Problem Set #6

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Problem Set #6 (Chapter 7) 24. U.S. Unemployment. Barron’s reported that the average number of weeks an individual is unemployed is 17.5 weeks. Assume that for the population of all unemployed individuals the population mean length of unemployment is 17.5 weeks and that the population standard deviation is four weeks. Suppose you would like to select a sample of 50 unemployed individuals for a follow-up study. LO 6, 7, 8 a. Describe the sampling distribution of z , the sample mean average for a sample of 50 unemployed individuals, by specifying its distribution form and parameters. b. What is the probability that a simple random sample of 50 unemployed individuals will provide a sample mean within one week of the population mean? c. What is the probability that a simple random sample of 50 unemployed individuals will provide a sample mean within 1/2 week of the population mean? 26. Federal Income Tax Returns. The Wall Street Journal reports that 33% of taxpayers with adjusted gross incomes between $30,000 and $60,000 itemized deductions on their federal income tax return. The mean amount of deductions for this population of taxpayers was $16,642. Assume the standard deviation is o = $2,400.L0 6,7, 8 a. What is the probability that a sample of taxpayers from this income group who have itemized deductions will show a sample mean within $200 of the population mean for each of the following sample sizes: 30, 50, 100, and 4007? b. What is the advantage of a larger sample size when attempting to estimate the population mean?

28. State Rainfalls. The state of California has a mean annual rainfall of 22 inches, whereas the state of New York has a mean annual rainfall of 42 inches. Assume that the standard deviation for both states is 4 inches. A sample of 30 years of rainfall for California and a sample of 45 years of rainfall for New York has been taken. LO 6, 7, 8 a. Describe the probability distribution of the sample mean annual rainfall for California by specifying its distribution form and parameters. b. What is the probability that the sample mean is within 1 inch of the population mean for California? c. What is the probability that the sample mean is within 1 inch of the population mean for New York? d. In which case, part (b) or part (c), is the probability of obtaining a sample mean within 1 inch of the population mean greater? Why?

38. Unnecessary Medical Care. According to Reader’s Digest, 42% of primary care doctors think their patients receive unnecessary medical care. LO 9, 10, 11 a. Suppose a sample of 300 primary care doctors was taken. Specify the distribution form and parameters of the sampling distribution of the proportion of the doctors who think their patients receive unnecessary medical care. b. What is the probability that the sample proportion will be within +0.03 of the population proportion? c. What is the probability that the sample proportion will be within +0.05 of the population proportion? d. What would be the effect of taking a larger sample on the probabilities in parts (b) and (c)? Why?

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39. Better Business Bureau Complaints. In 2016 the Better Business Bureau settled 80% of complaints they received in the United States. Suppose you have been hired by the Better Business Bureau to investigate the complaints they received this year involving new car dealers. You plan to select a sample of new car dealer complaints to estimate the proportion of complaints the Better Business Bureau is able to settle. Assume the population proportion of complaints settled for new car dealers is 0.80, the same as the overall proportion of complaints settled in 2016. L0 9, 10, 11 a. Suppose you select a sample of 200 complaints involving new car dealers. Specify the distribution form and parameters of the sampling distribution of p . b. Based upon a sample of 200 complaints, what is the probability that the sample proportion will be within 0.04 of the population proportion? c. Suppose you select a sample of 450 complaints involving new car dealers. Specify the distribution form and parameters of the sampling distribution of p . d. Based upon the larger sample of 450 complaints, what is the probability that the sample proportion will be within 0.04 of the population proportion? e. As measured by the increase in probability, how much do you gain in precision by taking the larger sample in part (d)?

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