16.3 We will use the Minitab macro NormMirPrnmac or the equivalent R function to find the posterior distribution of p given a random sample y1,. . . , yn from the normal(p, u2) distribution where we know the standard deviation u = 5.

when we use a mixture prior for p. Suppose that our prior experience in similar situations leads us to believe that the prior distribution should be norrnal(1000, 52). However, we consider that the prior experience may have been under different circumstances, so we decide to use a mixture prior where go(p)

is the normal(1000, 52) and gl(p) is the noma1(1000, 152) distribution, and the prior probability P(I = 0) = .95. We take a random sample of n = 10 observations. They are 1030 1023 1027 1022 1023 1023 1030 1018 1015 101 1

(a) Use NorrnMixp.rnac or the equivalent R function to find the posterior

(b) Find a 95% Bayesian credible interval for p.

(c) Test the hypothesis HO : p 5 1000 vs. HI : p > 1000 at the 5% level of