The accompanying data was obtained from a study of a certain method for preparing pure alcohol from refinery streams (“Direct Hydration of Olefins,” Industrial and Eng.

Chemistry, 1961: 209–211). The independent variable x is volume hourly space velocity, and the dependent variable y is the amount of conversion of iso-butylene.

x | 1124446 y | 23.0 24.5 28.0 30.9 32.0 33.6 20.0

a. Assuming that a quadratic probabilistic model is appropriate, estimate the regression function.

b. Determine the predicted values and residuals, and construct a residual plot. Does the plot look roughly as expected when the quadratic model is correct? Does the plot indicate that any observation has had a great influence on the fit? Does a scatter plot identify a point having large influence? If so, which point?

c. Obtain s2 and R2

. Does the quadratic model provide a good fit to the data?

d. In Exercise 11, it was noted that the predicted value Yˆ

j and the residual Yj Yˆ

j are independent of one another, so that 2
V(Yj)
V(Yˆj) V(Yj Yˆ

j). A computer printout gives the estimated standard deviations of the predicted values as .955, .955, .712, .777, .777, .777, and 1.407. Use these values along with s2 to compute the estimated standard deviation of each residual. Then compute the standardized residuals and plot them against x. Does the plot look much like the plot of part

(b)? Suppose you had standardized the residuals using just s in the denominator. Would the resulting values differ much from the correct values?

e. Using information given in part (d), compute a 90% PI for isobutylene conversion when volume hourly space velocity is 4.