Suppose that under H0, a measurement X is N(0,?2) and that under H1, X is N(1,?2) and that the prior probability P(H0) = P(H1). For ? = 1and x ? [0,3], plot and compare (i) the p-value for the test of H0 and (ii) P(H0|x). Can the p-value be interpreted as the probability that H0 is true? Choose another value of ? and repeat.
In the previous problem with ? = 1, what is the probability that the p-value is less than 0.05 if H0 is true? What is the probability if H1 is true?