O1 - Correlation hypothesis tests

Correlation measures straight-line closeness

Can a sample coefficient of $r=0$ accompany a perfectly predictable relationship? Yes: a U-shaped pattern can have no overall linear direction. A correlation coefficient measures closeness to a straight line, so the shape of the relationship matters.

For example, the paired values with $x=-2,-1,0,1,2$ and $y=x^2$ have supplied coefficient $r=0$. Each positive $x$ has a matching negative $x$ with the same $y$. Their upward and downward linear contributions cancel, even though knowing $x$ determines $y$ exactly.

Use a plot to check the shape before interpreting a single summary number. An outlier can also have a large influence on a coefficient.

The sign gives direction; the magnitude gives closeness to a line. Thus $r=-0.92$ represents closer alignment than $r=0.40$, because $0.92>0.40$. A negative number is not automatically a weak correlation. Nor is $r$ a gradient: converting a length from metres to centimetres changes a graph's gradient but leaves its correlation coefficient unchanged. These interpretations are not in the formula booklet: learn them.

The third panel answers the opening puzzle visually. It has a relationship, but a straight line cannot describe its U shape. Zero correlation therefore does not establish independence. For this course, correlation coefficients are given; you do not need a method for calculating them.

The next question separates linear strength from the direction of the relationship.

Hypotheses about population correlation

A sample coefficient usually differs from zero even when the population has no linear correlation. The test asks whether the observed coefficient is sufficiently unusual under the population claim $\rho=0$.

Write hypotheses using $\rho$, rather than the already observed $r$. As with a proportion test, choose the direction from the investigation before seeing the sample.

These hypothesis conventions are not in the formula booklet: learn them. The exact null distribution used for the usual PMCC test assumes that paired observations form an independent random sample from a bivariate Normal population. This means a joint Normal model for the two variables; it is stronger than merely having two separate bell-shaped histograms. You need to state or assess the assumption, not calculate a bivariate density.

Repeated measurements from the same person may be dependent. A volunteer sample may fail to represent the intended population. A numerical comparison can still be performed, but those weaknesses limit the resulting inference.

Example 2 changes only the investigative claim. There is now no predicted direction, so both tails matter.

The sample coefficient determines the evidence, not the value to put into a population alternative.

Use the purpose of the study to correct the following positive-tail test.

Using supplied critical values and p-values

A critical value tells you how far the sample coefficient must lie from zero for rejection. Use the value supplied for the sample size, significance level and tail. There is no universal cutoff such as $0.5$.

The Mathematics formula booklet has no statistical tables. AQA supplies the required critical value, a relevant extract, or a p-value. Do not calculate a PMCC or introduce an additional test statistic. The comparison rules themselves are not in the formula booklet: learn them.

For a positive one-tailed test, reject for sufficiently large positive $r$. For a negative one-tailed test, reject for sufficiently negative $r$. If a positive critical magnitude $c$ is supplied for that negative test, its signed boundary is $-c$. For a two-tailed test, compare $|r|$ with the supplied two-tailed critical magnitude.