Statistics for the Behavioral Sciences by Frederick J. Gravetter, Larry B. Wallnau ISBN 10: 1305504917 ISBN 13: 9781305504912

SECTION 16.2 | The Standard Error of Estimate and Analysis of Regression 543
Finally, the F-ratio for evaluating the significance of the regression equation is
F 5
MS regression
MS residual
5 112.01
7.33 5 15.28
The F-ratio is in the critical region, so we reject the null hypothesis and conclude that the
regression equation does account for a significant portion of the variance for the Y scores.
The complete analysis of regression is summarized in Table 16.1, which is a common
format for computer printouts of regression analysis.
TABLE 16.1
A summary table showing
the results of the analysis
of regression in
Example 16.5.
Source SS df MS F
Regression 112.01 1 112.01 15.28
Residual 43.99 6 7.33
Total 156 7
■
■ Significance of Regression and Significance of the Correlation
As noted earlier, in situation with a single X variable and a single Y variable, testing the significance
of the regression equation is equivalent to testing the significance of the Pearson
correlation. Therefore, whenever the correlation between two variables is significant, you
can conclude that the regression equation is also significant. Similarly, if a correlation is
not significant, the regression equation is also not significant. For the data in Example 16.5,
we concluded that the regression equation is significant.
To demonstrate the equivalence of the two tests, we will show that the t statistic used to
test the significance of a correlation (Chapter 15, p. 508) is equivalent to the F-ratio used
to test the significance of the regression equation (Equation 16.12). We begin with the
t statistic introduced in Chapter 15.
t 5
Î r 2r
(1 2 r 2 )
(n 2 2)
First, we remove the population correlation, ρ, from the t equation. This value is always
zero, as specified by the null hypothesis, and its removal does not affect the equation. Next,
we square the t statistic to produce the corresponding F-ratio.
t 2 5 F 5
r2
(1 2 r 2 )
(n 2 2)
Finally, multiply the numerator and the denominator by SS Y
to produce
t 2 5 F 5 r2 (SS Y
)
(1 2 r 2 )(SS Y
)
(n 2 2)
You should recognize the numerator as SS regression
, which is equivalent to MS regression
because
df regression
= 1. Also, the denominator is identical to MS residual
.