Chapter 6: Q26E (page 348)
Question: Let t0 be a specific value of t. Use Table III in Appendix D to find t0 values such that the following statements are true.
Answer
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Water pollution testing. The EPA wants to test a randomlyselected sample of n water specimens and estimate themean daily rate of pollution produced by a mining operation.If the EPA wants a 95% confidence interval estimatewith a sampling error of 1 milligram per liter (mg/L),how many water specimens are required in the sample?Assume prior knowledge indicates that pollution readingsin water samples taken during a day are approximately
normally distributed with a standard deviation equal to5 mg/L.
In each case, find the approximate sample size required to construct a 95% confidence interval for p that has sampling error of SE = .08.
a. Assume p is near .2.
b. Assume you have no prior knowledge about p, but you wish to be certain that your sample is large enough to achieve the specified accuracy for the estimate.
A sampling dispute goes to court. Sampling of Medicare and Medicaid claims by the federal and state agencies who administer those programs has become common practice to determine whether providers of those services are submitting valid claims. (See the Statistics in Action for this chapter.) The reliability of inferences based on those samples depends on the methodology used to collect the sample of claims. Consider estimating the true proportion, p, of the population of claims that are invalid. (Invalid claims should not have been reimbursed by the agency.) Of course, to estimate a binomial parameter, p, within a given level of precision we use the formula provided in Section 6.5 to determine the necessary sample size. In a recent actual case, the statistician determined a sample size large enough to ensure that the bound on the error of the estimate would not exceed 0.05, using a 95% confidence interval. He did so by assuming that the true error rate was, which, as discussed in Section 6.5, provides the maximum sample size needed to achieve the desired bound on the error.
a. Determine the sample size necessary to estimate p to within .05 of the true value using a 95% confidence interval.
b. After the sample was selected and the sampled claims were audited, it was determined that the estimated error rate was and a 95% confidence interval for p was (0.15, 0.25). Was the desired bound on the error of the estimate met?
c. An economist hired by the Medicare provider noted that, since the desired bound on the error of .05 is equal to 25% of the estimated invalid claim rate, the “true” bound on the error was .25, not .05. He argued that a significantly larger sample would be necessary to meet the “relative error” (the bound on the error divided by the error rate) goal of .05, and that the statistician’s use of the “absolute error” of .05 was inappropriate, and more sampling was required. The statistician argued that the relative error was a moving target, since it depends on the sample estimate of the invalid claim rate, which cannot be known prior to selecting the sample. He noted that if the estimated invalid claim rate turned out to be larger than .5, the relative error would then be lower than the absolute error bound. As a consequence, the case went to trial over the relative vs. absolute error dispute. Give your opinion on the matter. [Note: The Court concluded that “absolute error was the fair and accurate measure of the margin of error.” As a result, a specified absolute bound on the error continues to be the accepted method for determining the sample size necessary to provide a reliable estimate of Medicare and Medicaid providers’ claim submission error rates.]
Monitoring phone calls to a toll-free number. A largefood-products company receives about 100,000 phone callsa year from consumers on its toll-free number. A computermonitors and records how many rings it takes for an operatorto answer, how much time each caller spends “on hold,” andother data. However, the reliability of the monitoring systemhas been called into question by the operators and their labour unions. As a check on the computer system, approximatelyhow many calls should be manually monitored during thenext year to estimate the true mean time that callers spend onhold to within 3 seconds with 95% confidence? Answer thisquestion for the following values of the standard deviation ofwaiting times (in seconds): 10, 20, and 30.
Accounting and Machiavellianism. Refer to the Behavioral Research in Accounting (January 2008) study of Machiavellian traits in accountants, Exercise 1.33 (p. 52). Recall that Machiavellian describes negative character traits that include manipulation, cunning, duplicity, deception, and bad faith. A Machiavellian (“Mach”) rating score was determined for each in a sample of accounting alumni of a large southwestern university. Scores range from a low of 40 to a high of 160, with the theoretical neutral Mach rating score of 100. The 122 purchasing managers in the sample had a mean Mach rating score of 99.6, with a standard deviation of 12.6.
a. From the sample, estimate the true mean Mach rating score of all purchasing managers.
b. Form a 95% confidence interval for the estimate, part b.
c. Give a practical interpretation of the interval, part c.
d. A director of purchasing at a major firm claims that the true mean Mach rating score of all purchasing managers is 85. Is there evidence to dispute this claim? From the given information we have 122 purchasing managers in the sample had a mean Mach rating score of 99.6, with a standard deviation of 12.6.
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