Hypothesis tests for correlation using the PMCC

A hypothesis test for correlation decides whether a sample gives sufficient evidence of linear correlation in the population. At A level, the test statistic is the product moment correlation coefficient, or PMCC, denoted by $r$.

The central question is not simply whether $r$ is positive or negative. Random samples can produce non-zero values of $r$ even when there is no population correlation. We ask whether $r$ is far enough from zero to be statistically significant.

Prerequisites

You should be able to:

• interpret scatter diagrams and distinguish correlation from causation;
• calculate and interpret the PMCC;
• identify explanatory and response variables;
• understand null hypotheses, alternative hypotheses and significance levels.

Review correlation and regression and the language of hypothesis testing if these ideas are unfamiliar.

Sample correlation and population correlation

Two symbols must be kept separate:

Symbol	Meaning
$r$	PMCC calculated from a sample
$\rho$	true PMCC for the whole population

The sample value $r$ is known once the data have been collected. The unknown parameter $\rho$ is the subject of the hypotheses.

The null hypothesis is

$$\boxed{H_0:\rho=0}.$$

It says that there is no linear correlation in the population. The alternative hypothesis states the kind of correlation being investigated.

Use $\rho$ in the hypotheses

Writing $H_0:r=0$ treats the observed sample statistic as an unknown population parameter. Hypotheses must be about $\rho$, although the decision is made using $r$.

Choosing a one tailed or two tailed test

The wording of the investigation determines $H_1$.

Question	Alternative hypothesis	Extreme values
Is there positive correlation?	$H_1:\rho>0$	large positive $r$
Is there negative correlation?	$H_1:\rho<0$	large negative $r$
Is there any linear correlation?	$H_1:\rho\neq0$	large $

The first two tests are one tailed. The last is two tailed, because evidence in either direction counts against $H_0$.

Choose the alternative from the question before examining the sample result. A positive value of $r$ does not justify changing a planned two tailed test into a positive one tailed test.

Worked example 1: writing hypotheses

A teacher investigates whether pupils who spend longer completing an online revision task tend to obtain higher test marks.

Let $\rho$ be the population PMCC between completion time and test mark. The teacher is looking specifically for positive correlation, so

$$\boxed{H_0:\rho=0}, \qquad \boxed{H_1:\rho>0}.$$

By contrast, an investigation into whether age is correlated with reaction time, without predicting a direction, would use

$$H_0:\rho=0, \qquad H_1:\rho\neq0.$$

The critical value method

Under $H_0$, the sampling distribution of $r$ is centred on zero. Values near $1$ or $-1$ are increasingly difficult to explain by chance alone.

A PMCC critical values table gives a boundary depending on:

• the sample size $n$;
• the significance level $\alpha$;
• whether the test is one tailed or two tailed.

For a positive one tailed test, reject $H_0$ when

$$r\geq c.$$

For a negative one tailed test, symmetry gives the critical value $-c$, so reject $H_0$ when

$$r\leq-c.$$

For a two tailed test, reject $H_0$ when

$$r\leq-c\quad\text{or}\quad r\geq c,$$

equivalently when

$$\boxed{|r|\geq c}.$$

Here $c$ is always quoted as a positive number in the table. Equality is included in the critical region.

Read the table heading carefully

Some formula booklets label columns by one tailed probabilities and others identify one tailed and two tailed levels explicitly. Use the column matching the complete test. A $5\%$ two tailed test places $2.5\%$ in each tail.

A reliable testing procedure

1. Define $\rho$ in context.
2. Write $H_0$ and $H_1$.
3. State the significance level and identify the correct tail or tails.
4. Calculate $r$ from all $n$ paired observations, unless it is given.
5. Find the critical value using $n$, the significance level and the number of tails.
6. Compare $r$ with the correct critical region.
7. Reject or do not reject $H_0$, then write a conclusion about population correlation in context.

Worked example 2: positive correlation

For a random sample of $12$ towns, a researcher records mean commuting distance and mean journey time. The sample PMCC is

$$r=0.624.$$

Test at the $5\%$ significance level whether there is positive linear correlation in the population of towns. The $5\%$ one tailed critical value for $n=12$ is $0.497$.

Let $\rho$ be the population PMCC between mean commuting distance and mean journey time.

$$H_0:\rho=0, \qquad H_1:\rho>0.$$

This is a positive one tailed test. Its critical region is

$$r\geq0.497.$$

Since

$$0.624>0.497,$$

the observed value lies in the critical region. Reject $H_0$. There is sufficient evidence at the $5\%$ significance level to suggest positive linear correlation between mean commuting distance and mean journey time in the population of towns.

This conclusion concerns association, not causation. It does not establish that increasing commuting distance causes every journey time to increase.

Worked example 3: negative correlation

A biologist takes a random sample of $18$ plants and measures leaf damage and subsequent growth. The PMCC is $r=-0.571$. Test at the $1\%$ level whether greater leaf damage is associated with lower growth. The $1\%$ one tailed table value for $n=18$ is $0.543$.

Let $\rho$ be the population PMCC between leaf damage and growth.

$$H_0:\rho=0, \qquad H_1:\rho<0.$$

For a negative test, attach a negative sign to the tabulated magnitude. The critical region is

$$r\leq-0.543.$$

As

$$-0.571<-0.543,$$

reject $H_0$. There is sufficient evidence at the $1\%$ level to suggest negative linear correlation between leaf damage and subsequent growth in the population of plants.

Comparing negative numbers

$-0.571$ is more extreme than $-0.543$ because it is farther from zero. On the number line, it is also smaller.

Worked example 4: two tailed test

A company investigates whether customer age is linearly correlated with the amount spent per visit. From $12$ randomly selected customers it obtains

$$r=-0.532.$$

Test for correlation at the $5\%$ significance level. The $5\%$ two tailed critical value for $n=12$ is $0.576$.

No direction was specified, so

$$H_0:\rho=0, \qquad H_1:\rho\neq0.$$

The critical regions are

$$r\leq-0.576 \quad\text{or}\quad r\geq0.576.$$

Equivalently, compare magnitudes:

$$|r|=|-0.532|=0.532<0.576.$$

The result is not in either critical region, so do not reject $H_0$. There is insufficient evidence at the $5\%$ level to suggest linear correlation between customer age and amount spent in the population.

Notice that $r=-0.532$ describes moderate negative correlation in this sample, but it is not statistically significant with only $12$ observations under this test.

Calculating the test statistic

A calculator usually gives $r$ after paired data have been entered in two variable statistics mode. Algebraically,

$$r=\frac{S_{xy}}{\sqrt{S_{xx}S_{yy}}},$$

where

$$S_{xx}=\sum x^2-\frac{(\sum x)^2}{n}, \qquad S_{yy}=\sum y^2-\frac{(\sum y)^2}{n},$$

and

$$S_{xy}=\sum xy-\frac{(\sum x)(\sum y)}{n}.$$

Worked example 5: from summary statistics to a test

For $10$ paired observations, suppose

$$S_{xx}=50, \qquad S_{yy}=80, \qquad S_{xy}=48.$$

Then

$$\begin{aligned} r &=\frac{48}{\sqrt{50\times80}}\\ &=\frac{48}{\sqrt{4000}}\\ &\approx0.759. \end{aligned}$$

To test $H_0:\rho=0$ against $H_1:\rho\neq0$ at the $5\%$ level, use the $5\%$ two tailed critical value for $n=10$, which is $0.632$.

Since

$$|0.759|>0.632,$$

reject $H_0$. There is sufficient evidence of non-zero linear correlation between the two variables in the population.

Do not round $r$ prematurely. Store the calculator value and round only when presenting it, because a heavily rounded value could fall on the wrong side of a critical boundary.

Assumptions and interpretation

The standard PMCC significance test assumes that the paired observations are a random sample from a bivariate normal distribution. In practical terms, inspect the scatter diagram before testing:

• the relationship should be reasonably linear;
• strong outliers should not dominate $r$;
• the pairs should be independent;
• each variable should be quantitative.

A curved relationship can be strong while having $r$ close to zero. A single unusual point can make $|r|$ misleadingly large or small. A significant PMCC test is therefore evidence of linear association under the model assumptions, not a guarantee that the chosen model is sensible.

Significance and strength are also different ideas. Significance depends on both $|r|$ and $n$. With a large sample, a weak correlation can be significant. With a small sample, even a fairly substantial sample correlation may not be significant.

Common misconceptions

”Do not reject” means $H_0$ is true

No. It means the data do not provide sufficient evidence against $H_0$ at the chosen significance level. Absence of significant evidence is not proof that $\rho=0$.

A negative $r$ can never exceed the critical value

For a negative test, compare with the negative boundary, such as $r\leq-0.549$, or compare magnitudes when appropriate. Do not compare $-0.70$ directly with positive $0.549$ and conclude it is not significant.

A two tailed test uses the one tailed $5\%$ critical value

No. A $5\%$ two tailed test divides the rejection probability between two tails and therefore has a larger critical magnitude than a $5\%$ one tailed test for the same $n$.

Significant correlation proves causation

No. Confounding variables, reverse causation and the sampling design may all explain an association.

Self-checks

Self-check 1

A sample of $20$ paired observations gives $r=0.401$. A researcher is testing

$$H_0:\rho=0 \quad\text{against}\quad H_1:\rho>0$$

at the $5\%$ level. The critical value is $0.378$. State the conclusion.

Answer

Since $0.401>0.378$, the result lies in the critical region. Reject $H_0$. There is sufficient evidence at the $5\%$ significance level to suggest positive linear correlation in the population. A full answer should name the variables in context.

Self-check 2

For $15$ paired observations, $r=-0.500$. Test for negative correlation at the $1\%$ level, given that the one tailed critical value has magnitude $0.592$.

Answer

The hypotheses are $H_0:\rho=0$ and $H_1:\rho<0$. The negative critical region is $r\leq-0.592$.

Since $-0.500>-0.592$, do not reject $H_0$. There is insufficient evidence at the $1\%$ level to suggest negative linear correlation in the population.

Self-check 3

A two tailed test at the $5\%$ level has critical regions $r\leq-0.444$ and $r\geq0.444$. Decide whether each sample result is significant:

1. $r=0.444$
2. $r=-0.461$
3. $r=0.439$

Answer

1. Significant, because equality is included in the critical region.
2. Significant, because $|-0.461|=0.461>0.444$.
3. Not significant, because $|0.439|<0.444$.

Self-check 4

Explain why a scatter diagram should be examined even when a calculator has already produced $r=0.86$.

Answer

The PMCC measures linear association and is sensitive to outliers. A scatter diagram may reveal curvature, separate groups or an influential point, any of which can make $r=0.86$ a poor summary of the relationship. It also helps assess whether the model assumptions are plausible.

Exam checklist

Before completing a correlation test, check that you have:

• used $\rho$, not $r$, in the hypotheses;
• chosen the direction from the question;
• used the correct row for $n$ and the correct one tailed or two tailed column;
• reflected the sign for a negative test;
• included equality in the critical region;
• avoided premature rounding;
• stated both the decision and a contextual conclusion;
• written “insufficient evidence” rather than claiming that $H_0$ is true.

Next, consolidate how the PMCC and regression line describe the same paired data in correlation and regression, or compare this critical value method with binomial hypothesis tests and normal hypothesis tests.