Suppose you want to find the mean μ of the math SAT scores for the class of 2008 high school seniors in your town. You know the distribution of scores is normal with a standard deviation of 116, which is the standard deviation for the national distribution of scores. On the basis of the information on previous classes, you put a normal prior on μ, say μ ∼ N(600, 252). If a sample of size 60 yields a mean score of 538,
(a) find the posterior distribution of μ;
(b) find a 95% credible interval for the true μ; and (c) find the probability that the posterior mean math SAT score is greater than 600.