always capitalize that symbol. One thing to note is that Rhere is calculated as the square
root of R^2 , and as such it will always be positive, even if the relationship is negative. This is
a result of the fact that the procedure is applicable for multiple predictors.
The ANOVA table is a test of the null hypothesis that the correlation is .00 in the popu-
lation. We will discuss hypothesis testing next, but what is most important here is that the
test statistic is F, and that the significance level associated with that Fis p 5 .004. Since p
is less than .05, we will reject the null hypothesis and conclude that the variables are not
linearly independent. In other words, there is a linear relationship between how well stu-
dents score on a test that reflects test-taking skills, and how well they perform on the SAT.
The exact nature of this relationship is shown in the next part of the printout. Here we have
a table labeled “Coefficients,” and this table gives us the intercept and the slope. The inter-
cept is labeled here as “Constant,” because it is the constant that you add to every predic-
tion. In this case it is 373.736. Technically it means that if a student answered 0 questions
correctly on Katz’s test, we would expect them to have an SAT of approximately 370. Since
a score of 0 would be so far from the scores these students actually obtained (and it is hard
to imagine anyone earning a 0 even by guessing), I would not pay very much attention to
that value.
In this table the slope is labeled by the name of the predictor variable. (All software so-
lutions do this, because if there were multiple predictors we would have to know which
variable goes with which slope. The easiest way to do this is to use the variable name as
the label.) In this case the slope is 4.865, which means that two students who differ by 1
point on Katz’s test would be predicted to differ by 4.865 on the SAT. Our regression equa-
tion would now be written as.
The standardized regression coefficient is shown as .532. This means that a one stan-
dard deviation difference in test scores is associated with approximately a one-half stan-
dard deviation difference in SAT scores. Note that, because we have only one predictor,
this standardized coefficient is equal to the correlation coefficient.
To the right of the standardized regression coefficient you will see tand pvalues for
tests on the significance of the slope and intercept. We will discuss the test on the slope
shortly. The test on the intercept is rarely of interest, but its interpretation should be evi-
dent from what I say about testing the slope.9.11 Hypothesis Testing
We have seen how to calculate ras an estimate of the relationship between two variables
and how to calculate the slope (b) as a measure of the rate of change of Yas a function of
X. In addition to estimating rand b, we often wish to perform a significance test on the null
hypothesis that the corresponding population parameters equal zero. The fact that a value
of ror bcalculated from a sample is not zero is not in itself evidence that the correspon-
ding parameters in the population are also nonzero.Testing the Significance of r
The most common hypothesis that we test for a sample correlation is that the correlation be-
tween Xand Yin the population, denoted r(rho), is zero. This is a meaningful test because
the null hypothesis being tested is really the hypothesis that Xand Yare linearly independ-
ent. Rejection of this hypothesis leads to the conclusion that they are not independent and
that there is some linear relationship between them.
It can be shown that when r50, for large N, rwill be approximately normally distrib-
uted around zero.YN =4.865 3 Score 1 373.736Section 9.11 Hypothesis Testing 271