Hypothesis tests for correlationAQA A-Level Maths: Revision notes

The product moment correlation coefficient $r$ measures how close the points on a scatter diagram lie to a straight line. It always satisfies $-1\le r\le1$:

• $r$ close to $1$: strong positive linear correlation;
• $r$ close to $-1$: strong negative linear correlation;
• $r$ close to $0$: little or no linear correlation. You do not calculate $r$ in this topic: it is given in the question. Remember that $r$ measures linear association only, so $r\approx0$ does not rule out a non-linear relationship.

A sample $r$ varies from sample to sample, so we test a claim about the population correlation coefficient $\rho$ (rho). The null hypothesis is that there is no correlation: $H_0:\rho=0.$ The alternative hypothesis is one-tailed if the direction is known in advance, $H_1:\rho>0$ or $H_1:\rho<0$, and two-tailed if any correlation is of interest, $H_1:\rho\ne0$. Choose the direction from the question wording, before looking at $r$. The test assumes a random sample, and that the two variables are suitable for this test (the pairs are drawn from a population in which both variables vary).

A table gives the critical value of $r$ for each sample size $n$, significance level and type of test. Compare the sample $r$ with it:

• one-tailed upper test: reject $H_0$ if $r\ge$ critical value;
• one-tailed lower test: reject $H_0$ if $r\le-$ critical value;
• two-tailed test: reject $H_0$ if $\lvert r\rvert\ge$ critical value (use the two-tailed table, or half the significance level in each tail). Example: $n=12$, $r=0.62$, $H_1:\rho>0$ at $5\%$. The critical value is $0.4973$. $0.62>0.4973$, so reject $H_0$: there is sufficient evidence of positive correlation.

If a $p$-value is given, it is the probability of a sample correlation at least this extreme if $\rho=0$. Compare it with the significance level (halving the level for a two-tailed test if the $p$-value is for one tail):

• $p<$ significance level: reject $H_0$;
• $p\ge$ significance level: do not reject $H_0$. Example: for $r=-0.47$ with $n=10$ and $H_1:\rho<0$, the $p$-value is $0.085$. This is greater than $0.05$, so $H_0$ is not rejected at $5\%$, but it is less than $0.10$, so $H_0$ is rejected at $10\%$.

State the decision and the meaning in context: 'there is sufficient evidence, at the $5\%$ level, of positive correlation between hours revising and test score'. If $H_0$ is not rejected, say there is insufficient evidence of correlation. Do not say $H_0$ is proved. Cautions that earn marks in evaluation questions:

• correlation does not imply causation: another variable may explain both;
• $r$ measures only a linear relationship;
• predictions outside the range of the data (extrapolation) are unreliable;
• the same $r$ can be significant at $5\%$ but not at $1\%$, so say how strong the evidence is;
• a one-tailed test is only valid if the direction was decided before seeing the data.