PMCC Hypothesis Testing in A-Level Maths

A PMCC hypothesis test asks whether a sample of paired quantitative data provides enough evidence of linear correlation in the underlying population. Students compute a sample product moment correlation coefficient r, then compare it with a critical value from the Edexcel statistical tables to decide whether to reject the null hypothesis that the population correlation coefficient ρ is zero. Typical contexts include windspeed and gust readings, height and arm-span data, and density of population against distance from a city centre.

The marks that go missing on PMCC questions are predictable: writing the hypotheses in terms of the sample statistic r rather than the population parameter ρ, using a positive critical value in a negative-correlation test, mixing the signs of the observed and critical values, and misreading the direction of the alternative hypothesis. This guide focuses on getting all four right. If the data are counts of successes in a fixed number of trials, see binomial hypothesis testing; if the question is about whether a population mean has changed, see normal distribution hypothesis testing.

What a PMCC hypothesis test is

The product moment correlation coefficient r measures the strength and direction of the linear relationship between two quantitative variables in a sample. A PMCC hypothesis test uses that sample value to draw an inference about the unknown population correlation coefficient ρ. The null hypothesis is always ρ = 0, i.e. that there is no linear correlation in the population; the alternative depends on what the question claims.

When PMCC is the right test

Use a PMCC test when:

• The data consist of paired observations on two quantitative variables, typically in a table or scatter diagram.
• The question asks about correlation, evidence of a linear relationship, or whether two variables are linearly related.
• The sample size n is supplied, so that the corresponding critical value can be read off the formula booklet.

If the variable is a count of successes rather than paired data, use binomial hypothesis testing. If the question is about whether a population mean has changed from a known value, see normal distribution hypothesis testing.

Writing hypotheses with ρ

This is where the easiest marks are lost. The hypotheses must always be written in terms of the population parameter ρ, not the sample statistic r:

Three points to drill into students:

• Use the Greek letter ρ, not the Roman letter r.
• Always include the equality “= 0” in the null hypothesis.
• Define ρ in words: “ρ is the product moment correlation coefficient between … and … for the underlying population.”

Watch: hypothesis testing with the P.M.C.C.

An introduction to hypothesis testing with the product moment correlation coefficient using one and two tail tests.

Positive and negative correlation tests

Whether the test is one-tailed or two-tailed comes from the language of the claim. For PMCC, watch specifically for these phrases:

• Positive correlation, increases together, more X is associated with more Y → H₁: ρ > 0.
• Negative correlation, more X is associated with less Y, inverse relationship → H₁: ρ < 0.
• Correlation (no direction stated), there is a linear relationship → H₁: ρ ≠ 0.

How to use critical values

The Edexcel formulae and statistical tables booklet gives positive critical values of r indexed by sample size n and by tail probability. The interpretation depends on the test type:

• One-tailed test for positive correlation: reject H₀ if the observed r is greater than the table value.
• One-tailed test for negative correlation: the relevant critical value is the negative of the table value; reject H₀ if the observed r is less than this.
• Two-tailed test: use the column for half the significance level; reject H₀ if |r| exceeds the table value.

Number-line method for comparison

Inequality comparisons with negative numbers are where students go wrong. A more reliable approach is the number-line method:

1. Draw a horizontal line from −1 to +1.
2. Mark the critical value(s) — for a one-tailed negative test this is a negative number; for a one-tailed positive test it is positive; for a two-tailed test there are two symmetric values.
3. Mark the observed r.
4. Ask: is the observed value further from zero than the critical value? If yes, reject H₀.

Framing the question as “is the observed correlation stronger than the critical correlation?” removes the sign confusion that causes so many lost marks.

Every PMCC conclusion must be non-assertive (“there is/insufficient evidence to reject H₀”) and in context — name the two variables and the direction of correlation being tested. A conclusion that simply says “reject H₀” will lose the final mark.

Common PMCC mistakes

• Writing r instead of ρ. H₀: r = 0 is incorrect; hypotheses must refer to the population parameter ρ.
• Using positive critical values in a negative-correlation test. The table value must be negated; reject H₀ when the observed r is more negative than the negative critical value.
• Incompatible signs in the comparison. Comparing a negative observed r with a positive critical value, leading to “do not reject H₀” when the answer should be the opposite.
• Misidentifying one-tailed vs two-tailed tests. Phrases such as “correlation” without direction are two-tailed; phrases naming positive or negative correlation are one-tailed.

Worked PMCC hypothesis test question

A city council collects data on the population densities in different areas of the city, in people per hectare, and the distances of those areas from the city centre, in km. It calculates the product moment correlation coefficient between the two sets of data and finds it to be −0.51. The data come from 24 sample areas. Test, at the 1% level of significance, the claim that there is a negative correlation between population density and the distance from the city centre.

Teach this as a lesson

Support correlation hypothesis testing with a structured lesson that builds up from interpreting r on a scatter diagram, through using the critical values in the formula booklet, to writing fully contextual conclusions for both positive- and negative-correlation claims.

A Mr Mathematics membership includes the PowerPoint, differentiated worksheet and student PDF for this lesson, plus the full set of A-level statistics resources.

Related hypothesis testing guides

• Binomial hypothesis testing — how to test a claim about a probability using a count of successes in a fixed number of trials.
• Normal distribution hypothesis testing — how to test a claim about a population mean using a sample mean, X̄ and σ/√n.