A history professor is making up a final examination that is to be given to a very large group of students. His feelings about the average grade that they should get is expressed subjectively by a normal distribution with the mean µ0 = 65.2 and the standard deviation σ0 = 1.5.
(a) What prior probability does the professor assign to the actual average grade being somewhere on the interval from 63.0 to 68.0?
(b) What posterior probability would he assign to this event if the examination is tried on a random sample of 40 students whose grades have a mean of 72.9 and a standard deviation of 7.4? Use s = 7.4 as an estimate of σ.