In this problem, we will construct a narrower con- fidence

Chapter 11, Problem 25

(choose chapter or problem)

In this problem, we will construct a narrower con- fidence band for a regression function using Theorem 11.3.7. Let 0 and 1 be the least-squares estimators, and let be the estimator of used in this section. Let x0 < x1 be two possible values of the predictor X. a. Find formulas for the simultaneous coefficient 1 0 confidence intervals for 0 + 1x0 and 0 + 1x1. b. For each real number x, find the formula for the unique such that x = x0 + (1 )x1. Call that value (x). c. Call the intervals found in part (a) (A0, B0) and (A1, B1), respectively. Define the event C = {A0 < 0 + 1x0 < B0 and A1 < 0 + 1x1 < B1}. For each real x, define L(x) and U (x) to be, respectively, the smallest and largest of the following four numbers: (x)A0 + [1 (x)]A1, (x)B0 + [1 (x)]A1, (x)A0 + [1 (x)]B1, (x)B0 + [1 (x)]B1. If the event C occurs, prove that, for every real x, L(x) < 0 + 1x < U (x).