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10.4 EVALUATING THE MULTIPLE REGRESSION EQUATION 499
We say that about 37.1 percent of the total variation in the Y values is
explained by the fitted regression plane, that is, by the linear relationship
with age and education level.
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Testing the Regression Hypothesis To determine whether the overall
regression is significant (that is, to determine whether R 2 y.12 is significant), we may perform
a hypothesis test as follows.
1. Data. The research situation and the data generated by the research are examined
to determine if multiple regression is an appropriate technique for analysis.
2. Assumptions. We assume that the multiple regression model and its underlying
assumptions as presented in Section 10.2 are applicable.
3. Hypotheses. In general, the null hypothesis is H 0 : b 1 = b 2 = b 3 = . . . =
b k = 0 and the alternative is H A : not all b i = 0. In words, the null hypothesis
states that all the independent variables are of no value in explaining the variation
in the Y values.
4. Test statistic. The appropriate test statistic is V.R., which is computed as part
of an analysis of variance. The general ANOVA table is shown as Table 10.4.1.
In Table 10.4.1, MSR stands for mean square due to regression and MSE stands
for mean square about regression or, as it is sometimes called, the error mean
square.
5. Distribution of test statistic. When H 0 is true and the assumptions are met, V.R.
is distributed as F with k and n - k - 1 degrees of freedom.
6. Decision rule. Reject H 0 if the computed value of V.R. is equal to or greater than
the critical value of F.
7. Calculation of test statistic. See Table 10.4.1.
8. Statistical decision. Reject or fail to reject H 0 in accordance with the decision
rule.
9. Conclusion. If we reject H 0 , we conclude that, in the population from which the
sample was drawn, the dependent variable is linearly related to the independent variables
as a group. If we fail to reject H 0 , we conclude that, in the population from
which our sample was drawn, there may be no linear relationship between the
dependent variable and the independent variables as a group.
10. p value. We obtain the p value from the table of the F distribution.
We illustrate the hypothesis testing procedure by means of the following example.
TABLE 10.4.1
ANOVA Table for Multiple Regression
Source SS d.f. MS V.R.
Due to regression SSR k
About regression SSE n - k - 1
Total SST n - 1
MSR = SSR>k
MSE = SSE>1n - k - 12
MSR>MSE