12.3 The local consumer watchdog group was concerned about the cost of electricity to residential customers over the New Zealand winter months (Southern Hemisphere). They took a random sample of 25 residential electricity accounts and looked at the total cost of electricity used over the three months of June, July, and August. The costs were:

514 536 345 440 427 443 386 418 364 483 506 385 410 561 275 306 294 402 350 343 480 334 324 414 296 Assume that the amount of electricity used over the three months by aresidential account is normal (p, a2), where the known standard deviation = 80.

(a) Use a normal (325,802) prior for p. Find the posterior distribution for

(b) Find a 95% Bayesian credible interval for p.

(c) Perform a Bayesian test of the hypothesis P.

HO : p = 350 versus HI : p # 350 at the 5% leve