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

SECTION 15.4 | Hypothesis Tests with the Pearson Correlation 507
Although sample correlations are used to test hypotheses about population correlations,
you should keep in mind that samples are not expected to be identical to the populations
from which they come; there will be some discrepancy (sampling error) between a sample
statistic and the corresponding population parameter. Specifically, you should always
expect some error between a sample correlation and the population correlation it represents.
One implication of this fact is that even when there is no correlation in the population
(ρ = 0), you are still likely to obtain a nonzero value for the sample correlation. This
is particularly true for small samples. Figure 15.12 illustrates how a small sample from a
population with a near-zero correlation could result in a correlation that deviates from zero.
The colored dots in the figure represent the entire population and the three circled dots represent
a random sample. Note that the three sample points show a relatively good, positive
correlation even through there is no linear trend (ρ = 0) for the population.
When you obtain a nonzero correlation for a sample, the purpose of the hypothesis test
is to decide between the following two interpretations.
1. There is no correlation in the population (ρ = 0) and the sample value is the result
of sampling error. Remember, a sample is not expected to be identical to the population.
There always is some error between a sample statistic and the corresponding
population parameter. This is the situation specified by H 0
.
2. The nonzero sample correlation accurately represents a real, nonzero correlation in
the population. This is the alternative stated in H 1
.
The correlation from the sample will help to determine which of these two interpretations
is more likely. A sample correlation near zero supports the conclusion that the population
correlation is also zero. A sample correlation that is substantially different from zero
supports the conclusion that there is a real, nonzero correlation in the population.
■ The Hypothesis Test
The hypothesis test evaluating the significance of a correlation can be conducted using
either a t statistic or an F-ratio. The F-ratio is discussed later (pp. 541–543) and we focus
Y values
FIGURE 15.12
Scatter plot of a population of X and
Y values with a near-zero correlation.
However, a small sample of n = 3
data points from this population shows
a relatively strong, positive correlation.
Data points in the sample are circled.
X values