Computer-assisted learning. Data from a study of computer-assisted learning by 12 students, showing the total number of responses in completing a lesson (X) and the cost of computer time (Y, in cents), follow.
| i : | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
| Xj: | 16 | 14 | 22 | 10 | 14 | 17 | 10 | 13 | 19 | 12 | 18 | 11 |
| Yi: | 77 | 70 | 85 | 50 | 62 | 70 | 55 | 63 | 88 | 57 | 81 | 51 |
a. Fit a linear regression function by ordinary least squares, obtain the residuals, and plot the residuals against X. What does the residual plot suggest?
b. Divide the cases into two groups, placing the six cases with the smallest fitted values Yi into group 1 and the other six cases into group 2. Conduct the Brown-Forsythe test for constancy of the error variance, using a = .05. State the decision rule and conclusion.
c. Plot the absolute values of the residuals against X. What does this plot suggest about the relation between the standard deviation of the error term and X?
d. Estimate the standard deviation function by regressing the absolute values of the residuals against X, and then calculate the estimated weight for each case using (11.16a). Which case receives the largest weight? Which case receives the smallest weight?
e. Using the estimated weights, obtain the weighted least squares estimates of ß0 and ß1. Are these estimates similar to the ones obtained with ordinary least squares in part (a)?
f. Compare the estimated standard deviations of the weighted least squares estimates bw0and bw1 in part (e) with those for the ordinary least squares estimates in part (a). What do you find?
g. Iterate the steps in parts (d) and (e) one more time. Is there a substantial change in the estimated regression coefficients? If so, what should you do?
