The annual proportion of new restaurants that survive in business for at least 1 year in a U.S. city with population \(\geq 500,000\) people is assumed to be the outcome of some Beta population distribution. Part of the maintained hypothesis is that \(b=1\) in the Beta distribution, so that the population distribution is assumed to be Beta(a,1). A random sample of size 50 from the beta population distribution results in the geometric mean \(\bar{x}_{g}=.84\).

(a) Define a UMP level 05 test of the hypothesis that less than three-quarters of new restaurants are expected to survive at least 1 year in business in U.S. cities of size \(\geq 500,000\). Test the hypothesis.

(b) Plot the power function for the test. Interpret the power function both from the standpoint of a potential investor in a restaurant and from the perspective of the managing director of a chamber of commerce.

(c) Calculate and interpret the \(p\)-value for the test.