Suppose that the proportion p of defective items in a large manufactured lot is unknown, and it is desired to test the following simple hypotheses:
H0: p = 0.3,
H1: p = 0.4.
Suppose that the prior probability that p = 0.3 is 1/4, and the prior probability that p = 0.4 is 3/4; also suppose that the loss from choosing an incorrect decision is 1 unit, and the loss from choosing a correct decision is 0. Suppose that a random sample of n items is selected from the lot. Show that the Bayes test procedure is to reject H0 if and only if the proportion of defective items in the sample is greater than
log (7/6) + 1/n log (1/3)/log(14/9).