Читать книгу Introduction to Linear Regression Analysis - Douglas C. Montgomery - Страница 39

In addition to point estimates of β₀, β₁, and σ², we may also obtain confidence interval estimates of these parameters. The width of these confidence intervals is a measure of the overall quality of the regression line. If the errors are normally and independently distributed, then the sampling distribution of both and is t with n − 2 degrees of freedom. Therefore, a 100(1 − α) percent confidence interval (CI) on the slope β₁ is given by

(2.39)

and a 100(1 − α) percent CI on the intercept β₀ is

(2.40)

These CIs have the usual frequentist interpretation. That is, if we were to take repeated samples of the same size at the same x levels and construct, for example, 95% CIs on the slope for each sample, then 95% of those intervals will contain the true value of β₁.

If the errors are normally and independently distributed, Appendix C.3 shows that the sampling distribution of (n − 2)MS_Res/σ² is chi square with n − 2 degrees of freedom. Thus,

and consequently a 100(1 − α) percent CI on σ² is

(2.41)