The mathematics department, of a large university is designing a placement test, to be given to the incoming freshman classes. Members of the: department feel that the average grade for this test will vary from one freshman class to another. This variation of the average class grade is expressed subjectively by a normal distribution with mean μ₀ = 72 and variance: σ₀² = 5.7G.

(a) What, prior probability does the department assign to the actual average grade being somewhere between 71.8 and 73.4 for next year's freshman class?

(b) If the test is tried on a random sample of 100 freshman students from the next, incoming freshman class resulting in an average grade of 70 with a variance of 64, construct a 95% Bayesian interval for μ.

(c) What posterior probability should the department assign to the event of part (a)