In this problem, we will construct a narrower confidence band for a regression function using Theorem 11.3.7. Let be the least-squares estimators, and let σ' be the estimator of σ used in this section. Let x0 < x1 be two possible values of the predictor X.
a. Find formulas for the simultaneous coefficient 1− α0 confidence intervals for β0 + β1x0 and β0 + β1x1.
b. For each real number x, find the formula for the unique α such that x = αx0 + (1 − α)x1. Call that value α(x).
c. Call the intervals found in part (a) (A0, B0) and (A1, B1), respectively. Define the event
C = {A0 < β0 + β1x0 < B0 and A1 < β0 + β1x1 < B1}.
For each real x, define L(x) and U(x) to be, respectively, the smallest and largest of the following four numbers:
α(x)A0 + [1− α(x)]A1, α(x)B0 + [1− α(x)]A1,
α(x)A0 + [1− α(x)]B1, α(x)B0 + [1− α(x)]B1.
If the event C occurs, prove that, for every real x, L(x) < β0 + β1x < U(x).