520 Chapter 15 Multiple Regression
Two Variable Relationships
The most obvious thing to do with these data is to ask about the relationship between ex-
penditure and outcome. We will ignore the ACT data for now and concentrate on the rela-
tionship between performance on the SAT and expenditures for education. While we are
doing that we will also look at the correlations between other possible predictors of test
performance. We would presumably like to see that the more money we spend on educa-
tion, the better our students do. In addition, it would be of interest to ask whether the
pupil/teacher ratio is related to outcome, as many people have argued, and whether higher
salaries for teachers play a role. Keep in mind that the SAT score is our measure of educa-
tional performance, and it is a good measure for our purposes in this example, though it is
not a good general measure of school performance, nor was it ever intended as such.
The graphic in the upper right corner of Figure 15.1 is a scatterplot of SAT scores
against expenditures. In addition Table 15.2 shows the Pearson correlations between some
of our variables, the most interesting being the negative correlation of SAT and Expend.
The relationship is somewhat surprising, because it would suggest that the more money we
spend on educating our children the worse they do. The regression line is clearly decreas-
ing and the correlation is 2 .381. Although that correlation is not terribly large, it is statisti-
cally significant (p 5 .006) and cannot just be ignored. Those students who come from
wealthier schools tend to do worse. Why should this be? The other interesting thing that
we see from the table of correlations is that there appears to be no relationship between
pupil/teach ratio and performance. What are we to make of this?
An answer to our puzzle comes from what I said previously about the SAT test itself. Not
all colleges and universities require that students take the SAT, and there is a tendency for
those that do require it to be the more prestigious universities that take only the top students.Pearson Correlation
Sig. (2-tailed)
N
Pearson Correlation
Sig. (2-tailed)
N
Pearson Correlation
Sig. (2-tailed)
N
Pearson Correlation
Sig. (2-tailed)
N
Pearson Correlation
Sig. (2-tailed)
N
Pearson Correlation
Sig. (2-tailed)
NExpendPTratioSalaryPctSATSATLogPctSATExpend
150
2 .371**
.008
50
.870**
.000
50
.593**
.000
50
2 .381**
.006
50
.561**
.000
50PTratio
2 .371**
.008
50
150
2 .001
.994
50
2 .213
.137
50
.081
.575
50
2 .132
.361
50Salary
.870**
.000
50
2 .001
.994
50
150
.617**
.000
50
2 .440**
.001
50
.613**
.000
50PctSAT
.593**
.000
50
2 .213
.137
50
.617**
.000
50
150
2 .887**
.000
50
.961**
.000
50SAT
2 .381**
.006
50
.081
.575
50
2 .440**
.001
50
2 .887**
.000
50
150
2 .926**
.000
50LogPctSAT
.561**
.000
50
2 .132
.361
50
.613**
.000
50
.961**
.000
50
2 .926**
.000
50
150Correlations** Correlation is significant at the 0.01 level (2-tailed).Table 15.2 Correlations between selected variables