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For Exercises 8.47, 8.48, and 8.49, see page 506; for Exercises 8.50 and 8.51, see page 509; for Exercises 8.52 and 8.53, see page 514; for Exercise 8.54, see page 516; and for Exercise 8.55, see page 518.
8.56 Identify the key elements. For each of the following scenarios, identify the populations, the counts, and the sample sizes; compute the two proportions and find their difference.
(a) A study of tipping behaviors examined the relationship between the color of the shirt worn by the server and whether or not the customer left a tip.19 There were 418 male customers in the study; 40 of the 69 who were served by a server wearing a red shirt left a tip. Of the 349 who were served by a server wearing a different colored shirt, 130 left a tip.
(b) A sample of 40 runners will be used to compare two new routines for stretching. The runners will be randomly assigned to one of the routines which they will follow for two weeks. Satisfaction with the routines will be measured using a questionnaire at the end of the two-week period. For the first routine, nine runners said that they were satisfied or very satisfied. For the second routine, six runners said that they were satisfied or very satisfied.
8.57 Apply the confidence interval guidelines. Refer to the previous exercise. For each of the scenarios, determine whether or not the guidelines for using the large-sample method for a 95% confidence interval are satisfied. Explain your answers.
8.58 Find the 95% confidence interval. Refer to Exercise 8.56. For each scenario, find the large-sample 95% confidence interval for the difference in proportions and use the scenario to explain the meaning of the confidence interval.
8.59 Apply the significance test guidelines. Refer to Exercise 8.56. For each of the scenarios, determine whether or not the guidelines for using the large-sample significance test are satisfied. Explain your answers.
8.60 Perform the significance test. Refer to Exercise 8.56. For each scenario, perform the large-sample significance test and use the scenario to explain the meaning of the significance test.
8.61 Find the relative risk. Refer to Exercise 8.56. For each scenario, find the relative risk. Be sure to give a justification for your choice of proportions to use in the numerator and the denominator of the ratio. Use the scenarios to explain the meaning of the relative risk.
8.62 Teeth and military service. In 1898, the United States and Spain fought a war over the U.S. intervention in the Cuban War of Independence. At that time, the U.S. military was concerned about the nutrition of its recruits. Many did not have a sufficient number of teeth to chew the food provided to soldiers. As a result, it was likely that they would be undernourished and unable to fulfill their duties as soldiers. The requirements at that time specified that a recruit must have “at least four sound double teeth, one above and one below on each side of the mouth, and so opposed” so that they could chew food. Of the 58,952 recruits who were under the age of 20, 68 were rejected for this reason. For the 43,786 recruits who were 40 or over, 3801 were rejected.20
(a) Find the proportion of rejects for each age group.
(b) Find a 99% confidence interval for the difference in the proportions.
(c) Use a significance test to compare the proportions. Write a short paragraph describing your results and conclusions.
(d) Are the guidelines for the use of the large-sample approach satisfied for your work in parts (b) and (c)? Explain your answers.
8.63 Physical education requirements. In the 1920s, about 97% of U.S. colleges and universities required a physical education course for graduation. Today, about 40% require such a course. A recent study of physical education requirements included 354 institutions: 225 private and 129 public. Among the private institutions, 60 required a physical education course, while among the public institutions, 101 required a course.21
(a) What are the explanatory and response variables for this exercise? Justify your answers.
(b) What are the populations?
(c) What are the statistics?
(d) Use a 95% confidence interval to compare the private and the public institutions with regard to the physical education requirement.
(e) Use a significance test to compare the private and the public institutions with regard to the physical education requirement.
(f) For parts (d) and (e), verify that the guidelines for using the large-sample methods are satisfied.
(g) Summarize your analysis of these data in a short paragraph.
8.64 Exergaming in Canada. Exergames are active video games such as rhythmic dancing games, virtual bicycles, balance board simulators, and virtual sports simulators that require a screen and a console. A study of exergaming practiced by students from grades 10 and 11 in Montreal, Canada, examined many factors related to participation in exergaming.22 Of the 358 students who reported that they stressed about their health, 29.9% said that they were exergamers. Of the 851 students who reported that they did not stress about their health, 20.8% said that they were exergamers.
(a) Define the two populations to be compared for this exercise.
(b) What are the counts, the sample sizes, and the proportions?
(c) Are the guidelines for the use of the large-sample confidence interval satisfied?
(d) Are the guidelines for the use of the large-sample significance test satisfied?
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8.65 Confidence interval for exergaming in Canada. Refer to the previous exercise. Find the 95% confidence interval for the difference in proportions. Write a short statement interpreting this result.
8.66 Significance test for exergaming in Canada. Refer to Exercise 8.64. Use a significance test to compare the proportions. Write a short statement interpreting this result.
8.67 Adult gamers versus teen gamers. A Pew Internet Project Data Memo presented data comparing adult gamers with teen gamers with respect to the devices on which they play. The data are from two surveys. The adult survey had 1063 gamers, while the teen survey had 1064 gamers. The memo reports that 54% of adult gamers played on game consoles (Xbox, PlayStation, Wii, etc.), while 89% of teen gamers played on game consoles.23
(a) Refer to the table that appears on page 505. Fill in the numerical values of all quantities that are known.
(b) Find the estimate of the difference between the proportion of teen gamers who played on game consoles and the proportion of adults who played on these devices.
(c) Is the large-sample confidence interval for the difference between two proportions appropriate to use in this setting? Explain your answer.
(d) Find the 95% confidence interval for the difference.
(e) Convert your estimated difference and confidence interval to percents.
(f) The adult survey was conducted between October and December 2008, whereas the teen survey was conducted between November 2007 and February 2008. Do you think that this difference should have any effect on the interpretation of the results? Be sure to explain your answer.
8.68 Significance test for gaming on computers. Refer to the previous exercise. Test the null hypothesis that the two proportions are equal. Report the test statistic with the P-value and summarize your conclusion.
8.69 Gamers on computers. The report described in Exercise 8.67 also presented data from the same surveys for gaming on computers (desktops or laptops). These devices were used by 73% of adult gamers and by 76% of teen gamers. Answer the questions given in Exercise 8.67 for gaming on computers.
8.70 Significance test for gaming on consoles. Refer to the previous exercise. Test the null hypothesis that the two proportions are equal. Report the test statistic with the P-value and summarize your conclusion.
8.71 Can we compare gaming on consoles with gaming on computers? Refer to the previous four exercises. Do you think that you can use the large-sample confidence intervals for a difference in proportions to compare teens’ use of computers with teens’ use of consoles? Write a short paragraph giving the reason for your answer. (Hint: Look carefully at the assumptions needed for this procedure on page 512.)
8.72 What’s wrong? For each of the following, explain what is wrong and why.
(a) A z statistic is used to test the null hypothesis that .
(b) If two sample proportions are equal, then the sample counts are equal.
(c) A 95% confidence interval for the difference in two proportions includes errors due to nonresponse.
8.73 Find the power. Consider testing the null hypothesis that two proportions are equal versus the two-sided alternative with α = 0.05, 80% power, and equal sample sizes in the two groups.
(a) For each of the following situations, find the required sample size: (i) p1 = 0.1 and p2 = 0.2 (ii) p1 = 0.2 and p2 = 0.3, (iii) p1 = 0.3 and p2 = 0.4, (iv) p1 = 0.4 and p2 = 0.5, (v) p1 = 0.5 and p2 = 0.6, (vi) p1 = 0.6 and p2 = 0.7, (vii) p1 = 0.7 and p2 = 0.8, and (viii) p1 = 0.8 and p2 = 0.9.
(b) Write a short summary describing your results.