Write About It 401 9. Cost of repairs: Refer to Exercise 8. Assume the population is normal. Construct a 95% confidence interval for the population standard deviation ��. (4.03, 12.39) 10. Super Bowl: A simple random sample of 140 residents in a certain town was polled the week after the Super Bowl, and 75 of them said they had watched the game on television. a. Construct a 95% confidence interval for the proportion of people in the town who watched the Super Bowl on television. b. The 2013 Super Bowl between the Baltimore Ravens and the San Francisco 49ers was the third most watched television program in history, with 48.1% of television sets in the U.S. tuned to the game. Someone claims that the percentage of people who watched the game in this town was less than 48.1%. Does the confidence interval contradict this claim? Explain. No c. Use the results from the sample of 140 to estimate the sample size necessary for a 95% confidence interval to have a margin of error of 0.025. 1529 11. Testing math skills: In order to test the effectiveness of a program to improve mathematical skills, a simple random sample of 45 fifth-graders was chosen to participate in the program. The students were given an exam at the beginning of the program and again at the end. The sample mean increase in the exam score was 12.2 points, with a sample standard deviation of 4.7 points. a. Construct a 99% confidence interval for the mean increase in score. (10.3, 14.1) b. The developers of the program claim that the program will produce a mean increase of more than 15 points. Does the confidence interval contradict this claim? Explain. Yes 12. Sleep time: In a sample of 87 young adults, the average time per day spent in bed asleep was 7.06 hours. Assume the population sample standard deviation is 1.11 hours. a. Construct a 99% confidence interval for the mean time spent in bed asleep. (6.75, 7.37) b. Some health experts recommend that people get 8 hours or more of sleep per night. Based on the confidence interval, is it reasonable to believe that the mean number of hours of sleep for young adults is 8 or more? Explain. No c. How large would the sample have to be so that a 99% confidence interval would have a margin of error of 0.1? 818 Source: Behavioral Medicine 27:71–76 13. Leaking tanks: Leakage from underground fuel tanks has been a source of water pollution. In a random sample of 107 gasoline stations, 18 were found to have at least one leaking underground tank. a. Find a point estimate for the proportion of gasoline stations with at least one leaking underground tank. 0.168 b. Construct a 95% confidence interval for the proportion of gasoline stations with at least one leaking underground tank. c. Use the point estimate computed in part (a) to determine the number of stations that must be sampled so that a 95% confidence interval will have a margin of error of 0.03. 598 14. Waist size: According to the National Health Statistics Reports, a sample of 783 men aged 20–29 years had a mean waist size of 36.9 inches with a standard deviation of 8.8 inches. a. Construct a 95% confidence interval for the mean waist size. (36.3, 37.5) b. The results of another study suggest that the mean waist size for men aged 30–39 is 38.7 inches. Based on the confidence interval, is it reasonable to believe that the mean waist size for men aged 20–29 may be 38.7 inches? Explain. No 15. Don’t construct a confidence interval: A meteorology student examines precipitation records for a certain city and discovers that of the last 365 days, it rained on 46 of them. Explain why these data cannot be used to construct a confidence interval for the proportion of days in this city that are rainy. Write About It 1. When constructing a confidence interval for �� when �� is known, we assume that we have a simple random sample, that �� is known, and that either the sample size is large or the population is approximately normal. Why is it necessary for these assumptions to be met? 2. What factors can you think of that may affect the width of a confidence interval? In what way does each factor affect the width? 3. Explain the difference between confidence and probability. In Exercises 4 and 5, express the following survey results in terms of confidence intervals for p: 4. According to a survey of 1000 American adults, 55% of Americans do not have a will specifying the handling of their estate. The survey’s margin of error was plus or minus 3%. Source: FindLaw.com 5. In a survey of 5050 U.S. adults, 29% would consider traveling abroad for medical care because of medical costs. The survey’s margin of error was plus or minus 2%. Source: The Gallup Poll 6. When constructing a confidence interval for ��, how do you decide whether to use the t distribution or the normal distribution? Are there any circumstances when it is acceptable to use either distribution? 7. It is stated in the text that there are many different t distributions. Explain how this is so.
navidi_monk_elementary_statistics_2e_ch7-9
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