 0.10 and 9 degrees of freedom, we find t  /2 = Solution

42 © 2010 Pearson Prentice Hall. All rights reserved 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 $ Solution

43 © 2010 Pearson Prentice Hall. All rights reserved Section Inference about Two Means: Independent Samples 11.2

44 © 2010 Pearson Prentice Hall. All rights reserved Objectives 1.Test hypotheses regarding the difference of two independent means 2.Construct and interpret confidence intervals regarding the difference of two independent means

45 © 2010 Pearson Prentice Hall. All rights reserved Suppose that a simple random sample of size n 1 is taken from a population with unknown mean  1 and unknown standard deviation  1. In addition, a simple random sample of size n 2 is taken from a population with unknown mean  2 and unknown standard deviation  2. If the two populations are normally distributed or the sample sizes are sufficiently large (n 1 ≥ 30, n 2 ≥ 30), then 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)

46 © 2010 Pearson Prentice Hall. All rights reserved Objective 1 Test Hypotheses Regarding the Difference of Two Independent Means

47 © 2010 Pearson Prentice Hall. All rights reserved To test hypotheses regarding two population means,  1 and  2, with unknown population standard deviations, we can use the following steps, provided that: 1.the samples are obtained using simple random sampling; 2.the samples are independent; 3.the populations from which the samples are drawn are normally distributed or the sample sizes are large (n 1 ≥ 30, n 2 ≥ 30). Testing Hypotheses Regarding the Difference of Two Means

48 © 2010 Pearson Prentice Hall. All rights reserved Step 1: Determine the null and alternative hypotheses. The hypotheses are structured in one of three ways:

49 © 2010 Pearson Prentice Hall. All rights reserved Step 2: Select a level of significance, , based on the seriousness of making a Type I error.

50 © 2010 Pearson Prentice Hall. All rights reserved Step 3: Compute the test statistic which approximately follows Student’s t- distribution.

51 © 2010 Pearson Prentice Hall. All rights reserved Step 4: Use Table VI to determine the critical value using the smaller of n 1 -1 or n 2 -1 degrees of freedom. Classical Approach

52 © 2010 Pearson Prentice Hall. All rights reserved Classical Approach (critical value) Two-Tailed

53 © 2010 Pearson Prentice Hall. All rights reserved Classical Approach (critical value) Left-Tailed

54 © 2010 Pearson Prentice Hall. All rights reserved Classical Approach (critical value) Right-Tailed

55 © 2010 Pearson Prentice Hall. All rights reserved Step 5: Compare the critical value with the test statistic: Classical Approach

56 © 2010 Pearson Prentice Hall. All rights reserved Step 4: 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

57 © 2010 Pearson Prentice Hall. All rights reserved P-Value Approach Two-Tailed

58 © 2010 Pearson Prentice Hall. All rights reserved P-Value Approach Left-Tailed

59 © 2010 Pearson Prentice Hall. All rights reserved P-Value Approach Right-Tailed

60 © 2010 Pearson Prentice Hall. All rights reserved Step 5: If P-value < , reject the null hypothesis. P-Value Approach

61 © 2010 Pearson Prentice Hall. All rights reserved Step 6: State the conclusion.

62 © 2010 Pearson Prentice Hall. All rights reserved 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.

63 © 2010 Pearson Prentice Hall. All rights reserved 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

64 © 2010 Pearson Prentice Hall. All rights reserved

65 © 2010 Pearson Prentice Hall. All rights reserved Test the claim that “state” quarters have a mean weight that is more than “traditional” quarters at the  =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

66 © 2010 Pearson Prentice Hall. All rights reserved No outliers.

67 © 2010 Pearson Prentice Hall. All rights reserved Step 1: We want to determine whether state quarters weigh more than traditional quarters: H 0 :  1 =  2 versus H 1 :  1 >  2 Step 2: The level of significance is  =0.05. Step 3: The test statistic is Solution

68 © 2010 Pearson Prentice Hall. All rights reserved Solution: Classical Approach Step 4: This is a right-tailed test with  =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.

69 © 2010 Pearson Prentice Hall. All rights reserved Step 5: 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

70 © 2010 Pearson Prentice Hall. All rights reserved Step 4: 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) ≈ Solution: P-Value Approach

71 © 2010 Pearson Prentice Hall. All rights reserved Step 5: Since the P-value is less than the level of significance  =0.05, we reject the null hypothesis. Solution: P-Value Approach

72 © 2010 Pearson Prentice Hall. All rights reserved Step 6: There is sufficient evidence at the  =0.05 level to conclude that the state quarters weigh more than the traditional quarters. Solution

73 © 2010 Pearson Prentice Hall. All rights reserved 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:

74 © 2010 Pearson Prentice Hall. All rights reserved Objective 3 Construct and Interpret Confidence Intervals Regarding the Difference of Two Independent Means

75 © 2010 Pearson Prentice Hall. All rights reserved A simple random sample of size n 1 is taken from a population with unknown mean  1 and unknown standard deviation  1. Also, a simple random sample of size n 2 is taken from a population with unknown mean  2 and unknown standard deviation  2. If the two populations are normally distributed or the sample sizes are sufficiently large (n 1 ≥30 and n 2 ≥30), a (1-  )  100% confidence interval about  1 -  2 is given by Lower bound: Upper bound: Constructing a (1-  )  100% Confidence Interval for the Difference of Two Means

76 © 2010 Pearson Prentice Hall. All rights reserved 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

77 © 2010 Pearson Prentice Hall. All rights reserved We have already verified that the populations are approximately normal and that there are no outliers. Recall = 5.702, s 1 = , = and s 2 = From Table VI with  = 0.05 and 15 degrees of freedom, we find t  /2 = Solution

78 © 2010 Pearson Prentice Hall. All rights reserved Thus, Lower bound = Upper bound = Solution

79 © 2010 Pearson Prentice Hall. All rights reserved We are 95% confident that the mean weight of the “state” quarters is between and 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

80 © 2010 Pearson Prentice Hall. All rights reserved 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.

81 © 2010 Pearson Prentice Hall. All rights reserved Section Inference about Two Population Proportions 11.3

82 © 2010 Pearson Prentice Hall. All rights reserved Objectives 1.Test hypotheses regarding two population proportions 2.Construct and interpret confidence intervals for the difference between two population proportions 3.Use McNemar’s Test to compare two proportions from matched-pairs data 4.Determine the sample size necessary for estimating the difference between two population proportions within a specified margin of error.

83 © 2010 Pearson Prentice Hall. All rights reserved 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. Sampling Distribution of the Difference between Two Proportions

84 © 2010 Pearson Prentice Hall. All rights reserved The standardized version of is then written as which has an approximate standard normal distribution. Sampling Distribution of the Difference between Two Proportions

85 © 2010 Pearson Prentice Hall. All rights reserved Objective 1 Test Hypotheses Regarding Two Population Proportions

86 © 2010 Pearson Prentice Hall. All rights reserved The best point estimate of p is called the pooled estimate of p, denoted, where Test statistic for Comparing Two Population Proportions

87 © 2010 Pearson Prentice Hall. All rights reserved To test hypotheses regarding two population proportions, p 1 and p 2, we can use the steps that follow, provided that: 1.the samples are independently obtained using simple random sampling, 2. and 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. Hypothesis Test Regarding the Difference between Two Population Proportions

88 © 2010 Pearson Prentice Hall. All rights reserved Step 1: Determine the null and alternative hypotheses. The hypotheses can be structured in one of three ways:

89 © 2010 Pearson Prentice Hall. All rights reserved Step 2: Select a level of significance, , based on the seriousness of making a Type I error.

90 © 2010 Pearson Prentice Hall. All rights reserved Step 3: Compute the test statistic where

91 © 2010 Pearson Prentice Hall. All rights reserved Step 4: Use Table V to determine the critical value. Classical Approach

92 © 2010 Pearson Prentice Hall. All rights reserved Classical Approach (critical value) Two-Tailed

93 © 2010 Pearson Prentice Hall. All rights reserved Classical Approach (critical value) Left-Tailed

94 © 2010 Pearson Prentice Hall. All rights reserved Classical Approach (critical value) Right-Tailed

95 © 2010 Pearson Prentice Hall. All rights reserved Step 5: Compare the critical value with the test statistic: Classical Approach

96 © 2010 Pearson Prentice Hall. All rights reserved Step 4: Use Table V to estimate the P-value.. P-Value Approach

97 © 2010 Pearson Prentice Hall. All rights reserved P-Value Approach Two-Tailed

98 © 2010 Pearson Prentice Hall. All rights reserved P-Value Approach Left-Tailed

99 © 2010 Pearson Prentice Hall. All rights reserved P-Value Approach Right-Tailed

100 © 2010 Pearson Prentice Hall. All rights reserved Step 5: If P-value < , reject the null hypothesis. P-Value Approach

101 © 2010 Pearson Prentice Hall. All rights reserved Step 6: State the conclusion.

102 © 2010 Pearson Prentice Hall. All rights reserved 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  =0.05 level of significance. Parallel Example 1: Testing Hypotheses Regarding Two Population Proportions

103 © 2010 Pearson Prentice Hall. All rights reserved 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

104 © 2010 Pearson Prentice Hall. All rights reserved 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  = Solution

105 © 2010 Pearson Prentice Hall. All rights reserved Step 3: The pooled estimate of is: The test statistic is: Solution

106 © 2010 Pearson Prentice Hall. All rights reserved Step 4: This is a right-tailed test with  =0.05. The critical value is z 0.05 = Solution: Classical Approach

107 © 2010 Pearson Prentice Hall. All rights reserved Step 5: 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

108 © 2010 Pearson Prentice Hall. All rights reserved Step 4: 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) ≈ Solution: P-Value Approach

109 © 2010 Pearson Prentice Hall. All rights reserved Step 5: Since the P-value is greater than the level of significance  =0.05, we fail to reject the null hypothesis. Solution: P-Value Approach

110 © 2010 Pearson Prentice Hall. All rights reserved Step 6: There is insufficient evidence at the  =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

111 © 2010 Pearson Prentice Hall. All rights reserved Objective 2 Construct and Interpret Confidence Intervals for the Difference between Two Population Proportions

112 © 2010 Pearson Prentice Hall. All rights reserved To construct a (1-  )  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-  )  100% Confidence Interval for the Difference between Two Population Proportions

113 © 2010 Pearson Prentice Hall. All rights reserved Provided that these requirements are met, a (1-  )  100% confidence interval for p 1 -p 2 is given by Lower bound: Upper bound: Constructing a (1-  )  100% Confidence Interval for the Difference between Two Population Proportions

114 © 2010 Pearson Prentice Hall. All rights reserved 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

115 © 2010 Pearson Prentice Hall. All rights reserved We have already verified the requirements for constructing a confidence interval for the difference between two population proportions in the previous example. Recall Solution

116 © 2010 Pearson Prentice Hall. All rights reserved Thus, Lower bound = Upper bound = Solution

117 © 2010 Pearson Prentice Hall. All rights reserved 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 and 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

118 © 2010 Pearson Prentice Hall. All rights reserved Objective 3 Use McNemar’s Test to Compare Two Proportions from Matched-Pairs Data

119 © 2010 Pearson Prentice Hall. All rights reserved McNemar’s Test is a test that can be used to compare two proportions with matched-pairs data (i.e., dependent samples)

120 © 2010 Pearson Prentice Hall. All rights reserved 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

121 © 2010 Pearson Prentice Hall. All rights reserved 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.

122 © 2010 Pearson Prentice Hall. All rights reserved 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 )

123 © 2010 Pearson Prentice Hall. All rights reserved Step 2: Select a level of significance, , based on the seriousness of making a Type I error.

124 © 2010 Pearson Prentice Hall. All rights reserved Step 3: Compute the test statistic

125 © 2010 Pearson Prentice Hall. All rights reserved Step 4: 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  /2. Classical Approach

126 © 2010 Pearson Prentice Hall. All rights reserved Classical Approach

127 © 2010 Pearson Prentice Hall. All rights reserved Step 5: If z 0 > z  /2, reject the null hypothesis. Classical Approach

128 © 2010 Pearson Prentice Hall. All rights reserved Step 4: 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

129 © 2010 Pearson Prentice Hall. All rights reserved P-Value Approach

130 © 2010 Pearson Prentice Hall. All rights reserved Step 5: If P-value < , reject the null hypothesis. P-Value Approach

131 © 2010 Pearson Prentice Hall. All rights reserved Step 6: State the conclusion.

132 © 2010 Pearson Prentice Hall. All rights reserved 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

133 © 2010 Pearson Prentice Hall. All rights reserved The results of the survey are shown below: Do the proportions who agree with each scenario differ significantly? Use the  =0.05 level of significance. Parallel Example 4: Analyzing the Difference of Two Proportions from Matched-Pairs Data Detain Tap Phone AgreeDisagree Agree Disagree224450

134 © 2010 Pearson Prentice Hall. All rights reserved 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

135 © 2010 Pearson Prentice Hall. All rights reserved The samples are dependent and were obtained randomly. The total number of individuals who agree with one scenario, but disagree with the other is =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  = Solution

136 © 2010 Pearson Prentice Hall. All rights reserved Step 3: The test statistic is: Solution

137 © 2010 Pearson Prentice Hall. All rights reserved Step 4: The critical value with an  =0.05 level of significance is z =1.96. Solution: Classical Approach

138 © 2010 Pearson Prentice Hall. All rights reserved Step 5: 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

139 © 2010 Pearson Prentice Hall. All rights reserved Step 4: 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 = 2*P(Z > 0.56) ≈ Solution: P-Value Approach

140 © 2010 Pearson Prentice Hall. All rights reserved Step 5: Since the P-value is greater than the level of significance  =0.05, we fail to reject the null hypothesis. Solution: P-Value Approach

141 © 2010 Pearson Prentice Hall. All rights reserved Step 6: There is insufficient evidence at the  =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

142 © 2010 Pearson Prentice Hall. All rights reserved Objective 4 Determine the Sample Size Necessary for Estimating the Difference between Two Population Proportions within a Specified Margin of Error

143 © 2010 Pearson Prentice Hall. All rights reserved Sample Size for Estimating p 1 -p 2 The sample size required to obtain a (1-  )  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.

144 © 2010 Pearson Prentice Hall. All rights reserved A doctor wants to estimate the difference in the proportion of year old mothers that received prenatal care and the proportion of 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 year old mothers received prenatal care and 99.2% of year old mothers received prenatal care. b)she does not use any prior estimates. Parallel Example 5: Determining Sample Size

145 © 2010 Pearson Prentice Hall. All rights reserved We have E=0.02 and z  /2 =z =1.96. a)Letting, The doctor must sample 265 randomly selected year old mothers and 265 randomly selected year old mothers. Solution

146 © 2010 Pearson Prentice Hall. All rights reserved b)Without prior estimates of p 1 and p 2, the sample size is The doctor must sample 4802 randomly selected year old mothers and 4802 randomly selected 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

147 © 2010 Pearson Prentice Hall. All rights reserved Section Inference about Two Population Standard Deviations 11.4

148 © 2010 Pearson Prentice Hall. All rights reserved Objectives 1.Find critical values of the F-distribution 2.Test hypotheses regarding two population standard deviations

149 © 2010 Pearson Prentice Hall. All rights reserved Objective 1 Find Critical Values of the F-distribution

150 © 2010 Pearson Prentice Hall. All rights reserved Requirements for Testing Claims Regarding Two Population Standard Deviations 1. The samples are independent simple random samples.

151 © 2010 Pearson Prentice Hall. All rights reserved Requirements for Testing Claims Regarding Two Population Standard Deviations 1. The samples are independent simple random samples. 2. The populations from which the samples are drawn are normally distributed.

152 © 2010 Pearson Prentice Hall. All rights reserved CAUTION! If the populations from which the samples are drawn are not normal, do not use the inferential procedures discussed in this section.

153 © 2010 Pearson Prentice Hall. All rights reserved Notation Used When Comparing Two Population Standard Deviations : Variance for population 1 : Variance for population 2 : Sample variance for population 1 : Sample variance for population 2 n 1 : Sample size for population 1 n 2 : Sample size for population 2

154 © 2010 Pearson Prentice Hall. All rights reserved Fisher's F-distribution If and and are sample variances from independent simple random samples of size n 1 and n 2, respectively, drawn from normal populations, then follows the F-distribution with n 1 -1 degrees of freedom in the numerator and n 2 -1 degrees of freedom in the denominator.

155 © 2010 Pearson Prentice Hall. All rights reserved Characteristics of the F-distribution 1. It is not symmetric. The F-distribution is skewed right.

156 © 2010 Pearson Prentice Hall. All rights reserved Characteristics of the F-distribution 1. It is not symmetric. The F-distribution is skewed right. 2. The shape of the F-distribution depends upon the degrees of freedom in the numerator and denominator. This is similar to the  2 distribution and Student’s t- distribution, whose shapes depend upon their degrees of freedom.

157 © 2010 Pearson Prentice Hall. All rights reserved Characteristics of the F-distribution 1. It is not symmetric. The F-distribution is skewed right. 2. The shape of the F-distribution depends upon the degrees of freedom in the numerator and denominator. This is similar to the distribution and Student’s t- distribution, whose shape depends upon their degrees of freedom. 3. The total area under the curve is 1.

158 © 2010 Pearson Prentice Hall. All rights reserved Characteristics of the F-distribution 1. It is not symmetric. The F-distribution is skewed right. 2. The shape of the F-distribution depends upon the degrees of freedom in the numerator and denominator. This is similar to the distribution and Student’s t- distribution, whose shape depends upon their degrees of freedom. 3. The total area under the curve is The values of F are always greater than or equal to zero.

159 © 2010 Pearson Prentice Hall. All rights reserved

160 © 2010 Pearson Prentice Hall. All rights reserved is the critical F with n 1 – 1 degrees of freedom in the numerator and n 2 – 1 degrees of freedom in the denominator and an area of  to the right of the critical F.

161 © 2010 Pearson Prentice Hall. All rights reserved To find the critical F with an area of α to the left, use the following:

162 © 2010 Pearson Prentice Hall. All rights reserved Find the critical F-value: a)for a right-tailed test with  =0.1, degrees of freedom in the numerator = 8 and degrees of freedom in the denominator = 4. b)for a two-tailed test with  =0.05, degrees of freedom in the numerator = 20 and degrees of freedom in the denominator = 15. Parallel Example 1: Finding Critical Values for the F-distribution

163 © 2010 Pearson Prentice Hall. All rights reserved a)F 0.1,8,4 = 3.95 b)F.025,20,15 = 2.76; = 0.39 Solution

164 © 2010 Pearson Prentice Hall. All rights reserved NOTE: If the number of degrees of freedom is not found in the table, we follow the practice of choosing the degrees of freedom closest to that desired. If the degrees of freedom is exactly between two values, find the mean of the values.

165 © 2010 Pearson Prentice Hall. All rights reserved Objective 2 Test Hypotheses Regarding Two Population Standard Deviations

166 © 2010 Pearson Prentice Hall. All rights reserved To test hypotheses regarding two population standard deviations,  1 and  2, we can use the following steps, provided that 1.the samples are obtained using simple random sampling, 2.the sample data are independent, and 3.the populations from which the samples are drawn are normally distributed. Test Hypotheses Regarding Two Population Standard Deviations

167 © 2010 Pearson Prentice Hall. All rights reserved Step 1: Determine the null and alternative hypotheses. The hypotheses can be structured in one of three ways: Two-TailedLeft-TailedRight-Tailed H 0 :  1 =  2 H 1 :  1 ≠  2 H 1 :  1 <  2 H 1 :  1 >  2 Note:  1 is the population standard deviation for population 1 and  2 is the population standard deviation for population 2.

168 © 2010 Pearson Prentice Hall. All rights reserved Step 2: Select a level of significance, , based on the seriousness of making a Type I error.

169 © 2010 Pearson Prentice Hall. All rights reserved Step 3: Compute the test statistic which follows Fisher’s F-distribution with n 1 -1 degrees of freedom in the numerator and n 2 -1 degrees of freedom in the denominator.

170 © 2010 Pearson Prentice Hall. All rights reserved Step 4: Use Table VIII to determine the critical value(s) using n 1 -1 degrees of freedom in the numerator and n 2 -1 degrees of freedom in the denominator. Classical Approach

171 © 2010 Pearson Prentice Hall. All rights reserved Classical Approach (critical value) Two-Tailed

172 © 2010 Pearson Prentice Hall. All rights reserved Classical Approach (critical value) Left-Tailed

173 © 2010 Pearson Prentice Hall. All rights reserved Classical Approach (critical value) Right-Tailed

174 © 2010 Pearson Prentice Hall. All rights reserved Step 5: Compare the critical value with the test statistic: Classical Approach Two-TailedLeft-TailedRight-Tailed If or, reject the null hypothesis. If, reject the null hypothesis. If, reject the null hypothesis.

175 © 2010 Pearson Prentice Hall. All rights reserved Step 4: Use technology to determine the P-value. P-Value Approach

176 © 2010 Pearson Prentice Hall. All rights reserved Step 5: If P-value < , reject the null hypothesis. P-Value Approach

177 © 2010 Pearson Prentice Hall. All rights reserved Step 6: State the conclusion.

178 © 2010 Pearson Prentice Hall. All rights reserved CAUTION! The procedures just presented are not robust, minor departures from normality will adversely affect the results of the test. Therefore, the test should be used only when the requirement of normality has been verified.

179 © 2010 Pearson Prentice Hall. All rights reserved A researcher wanted to know whether “state” quarters had a standard deviation weight that is less than “traditional” quarters. He randomly selected 18 “state” quarters and 16 “traditional” quarters, weighed each of them and obtained the data on the next slide. A normal probability plot indicates that the sample data could come from a population that is normal. Test the researcher’s claim at the  = 0.05 level of significance. Parallel Example 2: Testing Hypotheses Regarding Two Population Standard Deviations

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181 © 2010 Pearson Prentice Hall. All rights reserved Step 1: The researcher wants to know if “state” quarters have a standard deviation weight that is less than “traditional” quarters. Thus H 0 :  1 =  2 versus H 1 :  1 <  2 This is a left-tailed test. Step 2: The level of significance is  = Solution

182 © 2010 Pearson Prentice Hall. All rights reserved Step 3: The standard deviation of “state” quarters was found to be and the standard deviation of “traditional” quarters was found to be The test statistic is then Solution

183 © 2010 Pearson Prentice Hall. All rights reserved Step 4: Since this is a left-tailed test, we determine the critical value at the 1-  = = 0.95 level of significance with n 1 -1=18-1=17 degrees of freedom in the numerator and n 2 -1=16-1=15 degrees of freedom in the denominator. Thus, Note: we used the table value F 0.05,15,15 for the above calculation since this is the closest to the required degrees of freedom available from Table VIII. Solution: Classical Approach

184 © 2010 Pearson Prentice Hall. All rights reserved Step 5: Since the test statistic F 0 = 0.52 is greater than the critical value F 0.95,17,15 =0.42, we fail to reject the null hypothesis. Solution: Classical Approach

185 © 2010 Pearson Prentice Hall. All rights reserved Step 4: Using technology, we find that the P-value is If the statement in the null hypothesis were true, we would expect to get the results obtained about 10 out of 100 times. This is not very unusual. Solution: P-Value Approach

186 © 2010 Pearson Prentice Hall. All rights reserved Step 5: Since the P-value is greater than the level of significance,  = 0.05, we fail to reject the null hypothesis. Solution: P-Value Approach

187 © 2010 Pearson Prentice Hall. All rights reserved Step 6: There is not enough evidence to conclude that the standard deviation of weight is less for “state” quarters than it is for “traditional” quarters at the  = 0.05 level of significance. Solution

188 © 2010 Pearson Prentice Hall. All rights reserved Section Putting It Together: Which Method Do I Use? 11.5

189 © 2010 Pearson Prentice Hall. All rights reserved Objectives 1.Determine the appropriate hypothesis test to perform

190 © 2010 Pearson Prentice Hall. All rights reserved Objective 1 Determine the Appropriate Hypothesis Test to Perform

191 © 2010 Pearson Prentice Hall. All rights reserved What parameter is addressed in the hypothesis? Proportion, p  or  2 Mean, 

192 © 2010 Pearson Prentice Hall. All rights reserved Proportion, p Is the sampling Dependent or Independent?

193 © 2010 Pearson Prentice Hall. All rights reserved 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

194 © 2010 Pearson Prentice Hall. All rights reserved 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

195 © 2010 Pearson Prentice Hall. All rights reserved  or  2 Provided the data are normally distributed, use the F-distribution with

196 © 2010 Pearson Prentice Hall. All rights reserved Mean,  Is the sampling Dependent or Independent?

197 © 2010 Pearson Prentice Hall. All rights reserved Mean,  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

198 © 2010 Pearson Prentice Hall. All rights reserved Mean,  Independent samples: Provided each sample size is greater than 30 or each population is normally distributed, use Student’s t-distribution


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