A mathematics test of 100 multiple-choice questions is to be given to all freshmen entering a large university. Initially, in a pilot study the test was given to a random sample of 20 freshmen. Suppose that, for the population of all entering freshmen, the distribution of the number of correct answers would be normal with a variance of 250.
a. What is the probability that the sample variance would be less than 100?
b. What is the probability that the sample variance would be more than 500?