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

Copyright © 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 11-3

Copyright © 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 11-4

Copyright © 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) 11-5

Copyright © 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 11-6

Copyright © 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 11-7

Copyright © 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 11-8

Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Hypothesis Test steps: 1.State hypothesis, left-tail, right-tail, two-tail. 2.State level of significance, α. 3.Compute the test statistic. 4.Find the critical value. 5.Find the P-value. 6.Compare test-stat with CV, and α with P- value. Reject or Fail to reject. 7.Write a conclusion statement. 11-9

Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Step 3: Compute the test statistic where Classical Approach 11-10

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 11-11

Copyright © 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 11-12

Copyright © 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 11-13

Copyright © 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 = Solution 11-14

Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Step 3: The pooled estimate of is: The test statistic is: Solution 11-15

Copyright © 2013, 2010 and 2007 Pearson Education, Inc. This is a right-tailed test with α = The critical value is z 0.05 = Solution: Classical Approach 11-16

Copyright © 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 11-17

Copyright © 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) ≈ Solution: P-Value Approach 11-18

Copyright © 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 11-19

Copyright © 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 11-20

Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Objective 3 Construct and Interpret Confidence Intervals for the Difference between Two Population Proportions 11-21

Copyright © 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 11-22

Copyright © 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 11-23

Copyright © 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 11-24

Copyright © 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 11-25

Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Thus, Lower bound = Upper bound = Solution 11-26

Copyright © 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 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 11-27

Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Objective 4 Test Hypotheses Regarding Two Proportions from Dependent Samples 11-28

Copyright © 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) 11-29

Copyright © 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

Copyright © 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 ≥

Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Hypothesis Test steps: 1.State hypothesis, left-tail, right-tail, two-tail. 2.State level of significance, α. 3.Compute the test statistic. 4.Find the critical value. 5.Find the P-value. 6.Compare test-stat with CV, and α with P- value. Reject or Fail to reject. 7.Write a conclusion statement

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 ) 11-33

Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Step 3: Compute the test statistic 11-34

Copyright © 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 11-35

Copyright © 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 Agree Disagree

Copyright © 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 11-37

Copyright © 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 = 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 = Solution 11-38

Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Step 3: The test statistic is: Solution 11-39

Copyright © 2013, 2010 and 2007 Pearson Education, Inc. The critical value with an α = 0.05 level of significance is z = Solution: Classical Approach 11-40

Copyright © 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 11-41

Copyright © 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) ≈ Solution: P-Value Approach 11-42

Copyright © 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 11-43

Copyright © 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 11-44

Copyright © 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 11-45

Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Sample Size for Estimating p 1 – p

Copyright © 2013, 2010 and 2007 Pearson Education, Inc. 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 11-47

Copyright © 2013, 2010 and 2007 Pearson Education, Inc. We have E = 0.02 and z α/2 = z = a)Letting, The doctor must sample 265 randomly selected year old mothers and 265 randomly selected year old mothers. Solution 11-48

Copyright © 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 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 11-49

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

Copyright © 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 11-51

Copyright © 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

Copyright © 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 11-53

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

Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Hypothesis Test steps: 1.State hypothesis, left-tail, right-tail, two-tail. 2.State level of significance, α. 3.Compute the test statistic. 4.Find the critical value. 5.Find the P-value. 6.Compare test-stat with CV, and α with P- value. Reject or Fail to reject. 7.Write a conclusion statement

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 11-56

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 11-57

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 11-58

Copyright © 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

Copyright © 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 11-60

Copyright © 2013, 2010 and 2007 Pearson Education, Inc. CityHampton InnLa Quinta Dallas Tampa Bay14996 St. Louis14949 Seattle San Diego Chicago16089 New Orleans14972 Phoenix12959 Atlanta12990 Orlando

Copyright © 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: – with = 51.4 and s d = Solution 11-62

Copyright © 2013, 2010 and 2007 Pearson Education, Inc. No violation of normality assumption. Solution 11-63

Copyright © 2013, 2010 and 2007 Pearson Education, Inc. No outliers. Solution 11-64

Copyright © 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 = Step 3: The test statistic is Solution 11-65

Copyright © 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 = –2.262 and t = Solution: Classical Approach 11-66

Copyright © 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 11-67

Copyright © 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 = 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 11-68

Copyright © 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 11-69

Copyright © 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 11-70

Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Objective 2 Construct and Interpret Confidence Intervals for the Population Mean Difference of Matched-Pairs Data 11-71

Copyright © 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 11-72

Copyright © 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 11-73

Copyright © 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 11-74

Copyright © 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 = From Table VI with α = 0.10 and 9 degrees of freedom, we find t α/2 = Solution 11-75

Copyright © 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 Hampton Inn versus La Quinta Inn is between $33.53 and $ Solution 11-76