6.
Consider the simple linear regression model y = 10 + 25x + \epsilon where the random error term is normally and independently distributed with mean zero and standard deviation 2. Use software to generate a sample of eight observations, one each at the levels x = 10, 12, 14, 16, 18, 20, 22, and 24.
a. Fit the linear regression model by least squares and find the estimates of the slope and intercept.
b. Find the estimate of $\sigma^2$
c. Find the standard errors of the slope and intercept.
d. Now use software to generate a sample of 16 observations, two each at the same levels of x used previously. Fit the model using least squares.
e. Find the estimate of $\sigma^2$ for the new model in part (d). Compare this to the estimate obtained in part (b). What impact has the increase in sample size had on the estimate?
f. Find the standard errors of the slope and intercept using the new model from part (d). Compare these standard errors to the ones that you found in part (c). What impact has the increase in sample size had on the estimated standard errors?
