Question: Suppose X is a normally distributed random variable with mean µ = 200 and standard deviation σ = 20

a. Sketch the probability density function of X¯. On the same graph, sketch the probability density function of the sample mean X from a sample of 16 observations.

b. Calculate P(180 ≤ X ≤ 220).

c. Calculate P(160 ≤ X ≤ 240).

d. Calculate P(195 ≤ X ≤ 205) for n = 4, 10, 20, 50, 100. Graph the calculated probabilities against n on a graph.

e. Calculate P(195 ≤ X ≤ 205) for a = 2, 4, 12, 16, 20, 30. Graph the calculated probabilities against a on a graph.

f. Suppose we wanted to conduct a survey. It is desired that we produce an interval estimate of the mean that is ± 5 from µ. with 99% confidence. Assume a value of σ = 15. How big should your sample be?