Introduction to Linear Regression Analysis. Douglas C. Montgomery

      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.

2.3.2 Testing Significance of Regression

      A very important special case of the hypotheses in Eq. (2.23) is

      (2.30)

These hypotheses relate to the significance of regression. Failing to reject H₀: β₁ = 0 implies that there is no linear relationship between x and y. This situation is illustrated in Figure 2.2. Note that this may imply either that x is of little value in explaining the variation in y and that the best estimator of y for any x is (Figure 2.2a) or that the true relationship between x and y is not linear (Figure 2.2b). Therefore, failing to reject H₀: β₁ = 0 is equivalent to saying that there is no linear relationship between y and x.

Alternatively, if H₀: β₁ = 0 is rejected, this implies that x is of value in explaining the variability in y. This is illustrated in Figure 2.3. However, rejecting H₀: β₁ = 0 could mean either that the straight-line model is adequate (Figure 2.3a) or that even though there is a linear effect of x, better results could be obtained with the addition of higher order polynomial terms in x (Figure 2.3b).

Figure 2.2 Situations where the hypothesis H₀: β₁ = 0 is not rejected.

Figure 2.3 Situations where the hypothesis H₀: β₁ = 0 is rejected.

      The test procedure for H₀: β₁ = 0 may be developed from two approaches. The first approach simply makes use of the t statistic in Eq. (2.27) with β₁₀ = 0, or

      The null hypothesis of significance of regression would be rejected if |t₀| > t_α/2,n−2.

      Example 2.3 The Rocket Propellant Data

      We test for significance of regression in the rocket propellant regression model of Example