Statistics for the Behavioral Sciences by Frederick J. Gravetter, Larry B. Wallnau ISBN 10: 1305504917 ISBN 13: 9781305504912

508 CHAPTER 15 | Correlation
on the t statistic here. The t statistic for a correlation has the same general structure as t
statistics introduced in Chapters 9, 10, and 11.
sample statistic 2 population parameter
t 5
standard error
In this case, the sample statistic is the sample correlation (r) and the corresponding parameter
is the population correlation (ρ). The null hypothesis specifies that the population correlation
is ρ = 0. The final part of the equation is the standard error, which is determined by
standard error for r 5 s r
5Î
1 2 r2
n 2 2
(15.7)
Thus, the complete t statistic is
t 5
Î r 2r
(1 2 r 2 )
(n 2 2)
(15.8)
Degrees of Freedom for the t Statistic The t statistic has degrees of freedom defined
by df = n – 2. An intuitive explanation for this value is that a sample with only n = 2 data
points has no degrees of freedom. Specifically, if there are only two points, they will fit perfectly
on a straight line, and the sample produces a perfect correlation of r = +1.00 or r =
–1.00. Because the first two points always produce a perfect correlation, the sample correlation
is free to vary only when the data set contains more than two points. Thus, df = n – 2.
The following examples demonstrate the hypothesis test.
EXAMPLE 15.7
A researcher is using a regular, two-tailed test with α = .05 to determine whether a nonzero
correlation exists in the population. A sample of n = 30 individuals is obtained and
produces a correlation of r = 0.35. The null hypothesis states that there is no correlation
in the population.
H 0
: ρ = 0
For this example, df = 28 and the critical values are t = ±2.048. With r 2 = 0.35 2 = 0.1225,
the data produce
0.35 2 0
t 5
Ï(1 2 0.1225)/28 5 0.35
0.177 5 1.97
The t value is not in the critical region so we fail to reject the null hypothesis. The sample
correlation is not large enough to reject the null hypothesis.
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EXAMPLE 15.8
With a sample of n = 30 and a correlation of r = 0.35, this time we use a directional, onetailed
test to determine whether there is a positive correlation in the population.
H 0
: ρ ≤ 0
H 1
: ρ > 0
(There is not a positive correlation.)
(There is a positive correlation.)
The sample correlation is positive, as predicted, so we simply need to determine whether it
is large enough to be significant. For a one-tailed test with df = 28 and α = .05, the critical
value is t = 1.701. In the previous example, we found that this sample produces t = 1.97,
which is beyond the critical boundary. For the one-tailed test, we reject the null hypothesis
and conclude that there is a significant positive correlation in the population.
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Although it is possible to conduct the hypothesis test by computing either a t statistic
or an F-ratio, the computations have been completed and are summarized in Table B.6 in