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 541
Similarly, the unpredicted variability is
SS residual
= (1 – r 2 )SS Y
= (1 – 0.847 2 )(156) = 0.282(156) = 43.99
Notice that the new formula for SS residual
produces the same value, within rounding error,
that we obtained by adding the squared residuals in Example 16.3. Also note that this
new formula is generally much easier to use because it requires only the correlation value
(r) and the SS for Y. The primary point of this example, however, is that SS residual
and the
standard error of estimate are closely related to the value of the correlation. With a large
correlation (near +1.00 or –1.00), the data points are close to the regression line, and the
standard error of estimate is small. As a correlation gets smaller (near zero), the data points
move away from the regression line, and the standard error of estimate gets larger. ■
Because it is possible to have the same regression line for sets of data that have different
correlations, it is also important to examine r 2 and the standard error of estimate. The regression
equation simply describes the best-fitting line and is used for making predictions. However,
r 2 and the standard error of estimate indicate how accurate these predictions will be.
■ Analysis of Regression
As we noted in Chapter 15, a sample correlation is expected to be representative of its
population correlation. For example, if the population correlation is zero, the sample correlation
is expected to be near zero. Note that we do not expect the sample correlation to be
exactly equal to zero. This is the general concept of sampling error that was introduced in
Chapter 1 (p. 6). The principle of sampling error is that there is typically some discrepancy
or error between the value obtained for a sample statistic and the corresponding population
parameter. Thus, when there is no relationship whatsoever in the population, a correlation
of ρ = 0, you are still likely to obtain a nonzero value for the sample correlation. In this
situation, however, the sample correlation is meaningless and a hypothesis test usually
demonstrates that the correlation is not significant.
Whenever you obtain a nonzero value for a sample correlation, you will also obtain real,
numerical values for the regression equation. However, if there is no real relationship in the
population, both the sample correlation and the regression equation are meaningless—they
are simply the result of sampling error and should not be viewed as an indication of any
relationship between X and Y. In the same way that we tested the significance of a Pearson
correlation, we can test the significance of the regression equation. In fact, when a single
variable X is being used to predict a single variable Y, the two tests are equivalent. In each
case, the purpose for the test is to determine whether the sample correlation represents
a real relationship or is simply the result of sampling error. For both tests, the null
hypothesis states that there is no relationship between the two variables in the population.
For a correlation,
For the regression equation,
H 0
: the population correlation is ρ = 0
H 0
: the slope of the regression equation (b or beta) is zero
The process of testing the significance of a regression equation is called analysis of
regression and is very similar to the analysis of variance (ANOVA) presented in Chapter 12.
As with ANOVA, the regression analysis uses an F-ratio to determine whether the variance
predicted by the regression equation is significantly greater than would be expected
if there were no relationship between X and Y. The F-ratio is a ratio of two variances, or
mean square (MS) values, and each variance is obtained by dividing an SS value by its