Hypothesis Testing for Zero Correlation
A sample's r is never quite zero by chance alone -- a formal test decides whether it's too big to be a coincidence.
Testing whether ρ is really nonzero
A sample's PMCC, $r$, can come out nonzero purely by chance, even when the two variables have no correlation at all in the wider population. A hypothesis test decides whether a sample's $r$ is large enough to be good evidence that the population correlation coefficient, $\rho$ (rho), is genuinely different from zero — rather than just sampling variation.
$r$ is calculated from a sample and can be found exactly. $\rho$ is the true correlation coefficient for the whole population, and is (almost) never known — hypotheses are always written in terms of $\rho$, and $r$ is the evidence used to test them.
One-tailed and two-tailed tests
As with any hypothesis test, the null hypothesis states "nothing interesting is happening":
The alternative hypothesis depends on what's being tested:
- One-tailed — testing specifically for positive or specifically for negative correlation: $H_1:\rho>0$, or $H_1:\rho<0$.
- Two-tailed — testing for correlation in either direction, with no assumed sign in advance: $H_1:\rho\neq0$.
Which one to use is decided entirely by the wording of the question — a claim like "sales increase as advertising spend increases" specifies a direction and calls for a one-tailed test; "there is a relationship between the variables" does not, and calls for a two-tailed test.
Critical values
The test statistic is the sample's own PMCC, $r$. Rather than a formula, its critical value is read directly from a table of critical values of the PMCC, based on the sample size $n$ and the significance level — this table is provided in the exam's formulae booklet.
| n | 0.10 | 0.05 | 0.025 | 0.01 | 0.005 |
|---|---|---|---|---|---|
| 4 | 0.8000 | 0.9000 | 0.9500 | 0.9800 | 0.9900 |
| 5 | 0.6870 | 0.8054 | 0.8783 | 0.9343 | 0.9587 |
| 6 | 0.6084 | 0.7293 | 0.8114 | 0.8822 | 0.9172 |
| 7 | 0.5509 | 0.6694 | 0.7545 | 0.8329 | 0.8745 |
| 8 | 0.5067 | 0.6215 | 0.7067 | 0.7887 | 0.8343 |
| 9 | 0.4716 | 0.5822 | 0.6664 | 0.7498 | 0.7977 |
Each column is a (one-tailed) significance level; for a two-tailed test at overall level $\alpha$, use the column headed $\alpha/2$. For $n=8$ at the $5\%$ level, the highlighted critical value is $0.6215$: any sample $r$ greater than $0.6215$ is significant evidence, at that level, that $\rho>0$.
Reaching a conclusion
Compare the sample's $r$ (one-tailed, matching the direction of $H_1$) or $|r|$ (two-tailed) against the table's critical value. If it's exceeded, reject $H_0$: there's evidence of correlation in the population. Otherwise, the evidence is insufficient.
Reading down any column of the table, the critical value decreases as $n$ increases. A modest correlation is much stronger evidence with a large sample than the same correlation would be with a tiny one, since a small sample can easily produce a large $r$ by chance alone.
Worked example
Just as with any other hypothesis test, failing to reject $H_0$ only means the sample didn't provide strong enough evidence against it — it is never proof that $\rho=0$. A real (perhaps weaker, or non-linear) relationship could still exist without producing a large enough sample $r$ to be detected, especially with a small sample.
Never write a hypothesis in terms of $r$ — hypotheses describe the unknown population parameter $\rho$, never the sample statistic. Writing "$H_0:r=0$" is a common but incorrect notational slip that loses marks.
Check yourself
A sample of $n=6$ pairs gives $r=0.72$. Test, at the 5% significance level, $H_0:\rho=0$ against $H_1:\rho\neq0$ (two-tailed), using the table above.