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.