Testing for zero correlationEdexcel A-Level Further Maths: Revision notes

A sample coefficient $r$ or $r_s$ is calculated from sample data, but we want to know whether there is correlation in the population. Use $\rho$ for the population product moment correlation coefficient and $\rho_s$ for the population Spearman coefficient. The null hypothesis is always that there is no correlation: $H_0:\rho=0$ (or $\rho_s=0$). The alternative hypothesis depends on the question:

• any correlation: $H_1:\rho\ne0$ (two-tailed);
• positive correlation: $H_1:\rho>0$ (one-tailed);
• negative correlation: $H_1:\rho<0$ (one-tailed). Hypotheses are always about the population parameter, never the sample value.

Tables give critical values for each sample size $n$, for one-tailed tests at several significance levels. For a two-tailed test at the 5% level, use the column for a one-tailed 2.5% test (half the significance level in each tail). Examples: product moment, $n=10$: $0.5494$ (one-tailed 5%) and $0.6319$ (two-tailed 5%). Spearman, $n=8$: $0.6429$ (one-tailed 5%) and $0.7381$ (two-tailed 5%). The critical region is $|r|\ge$ critical value for a two-tailed test; $r\ge$ critical value for $H_1:\rho>0$; $r\le-$ critical value for $H_1:\rho<0$.

1. State $H_0$ and $H_1$.
2. Find the critical value for the sample size $n$ and significance level.
3. Compare the sample coefficient with it.
4. Write a conclusion in two parts: whether $H_0$ is rejected, and what that means in context. Example: $n=10$, $r=-0.60$, test for any correlation at 5%. The critical value is $0.6319$. Since $0.60<0.6319$, do not reject $H_0$: there is insufficient evidence of correlation between exercise and resting heart rate. For a test for negative correlation at 5%, the critical value is $-0.5494$, and $-0.60<-0.5494$, so reject $H_0$.

The product moment test requires that the data come from a population with a bivariate normal distribution; formal verification is not required, but a scatter diagram should look roughly elliptical. The Spearman test makes no such assumption and only needs a monotonic relationship, so it suits ranked data and small samples where normality cannot be checked. The two tests can give different conclusions, especially when the result is close to a critical value. Example: for $n=8$, $r_s=0.786$ exceeds $0.7381$ (reject $H_0$), but $r=0.703$ is below $0.7067$ (do not reject). When results are borderline, say so and say which assumptions are secure.