Читать книгу Introduction to Linear Regression Analysis - Douglas C. Montgomery - Страница 41

A major use of a regression model is to estimate the mean response E(y) for a particular value of the regressor variable x. For example, we might wish to estimate the mean shear strength of the propellant bond in a rocket motor made from a batch of sustainer propellant that is 10 weeks old. Let x₀ be the level of the regressor variable for which we wish to estimate the mean response, say E(y|x₀). We assume that x₀ is any value of the regressor variable within the range of the original data on x used to fit the model. An unbiased point estimator of E(y|x₀) is found from the fitted model as

(2.42)

To obtain a 100(1 − α) percent CI on E(y|x₀), first note that is a normally distributed random variable because it is a linear combination of the observations yi. The variance of is

since (as noted in Section 2.2.4) . Thus, the sampling distribution of

is t with n − 2 degrees of freedom. Consequently, a 100(1 − α) percent CI on the mean response at the point x = x0 is

(2.43)

Note that the width of the CI for E(y|x₀) is a function of x₀. The interval width is a minimum for and widens as increases. Intuitively this is reasonable, as we would expect our best estimates of y to be made at x values near the center of the data and the precision of estimation to deteriorate as we move to the boundary of the x space.