13.5 A thermal power station discharges its cooling water into a 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:

Upstream Downstream 10.1 9.7 10.2 10.3 13.4 6.4 8.2 7.3 9.8 11.7 8.9

(a) We will assume that the observations come from normal(p.1, u2) and normal(p2, a2), where u2 = 22. Use independent normal (m, s2) prior distributions for 1-11 and p2, respectively, where m = 10 and s2 = 22.

Find the posterior distributions of p1 and p2, respectively.

(b) Find the posterior distribution of p1 - pz.

(c) Find a 95% Bayesian credible interval for p1 - p2.

(d) Perform a Bayesian test of the hypothesis Ho : p1 - p2 5 0 versus HI : p1 - p2 > 0 at the 5% level of significance. What conclusion can we draw?