Galileo's Data. Use Galileo's data in Display 10.1 (data file: case1001) to perform the following operations.

(a) Fit the regression of distance on height and height-squared. Obtain the estimates, their standard errors, the estimate of a 2, and the variance-covariance matrix of the estimated coefficients.

(b) Verify that the square roots of the diagonal elements are equal to the standard errors reported with the estimated coefficients.

(c) Compute the estimated mean distance when the initial height is 500 punti.

(d) Calculate the standard error for the estimated mean in part (c).

(e) Use the answer to parts (a) and (d) and the relationship between the variance of the estimated mean and the variance of prediction to obtain the standard error of prediction at an initial height of 500 punti.