388 Chapter 8 Confidence Intervals 20. Working at home: According to the U.S. Census Bureau, 43% of men who worked at home were college graduates. In a sample of 500 women who worked at home, 162 were college graduates. a. Find a point estimate for the proportion of college graduates among women who work at home. 0.324 b. Construct a 98% confidence interval for the proportion of women who work at home who are college graduates. (0.275, 0.373) c. Based on the confidence interval, is it reasonable to believe that the proportion of college graduates among women who work at home is the same as the proportion of college graduates among men who work at home? Explain. No 21. Sleep apnea: Sleep apnea is a disorder in which there are pauses in breathing during sleep. People with this condition must wake up frequently to breathe. In a sample of 427 people aged 65 and over, 104 of them had sleep apnea. a. Find a point estimate for the population proportion of those aged 65 and over who have sleep apnea. 0.244 b. Construct a 99% confidence interval for the proportion of those aged 65 and over who have sleep apnea. (0.190, 0.297) c. In another study, medical researchers concluded that more than 9% of elderly people have sleep apnea. Based on the confidence interval, does it appear that more than 9% of people aged 65 and over have sleep apnea? Explain. Yes Sources: Sleep 14:486–495; Mayo Clinic Proceedings 76:897–905 22. Internet service: An Internet service provider sampled 540 customers and found that 75 of them experienced an interruption in high-speed service during the previous month. a. Find a point estimate for the population proportion of all customers who experienced an interruption. 0.139 b. Construct a 90% confidence interval for the proportion of all customers who experienced an interruption. (0.114, 0.163) c. The company’s quality control manager claims that no more than 10% of its customers experienced an interruption during the previous month. Does the confidence interval contradict this claim? Explain. Yes 23. Volunteering: In 2012 the General Social Survey asked 1294 people whether they performed any volunteer work during the past year. A total of 517 people said they did. a. Find a point estimate for the proportion of people who performed volunteer work during the past year. 0.400 b. Construct a 95% confidence interval for the proportion of people who performed volunteer work during the past year. (0.373, 0.426) c. A sociologist states that 50% of Americans perform volunteer work in a given year. Does the confidence interval contradict this statement? Explain. Yes 24. SAT scores: A college admissions officer sampled 120 entering freshmen and found that 42 of them scored more than 550 on the math SAT. a. Find a point estimate for the proportion of all entering freshmen at this college who scored more than 550 on the math SAT. 0.350 b. Construct a 98% confidence interval for the proportion of all entering freshmen at this college who scored more than 550 on the math SAT. (0.249, 0.451) c. According to the College Board, 39% of all students who took the math SAT in 2009 scored more than 550. The admissions officer believes that the proportion at her university is also 39%. Does the confidence interval contradict this belief? Explain. No 25. WOW: In the computer game World of Warcraft, some of the strikes are critical strikes, which do more damage. Assume that the probability of a critical strike is the same for every attack, and that attacks are independent. During a particular fight, a character has 242 critical strikes out of 595 attacks. a. Construct a 95% confidence interval for the proportion of strikes that are critical strikes. (0.367, 0.446) b. Construct a 98% confidence interval for the proportion of strikes that are critical strikes. (0.360, 0.454) c. What is the effect of increasing the level of confidence on the width of the interval? Makes it wider 26. Contaminated water: In a sample of 42 water specimens taken from a construction site, 26 contained detectable levels of lead. a. Construct a 90% confidence interval for the proportion of water specimens that contain detectable levels of lead. (0.496, 0.742) b. Construct a 95% confidence interval for the proportion of water specimens that contain detectable levels of lead. (0.472, 0.766) c. What is the effect of increasing the level of confidence on the width of the interval? Makes it wider Source: Journal of Environmental Engineering 128:237–245 27. Call me: A sociologist wants to construct a 95% confidence interval for the proportion of children aged 8–10 living in New York who own a cell phone. a. A survey by the National Consumers League taken in 2012 estimated the nationwide proportion to be 0.32. Using this estimate, what sample size is needed so that the confidence interval will have a margin of error of 0.02? 2090 b. Estimate the sample size needed if no estimate of p is available. 2401 c. If the sociologist wanted to estimate the proportion in the entire United States rather than in New York, would the necessary sample size be larger, smaller, or about the same? Explain. About the same 28. Reading proficiency: An educator wants to construct a 98% confidence interval for the proportion of elementary schoolchildren in Colorado who are proficient in reading. a. The results of a recent statewide test suggested that the proportion is 0.70. Using this estimate, what sample size is needed so that the confidence interval will have a margin of error of 0.05? 455 b. Estimate the sample size needed if no estimate of p is available. 542 c. If the educator wanted to estimate the proportion in the entire United States rather than in Colorado, would the necessary sample size be larger, smaller, or about the same? Explain. About the same 29. Surgical complications: A medical researcher wants to construct a 99% confidence interval for the proportion of knee replacement surgeries that result in complications. a. An article in the Journal of Bone and Joint Surgery suggested that approximately 8% of such operations
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