Magic Valley Memorial Hospital administrators have recently received an internal audit report that indicates that 15% of all patient bills contain an error of one form or another. After spending considerable effort to improve the hospital's billing process, the administrators are convinced that things have improved. They believe that the new error rate is somewhere closer to 0.05.
a) Suppose that recently the hospital randomly sampled 10 patient bills and conducted a thorough study to determine whether an error exists. It found 3 bills with errors. Assuming that managers are correct that they have improved the error rate to 0.05, what is the probability that they would find 3 or more errors in a sample of 10 bills?
b) What is the probability that they would find 3 or more errors in a sample of 10 bills if the administrators had actually failed in improving the billing process, and the error rate was still 15%.
c) What conclusion would you reach about the accuracy of the hospital administrators' claim to have improved the error rate to 5%?