Section 8.2 Confidence Intervals for a Population Mean, Standard Deviation Unknown 375 b. If the confidence level were 99%, would the confidence interval be narrower or wider? Wider 20. A sample of size n = 35 has sample mean ̄x = 34.85 and sample standard deviation s = 17.9. a. Construct a 98% confidence interval for the population mean ��. (27.46, 42.24) b. If the confidence level were 95%, would the confidence interval be narrower or wider? Narrower Working with the Concepts 21. Online courses: A sample of 263 students who were taking online courses were asked to describe their overall impression of online learning on a scale of 1–7, with 7 representing the most favorable impression. The average score was 5.53, and the standard deviation was 0.92. a. Construct a 95% confidence interval for the mean score. (5.42, 5.64) b. Assume that the mean score for students taking traditional courses is 5.55. A college that offers online courses claims that the mean scores for online courses and traditional courses are the same. Does the confidence interval contradict this claim? Explain. No Source: Innovations in Education and Teaching International 45:115–126 22. Get an education: In 2012 the General Social Survey asked 1972 adults how many years of education they had. The sample mean was 13.37 years with a standard deviation of 3.13 years. a. Construct a 98% confidence interval for the mean length of time workers have been employed in their current job. (13.21, 13.53) b. Data collected in an earlier study suggests that the mean in 2000 was 13.26 years. A sociologist believes that the mean 2012 is the same. Does the confidence interval contradict this belief? Explain. No 23. Fake Twitter followers: Many celebrities and public figures have Twitter accounts with large numbers of followers. However, some of these followers are fake, resulting from accounts generated by spamming computers. In a sample of 46 twitter audits conducted in July 2013, the mean percentage of fake followers was 14.1 with a standard deviation of 9.6. a. Construct a 90% confidence interval for the mean percentage of fake Twitter followers. (11.7, 16.5) b. Based on the confidence interval, is it reasonable that the mean percentage of fake Twitter followers is less than 10? No Source: www.twitteraudit.com 24. Let’s go to the movies: A random sample of 35 Hollywood movies made since the year 2010 had a mean length of 121.6 minutes, with a standard deviation of 20.4 minutes. a. Construct a 95% confidence interval for the true mean length of all Hollywood movies made since 2000. (114.6, 128.6) b. As of June 2013, three Ironman movies have been released, and their mean length is 127 minutes. Someone claims that the mean length of Ironman movies is actually less than the mean length of all Hollywood movies. Does the confidence interval contradict this claim? Explain. No Based on data from Box Office Mojo. 25. Hip surgery: In a sample of 123 hip surgeries of a certain type, the average surgery time was 136.9 minutes with a standard deviation of 22.6 minutes. a. Construct a 95% confidence interval for the mean surgery time for this procedure. (132.9, 140.9) b. If a 99% confidence interval were constructed with these data, would it be wider or narrower than the interval constructed in part (a)? Explain. Wider Source: Journal of Engineering in Medicine 221:699–712 26. Sound it out: Phonics is an instructional method in which children are taught to connect sounds with letters or groups of letters. A sample of 134 first-graders who were learning English were asked to identify as many letter sounds as possible in a period of one minute. The average number of letter sounds identified was 34.06 with a standard deviation of 23.83. a. Construct a 98% confidence interval for the mean number of letter sounds identified in one minute. (29.19, 38.93) Tech: (29.21, 38.91) b. If a 95% confidence interval were constructed with these data, would it be wider or narrower than the interval constructed in part (a)? Explain. Narrower Source: School Psychology Review 37:5–17 27. Software instruction: A hybrid course is one that contains both online and classroom instruction. In a study performed at Middle Georgia State College, a software package was used as the main source of instruction in a hybrid college algebra course. The software tracked the number of hours it took for each student to meet the objectives of the course. In a sample of 45 students, the mean number of hours was 80.5, with a standard deviation of 51.2. a. Construct a 95% confidence interval for the mean number of hours it takes for a student to meet the course objectives. (65.1, 95.9) b. If a sample of 90 students had been studied, would you expect the confidence interval to be wider or narrower than the interval constructed in part (a)? Explain. Narrower 28. Baby talk: In a sample of 77 children, the mean age at which they first began to combine words was 16.51 months, with a standard deviation of 9.59 months. a. Construct a 95% confidence interval for the mean age at which children first begin to combine words. (14.32, 18.70) Tech: (14.33, 18.69) b. If a sample of 50 children had been studied, would you expect the confidence interval to be wider or narrower than the interval constructed in part (a)? Explain. Wider Source: Proceedings of the 4th International Symposium on Bilingualism, pp. 58–77 29. Baby weights: Following are weights, in pounds, of 12 two-month-old baby girls. It is reasonable to assume that the population is approximately normal.
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