A pharmaceutical company is analyzing the effectiveness of a new drug that it claims can stimulate hair growth in balding men. For the purposes of an advertising campaign, the marketing department would like to be able to claim that the drug will be effective for at least 50 percent of the balding men who use it. To test the claim, a random sample of 25 balding men are given the drug treatment, and it is found that 10 applications were effective in stimulating hair growth. The population of balding men is sufficiently large that you may treat this as a problem of random sampling with replacement.

(a) Test the null hypothesis \(H_{0}: p \geq .50\) using as close to a .10-size test as you can.

(b) Plot the power function for this test. Interpret the power function from the standpoint of both the pharmaceutical company and the consuming public.

(c) Is the test you used in

(a) a UMP test? Is it a UMPU test? Is it an unbiased test? Is it a consistent test?

(d) Calculate and interpret the \(P\)-value for the test.