An automobile manufacturer claims that the average mileage per gallon of gas for a particular model is normally distributed with M' = 20 and a' = 4, provided that the car is driven on a level road at a constant speed of 30 miles per hour. A rival manufacturer decides to use this as a prior distribution and to obtain additional information by conducting an experiment. In the experiment, it is assumed that the variance of mileage is a2 = 96 and the mileage is normally distributed. The experiment is conducted on 10 randomly chosen cars, with the sample mean (the average mileage in the sample)

equaling 18. Find the posterior distribution for fi, the average mileage per gallon of gas. On the basis of this posterior, what is the probability that /x is greater than or equal to 20? What is the probability that /x is between 19 and 21 ?