Reference no: EM131093822

Refer to Machine speed Problem 11.7.

a. On the basis of the weighted least squares fit in Problem II. 7e, construct an approximate 90 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 SOO bootstrap samples of size 12. For each bootstrap sample, (I) use ordinary least squares to regress Yon 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 (l1.I6a) to obtain weights, and (3) use weighted least squares to regression X and obtain the bootstrap estimated regression coefficient b; . (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 800 bootstrap estimates bf. Does the bootstrap sampling distribution of bf appear to approximate a normal distribution?

d. Calculate the sample standard deviation of the 800 bootstrap estimates . How does this value compare to the estimated standard deviation s{b_w1} used in part (a)?

e. Construct a 90 percent bootstrap confidence interval for ß₁ using reflection method (11.59). How does this confidence interval compare with that obtained in part (a)7 Does the approximate interval in part (a) appear to be useful for this data set?

Problem 11.7

Machine speed. The number of defective items produced by a machine (Y) is known to be linearly related to the speed setting of the machine (X). The data below were collected from recent quality control records.

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. Conduct the Breusch-Pagan test for constancy of the error variance, assuming log σ²₁? = Y_o + Y_l X_i use  α = .10. State the alternatives, decision rule, and conclusion.

c. Plot the squared residuals against X. What does the plot suggest about the relation between the variance of the error term and X?

d. Estimate the variance function by regressing the squared residuals against-X, and then calculate the estimated weight for each case using (11.16b).

e. Using the estimated weights, obtain the weighted least squares estimates of ß₀ and ß₁. Are the weighted least squares 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 b_wo and b_W1 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?