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156 Chapter 4 Linear Regression with One Regressor

In many school districts, student performance is measured by standardized
tests, and the job status or pay of some administrators can depend in part on how
well their students do on these tests. We therefore sharpen the superintendent’s
question: If she reduces the average class size by two students, what will the effect
be on standardized test scores in her district?

A precise answer to this question requires a quantitative statement about changes.
If the superintendent changes the class size by a certain amount, what would she
expect the change in standardized test scores to be? We can write this as a math-
ematical relationship using the Greek letter beta, bClassSize, where the subscript
ClassSize distinguishes the effect of changing the class size from other effects. Thus,

bClassSize = change in TestScore = ∆∆TCelastsSscSoizree, (4.1)
change in ClassSize

where the Greek letter ∆ (delta) stands for “change in.” That is, bClassSize is the
change in the test score that results from changing the class size divided by the

change in the class size.

If you were lucky enough to know bClassSize, you would be able to tell the
superintendent that decreasing class size by one student would change district-

wide test scores by bClassSize. You could also answer the superintendent’s actual
question, which concerned changing class size by two students per class. To do so,

rearrange Equation (4.1) so that

∆TestScore = bClassSize * ∆ClassSize. (4.2)

Suppose that bClassSize = - 0.6. Then a reduction in class size of two students per
class would yield a predicted change in test scores of ( -0.6) * ( -2) = 1.2; that
is, you would predict that test scores would rise by 1.2 points as a result of the
reduction in class sizes by two students per class.

Equation (4.1) is the definition of the slope of a straight line relating test
scores and class size. This straight line can be written

TestScore = b0 + bClassSize * ClassSize, (4.3)

where b0 is the intercept of this straight line and, as before, bClassSize is the slope.
According to Equation (4.3), if you knew b0 and bClassSize, not only would you be
able to determine the change in test scores at a district associated with a change
in class size, but you also would be able to predict the average test score itself for
a given class size.