Читать книгу Introduction to Linear Regression Analysis - Douglas C. Montgomery - Страница 31

Suppose that we wish to test the hypothesis that the slope equals a constant, say β₁₀. The appropriate hypotheses are

(2.23)

where we have specified a two-sided alternative. Since the errors εi are NID(0, σ²), the observations yi are NID(β₀ + β₁xi, σ²). Now is a linear combination of the observations, so is normally distributed with mean β₁ and variance σ²/Sxx using the mean and variance of found in Section 2.2.2. Therefore, the statistic

is distributed N(0, 1) if the null hypothesis H₀: β₁ = β₁₀ is true. If σ² were known, we could use Z₀ to test the hypotheses (2.23). Typically, σ² is unknown. We have already seen that MS_Res is an unbiased estimator of σ². Appendix C.3 establishes that (n − 2)MS_Res/σ² follows a distribution and that MS_Res and are independent. By the definition of a t statistic given in Section C.1,

(2.24)

follows a t_n−2 distribution if the null hypothesis H₀: β₁ = β₁₀ is true. The degrees of freedom associated with t₀ are the number of degrees of freedom associated with MS_Res. Thus, the ratio t₀ is the test statistic used to test H₀: β₁ = β₁₀. The test procedure computes t₀ and compares the observed value of t₀ from Eq. (2.24) with the upper α/2 percentage point of the t_n−2 distribution (t_α/2,n−2). This procedure rejects the null hypothesis if

(2.25)

Alternatively, a P-value approach could also be used for decision making.

The denominator of the test statistic, t₀, in Eq. (2.24) is often called the estimated standard error, or more simply, the standard error of the slope. That is,

(2.26)

Therefore, we often see t₀ written as

(2.27)

A similar procedure can be used to test hypotheses about the intercept. To test

(2.28)

we would use the test statistic

(2.29)

where is the standard error of the intercept. We reject the null hypothesis H₀: β₀ = β₀₀ if |t₀| > t_α/2,n−2.