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Chapter 4: Getting in Line with Simple Linear Regression
and that the model fits well in more specific ways than the scatterplot and
correlation measure. This section presents methods for defining and assess-
ing the fit of a simple linear regression model.
Defining the conditions
Two major conditions must be met before you apply a simple linear regres-
sion model to a data set:
The y’s have to have a normal distribution for each value of x.
The y’s have to have a constant amount of spread (standard deviation)
for each value of x.
In the following sections, you look at these important conditions in depth.
Normal y’s for every x 77
For any value of x, the population of possible y-values must have a normal
distribution. The mean of this distribution is the value for y that is on the
best-fitting line for that x-value. That is, some of your data falls above the
best-fitting line, some data falls below the best fitting line, and a few may
actually land right on the line.
If the regression model is fitting well, the data values should be scattered
around the best-fitting line in such a way that about 68 percent of the values
lie within one standard deviation of the line, about 95 percent of the values
should lie within two standard deviations of the line, and about 99.7 percent
of the values should lie within three standard deviations of the line. This
specification, as you may recall from your intro stats course, is called the
68-95-99.7 rule, and it applies to all bell-shaped data (for which the normal
distribution applies).
You can see in Figure 4-3 how for each x-value, the y-values you may observe
tend to be located near the best-fitting line in greater numbers, and as you
move away from the line, you see fewer and fewer y-values, both above and
below the line. More than that, they’re scattered around the line in a way that
reflects a bell-shaped curve, the normal distribution.
Why does this condition makes sense? The data you collect on y for any partic-
ular x-value varies from individual to individual (for example, not all students’
textbooks weigh the same, even for students who weigh the exact same
amount). But those values aren’t allowed to vary any way they want to. To fit
the conditions of a linear regression model, for each given value of x, the data
should be scattered around the line according to a normal distribution. Most of
the points should be close to the line, and as you get farther and farther from
the line, you can expect fewer and fewer data points to occur. So condition
number one is that the data have a normal distribution for each value of x.