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 .05, using a 95% confidence interval. He did so by assuming that the true error rate was p = .5, 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 p̂ = .20 and a 95% confidence interval for p was (.15, .25). Was the desired bound on the error of the estimate met?