A based on the weighted least squares fit in problem 116e

Question

Refer to Computer-assisted learning Problem I 1.6

a. Based on the weighted least squares fit in Problem 11.6e, construct an approximate 95 percent confidence interval for ß₁ by means of (6.50), using the estimated standard deviation s{b_n-1}

b. Using random X sampling, obtain 750 bootstrap samples of size 12. For each bootstrap sample, (I) use ordinary least squares to regress Y on X and obtain the residuals, (2) estimate the standard deviation function by regressing the absolute residuals on X and then use the fitted standard deviation function and (11.16a) to obtain weights. and (3) use weighted least squares to regress Y on X and obtain the bootstrap estimated regression coefficient b₁. (Note that for each bootstrap sample, only one iteration of the iteratively reweighted least squares procedure is to be used.)

c. Construct a histogram of the 750 bootstrap estimates b₁. Does the bootstrap sampling distribution of b₁' appear to approximate a normal distribution?

d. Calculate the sample standard deviation of the 750 bootstrap estimates . How does this value compare to the estimated standard deviation s{b_m-1} used in part (a)?

e. Construct a 95 percent bootstrap confidence interval for ß₁ using reflection method(1 1.59). How does this confidence interval compare with that obtained in part (a)? Does the approximate interval in part (a) appear to be useful for this data set?

Problem 11.6

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) und the cost of computer time (Y. in cents), follow.

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  into gl1lup I and the other six cases into group 2. Conduct the Bl1lwn-Forsythe test for constancy of the error variance. using α = .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.I6a). 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 square estimates b_wo and hl/'I 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?