Let X be a Gaussian random variable with mean 10 and variance 4. A sample of size 9 is obtained and the sample mean, minimum, and maximum of the sample are calculated. (a) Find the probability that the sample mean is less than 9.
(b) Find the probability that the minimum is greater than 8.
(c) Find the probability that the maximum is less than 12.
(d) Find n so the sample mean is within 1 of the true mean with probability 0.95.
(e) Generate 100 random samples of size 9. Compare the probabilities obtained in parts a, b, and c to the observed relative frequencies.