Page 161 - Intermediate Statistics for Dummies

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Part II: Making Predictions by Using Regression
In order for the model to fit well, the residuals need to meet two conditions:
The residuals are independent. The independence of residuals means
that as you plot the residuals you don’t see any pattern; they don’t affect
each other and should be random.
The residuals have a normal distribution centered at zero, and the
standardized residuals follow suit. Having a normal distribution with
mean zero means that most of the residuals should be centered around
zero, with fewer of them occurring the farther from zero you get. You
should observe about as many residuals above the zero line as below it.
If the residuals are standardized, their standard deviation is one; you
should expect about 95 percent of them to lie between –2 and +2, follow-
ing the 68-95-99.7 Rule (see your intro stats text).
The way to determine whether or not these two conditions are met for the
residuals is by using a series of four graphs called residual plots. (The resid-
uals are the distances between the predicted values in the model and the
observed values of the data themselves.) Most statisticians prefer to stan-
dardize the residuals (convert them to Z-scores by subtracting their mean
and dividing by their standard deviation) before looking at them, because
then you can compare them with values on a Z-distribution. Hence, you
can ask Minitab to give you a series of four standardized residual plots with
which to check the conditions.
Figure 7-7 shows the standardized residual plots for the quadratic model,
using the quiz-score data from previous sections. The upper-left plot shows
that the standardized residuals follow one-to-one with a normal distribution.
The upper-right plot shows that most of the standardized residuals fall
between –2 and +2 (see Chapter 4 for more on standardized residuals). The
lower-left plot shows that the residuals bear some resemblence to a normal
distribution, and the lower-right plot demonstrates how the residuals have
no pattern. They appear to occur at random. All of these plots together sug-
gest that the conditions on the residual are met to apply the selected qua-
dratic regression model.
Making predictions
After you’ve found the model that fits well, you can now use that model to
make predictions for y given x by simply plugging in the desired x-value, and
out comes your predicted value for y. (Make sure any values you plug in for x
occur within the range of where data was collected; if not, you can’t guaran-
tee the model holds.)