Chapter B.2, Problem 44E

Stopping Distance. In their text Methods of Correlation and Regression Analysis (New York. Wiley, 1959). M. Ezekiel and K. Fox present data on car speed and the distance required to stop the car. The table below provides 63 observations on speed, in miles per hour (mph). and stopping distance, in feet. The scatterplot of stopping distance versus speed, given in Output B.33 below, shows curvature. We will use polynomial regression in an attempt to find a suitable regression equation to fit the data.

1. a. Based on the scatterplot in Output B.33. what degree polynomial would you use to fit the data?
2. b. We obtained Output B.34 on page B-55. showing the results of fitting a sequence of polynomial regression equations in the centered predictor variable $spee{d}_{{}_c}=speed-\overline{speed}$. We start with a first-degree (linear) equation in speed_c. then a second-degree (quadratic) equation, and so on. up to a fifth-degree (quintic) equation. Apply the forward selection method to determine a polynomial regression equation for these data. Use a 5% level of significance.

OUTPUT B.33 Scatterplot of stopping distance versus speed for Exercise B.44

1. c. Refer to Output B.34. Starting with a fifth-degree polynomial regression equation, apply the backward elimination method to determine a polynomial regression equation for these data. Use a 5% level of significance.
2. d. Do the forward selection and backward elimination methods yield the same results? If so, is that always the case?
3. e. Outputs B.35(a)-(c) on page B-56 show plots of residuals versus fitted values for the linear, quadratic, and cubic polynomial regression equations. Interpret these residual plots with regard to the appropriateness of each of the three polynomial regression equations. Do your interpretations agree with the regression models obtained in parts (b) and (c)? Which regression model would you select?
4. f. Using Output B.35. assess the assumption of equal conditional standard deviations.
5. g. In view of part (f). it might be more appropriate to use the method of transformations to determine a suitable regression model. Based on the results you obtained above, what transformation would you try? Explain your answer.

OUTPUT E.34

Output of polynomial regressions for Exercise B.44