16.4 We are taking a random sample from the nonnal(p,u2) distribution where we know the standard deviation u = 4. Suppose that our prior experience in similar situations leads us to believe that the prior distribution should be norm~1(255,4~). 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 norm~1(255,4~) and gI(p) is the n0nna1(255,12~) distribution, and the prior probability P(I = 0) = .95. We take a random sample of n = 10 observations. They are 249 258 255 26 1 259 254 26 1 256 253 254

(a) Use NormMixp.mac 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 distribution g(p1 y).

significance.