Consider the simple linear regression model y = 50 + 10 x + where is NID (0, 16). Suppose that n = 20

Consider the simple linear regression model y = 50 + 10 x + ε where ε is NID

(0, 16). Suppose that n = 20 pairs of observations are used to fi t this model.

Generate 500 samples of 20 observations, drawing one observation for each level of x = 1, 1.5, 2, . . . , 10 for each sample.

a. For each sample compute the least - squares estimates of the slope and intercept. Construct histograms of the sample values of ˆ
β0 and ˆ
β1. Discuss the shape of these histograms.

b. For each sample, compute an estimate of E ( y | x = 5). Construct a histogram of the estimates you obtained. Discuss the shape of the histogram.

c. For each sample, compute a 95% CI on the slope. How many of these intervals contain the true value β1 = 10? Is this what you would expect?

d. For each estimate of E ( y | x = 5) in part

b, compute the 95% CI. How many of these intervals contain the true value of E ( y | x = 5) = 100? Is this what you would expect?