Hypothesis testing for zero correlation

In a nutshell

Based on the product moment correlation for a given sample, $r$ , you can predict that measure for the population, $\rho$ , using a hypothesis test.

One-tailed test

This is used to test if $\rho$ is greater than zero or less than zero (i.e. testing for positive correlation or testing for negative correlation). One of two cases can be tested:

$H_0:\rho =0; H_1: \rho\gt 0$
$H_0:\rho =0; H_1: \rho\lt 0$

Two-tailed test

This is used when you only want to test if $\rho$ is not equal to $0$ (i.e. testing whether there is any correlation at all):

$H_0:\rho =0; H_1: \rho\neq 0$

Critical region

The critical region for $r$ for your hypothesis can be determined using a given table of critical values. This region will depend on the significance level and the sample size:

The bigger the significance level, the smaller the critical value.
The bigger the sample size, the smaller the critical value.
The smaller the critical value, the larger the critical region. This is because for a critical value $c$ , the corresponding critical region is $\vert r \vert \gt c$ .

Example 1

For a given sample of $8$ and a significance level of $10\%$ , if $H_1: \rho\gt 0$ , the critical region is $r\gt 0.5067$ . This means that:

If the sample $r\lt 0.5067$ , there is not enough evidence to reject $H_0$ .
If the sample $r \gt 0.5067$ , you can reject $H_0$ .

Example 2

The marks on two different subjects were taken for $30$ students, which produced a product moment correlation coefficient of $-0.51$ . Test, with a significance level of $10\%$ , whether or not there is a negative correlation between the marks in these two subjects for all the students.

For the given sample and a significance value of $10\%$ , you have:

$H_0: \rho=0\\H_1: \rho\lt0$ 

You also have, by the table, $r\lt -0.3061$ .

Because $-0.51 \lt -0.3061$ , you can reject $H_0$ .

There is evidence, at the 10% level of significance, that a greater mark in the first subject means a smaller one in the second.