A large metropolitan branch of a savings and loan is examining staffing issues and wants to test the hypothesis that the expected number of customers requiring the services of bank personnel during the midweek (TuesdayThursday) noon hour is \(\leq 50\). The bank has obtained the outcome of a random sample consisting of 100 observations on the number of noon hour customers requiring service from bank personnel. It was observed that \(\overline{\mathbf{x}}=54\). You may assume that the population distribution is Poisson in this case.

(a) Design a UMP level . 05 test of the null hypothesis having size as close to 05 as possible. Test the hypothesis.

(b) Plot the power function for the test. Interpret the power function both from the standpoint of management's desire for staff reductions and the need to provide quality customer service to bank customers.