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9.4 EVALUATING THE REGRESSION EQUATION 425
Y
(a)
X
Y
(b)
FIGURE 9.4.2 Population conditions relative to X and Y that may cause rejection of the null
hypothesis that b 1 = 0. (a) The relationship between X and Y is linear and of sufficient strength
to justify the use of a sample regression equation to predict and estimate Y for given values
of X. (b) A linear model provides a good fit to the data, but some curvilinear model would
provide an even better fit.
X
Unexplained Deviation Finally, we measure the vertical distance of the
observed point from the regression line to obtain 1y i - yN i 2, which is called the unexplained
deviation, since it represents the portion of the total deviation not “explained”
or accounted for by the introduction of the regression line. These three quantities are
shown for a typical value of Y in Figure 9.4.4. The difference between the observed value
of Y and the predicted value of Y, 1y i - yN i 2, is also referred to as a residual. The set of
residuals can be used to test the underlying linearity and equal-variances assumptions
of the regression model described in Section 9.2. This procedure is illustrated at the end
of this section.
It is seen, then, that the total deviation for a particular y i is equal to the sum of
the explained and unexplained deviations. We may write this symbolically as
1y i - y2 = 1yN i - y2 + 1y i - yN i 2
total explained unexplained
deviation deviation deviation
(9.4.1)