alt="in27-3"/>. Note that the degrees of freedom have an additive property:
We can use the usual analysis-of-variance F test to test the hypothesis H0: β1 = 0. Appendix C.3 shows that (1) SSRes = (n − 2)MSRes/σ2 follows a
follows the F1,n−2 distribution. Appendix C.3 also shows that the expected values of these mean squares are
These expected mean squares indicate that if the observed value of F0 is large, then it is likely that the slope β1 ≠ 0. Appendix C.3 also shows that if β1 ≠ 0, then F0 follows a noncentral F distribution with 1 and n − 2 degrees of freedom and a non-centrality parameter of
This noncentrality parameter also indicates that the observed value of F0 should be large if β1 ≠ 0. Therefore, to test the hypothesis H0: β1 = 0, compute the test statistic F0 and reject H0 if
The test procedure is summarized in Table 2.4.
TABLE 2.4 Analysis of Variance for Testing Significance of Regression
| Source of Variation | Sum of Squares | Degrees of Freedom | Mean Square | F 0 |
| Regression |
|
1 | MS R | MSR/MSRes |
| Residual |
|
n − 2 | MS Res | |
| Total | SS T | n − 1 |
Example 2.4 The Rocket Propellant Data
We will test for significance of regression in the model developed in Example 2.1 for the rocket propellant data. The fitted model is
The analysis of variance is summarized in Table 2.5. The computed value of F0 is 165.21, and from Table A.4, F0.01,1,18 = 8.29. The P value for this test is 1.66 × 10−10. Consequently, we reject H0: β1 = 0.
The Minitab output in Table 2.3 also presents the analysis-of-variance test significance of regression. Comparing Tables 2.3 and 2.5, we note that there are some slight differences between the manual calculations and those performed by computer for the sums of squares. This is due to rounding the manual calculations to two decimal places. The computed values of the test statistics essentially agree.
More About the t Test
We noted in Section 2.3.2 that the t statistic
could be used for testing for significance of regression. However, note that on squaring both sides of Eq. (2.37), we obtain
Thus,
TABLE 2.5 Analysis-of-Variance Table for
