Independent random samples of ceramic produced by two different processes were tested for hardness. The results were:

(a) We will assume that the observations come from normal(µ1, σ2) and normal(µ2, σ2), where σ2 = .42. Use independent normal (m, s2) prior Find the posterior distributions of µ1 and µ2, respectively.

(b) Find the posterior distribution of µ1 − µ1.

(c) Find a 95% Bayesian credible interval for µ1 − µ2.

(d) Perform a Bayesian test of the hypothesis H0 : µ1 − µ2 ≥ 0 versus H1 : µ1 − µ2 < 0 at the 5% level of significance. What conclusion can we draw?