Suppose that a statistician is interested in the difference in the means of two normal populations, pi and p2. Let p = pi — p2. Each population has vari¬

ance 100, and the two populations are independent. A sample of size 25 is taken from the first population, with sample mean Mi = 80, and a sample of size 25 is taken from the second population, with sample mean M2 = 60.

(a) Using the classical approach to hypothesis testing presented in Chapter 7, test Ho 'pi — P2 < 0 against the alternative Hi'.pi — p2> 0, with a = .10.

(b) Let p = pi — p2, and suppose that the prior distribution of p is normally distributed with mean M' — 10 and variance a'2 = 50. Find P' (Ho)

and P' (Hi).

(c) Find the posterior distribution of p = pi — p2.

(d) From the posterior distribution, find the posterior odds ratio of Ho to Hi.