Testing for correlation

The idea

A correlation test asks whether the sample's correlation coefficient $r$ is strong enough to show correlation in the population, $\rho$ . Compare $r$ with the critical value for the sample size and significance level.

$H_0: \rho = 0$ ; $H_1: \rho > 0$ , $\rho < 0$ or $\rho \ne 0$ , chosen before looking at $r$ .
Find the critical value from the table for $n$ and the level (split the level between the tails if two-tailed).
If $|r|$ is beyond the critical value, reject $H_0$ ; conclude in context.

Watch out A significant result shows correlation, not that one variable causes the other.

See it worked

Worked example

For $n = 15$ , $r = 0.62$ . Test for positive correlation at the $5\%$ level, given the critical value $0.4409$ .

$H_0: \rho = 0$ , $H_1: \rho > 0$ .
$0.62 > 0.4409$ .
Reject $H_0$ : there is evidence of positive correlation.

Answer Reject $H_0$

Your turn

Question 1Non-calculator3 marks

A two-tailed correlation test is carried out at the $10\%$ level. What percentage is in each tail?

Question 2Non-calculator3 marks

Write the hypotheses to test whether hours of sunshine and ice-cream sales are positively correlated.

Write your answer, then show the working

Hypotheses are about the population correlation $\rho$ .
$H_0: \rho = 0$ (no correlation).
$H_1: \rho > 0$ (positive correlation).

Answer $H_0: \rho = 0$ , $H_1: \rho > 0$

Question 3Non-calculator3 marks

For $n = 12$ , $r = -0.61$ . Test $H_0: \rho = 0$ against $H_1: \rho \ne 0$ at the $5\%$ level, given the critical value $0.5760$ .

Write your answer, then show the working

Two-tailed, so compare $|r| = 0.61$ with $0.5760$ .
$0.61 > 0.5760$ .
Reject $H_0$ : there is evidence of correlation (negative).

Answer Reject $H_0$ : evidence of correlation

In past papers

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