A thermal power station discharges its cooling water into river. An environmental scientist wants to determine if this has adversely affected the dissolved oxygen level. She takes samples of water one kilometer upstream from the power station, and one kilometer downstream from the power station, and measures the dissolved oxygen level. The data are:

(a) We will assume that the observations come from normal(µ1, σ2) and normal(µ2, σ2), where σ2 = 22. Use independent normal (m, s2) prior distributions for µ1 and µ2, respectively, where m = 10 and s2 = 22.
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?