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4.4 Inference on Two Populations 127

correlated and their correlation is high, one may contemplate the possibility of
discarding one of the variables, since a highly correlated variable only conveys
redundant information.
Let ρ represent the true value of the Pearson correlation mentioned in section
2.3.4. The correlation test is formalised as:

H 0: ρ = 0, H 1: ρ ≠ 0, for a two-sided test.

For a one-sided test the alternative hypothesis is:

H 1: ρ > 0 or ρ < 0.

Let r represent the sample Pearson correlation when the null hypothesis is
verified and the sample size is n. Furthermore, assume that the random variables
are normally distributed. Then, the (r.v. corresponding to the) following test
statistic:

n − 2
t = r , 4.6
*
−
1 r 2

has a Student’s t distribution with n – 2 degrees of freedom.
The Pearson correlation test can be performed as part of the computation of
correlations with SPSS and STATISTICA. It can also be performed using the
Correlation Test sheet of Tools.xls (see Appendix F) or the
Probability Calculator; Correlations of STATISTICA (see also
Commands 4.2).

Example 4.5
Q: Consider the variables PMax and T80 of the meteorological dataset ( Meteo )
for the “moderate” category of precipitation (PClass = 2) as defined in 2.1.2. We
then have n = 16 measurements of the maximum precipitation and the maximum
temperature during 1980, respectively. Is there evidence, at α = 0.05, of a negative
correlation between these two variables?
A: The distributions of PMax and T80 for “moderate” precipitation are reasonably
well approximated by the normal distribution (see section 5.1). The sample
correlation is r = –0.53. Thus, the test statistic is:

*
r = –0.53, n = 16 ⇒ t = –2.33.

*
Since t 14 . 0 , 05 = − 1. 76 , the value of t falls in the critical region ] –∞, –1.76];
therefore, the null hypothesis is rejected, i.e., there is evidence of a negative
correlation between PMax and T80 at that level of significance. Note that the
*
observed significance of t is 0.0176, below α.