Let X1, ..., X16 be i.i.d from N(, 2 ) with 2 = 16 given and P16 i=1 xi = 128.

1. Let X1, ...,X16 be gland from New?) with 0'2 = 16 given and 23:1 9:.- = 128. (a) Use a Bayes Test to test H0 : p: = 7 vs H1 : ,u = 9 if Q = {it} consists of only these 2 points with prior probabilities: 7r(,u = 7) = g and 'iT(f.L = 9) = i. (b) Suppose ,u can be any real value with a prior distribution 7r(,u,) = N(6, 4). (i) Find the Bayes Estimate of ,u. (ii) Find the 95% Highest Posterior Density (HPD) credible interval for ,u. (iii) Use a Bayes Test to test H0 : p S 9 vs H1 : p > 9