OCR A Level Mathematics A: Hypothesis testing for correlation: Pearson's product-moment coefficient (H240) – Revision Notes

These notes condense references 2.05f and 2.05g of the OCR A Level Mathematics A (H240) specification, Version 3.1 (August 2026), for first assessment in 2018. Both are Stage 2, so A Level only. For full explanations and worked examples, read the study guide. Then test yourself with the practice questions. Other links: the OCR A Level Mathematics hub, the printable checklist, the wider Statistics revision notes and the free diagnostics.

Key facts (2.05f)

Idea	Remember
What r measures	How close the points lie to a straight line
Range	−1 ≤ r ≤ 1
r = 1 or r = −1	All points exactly on a line, upward or downward
Sign	Direction of the trend
Size of r	Tightness of the points, not the gradient
r near 0	No linear pattern; a curve is still possible
Change of units	Positive scaling or shifting leaves r unchanged; a negative multiplier flips its sign
Sample and population	r is for the sample, ρ is for the population

The specification excludes calculating r. You are given it, or you are given a p-value.

The test (2.05g) at a glance

Item	Statement
Null hypothesis	H₀: ρ = 0
Alternatives	H₁: ρ > 0, or ρ < 0, or ρ ≠ 0
Test statistic	The sample coefficient r
Sample size n	Number of pairs
Critical value c	From the table: n, significance level and tails
Assumption	Data may be assumed to come from a bivariate normal distribution
Table	The specification says a table of critical values will be provided

Method in steps: critical value

1. Define ρ in context and write H₀ and H₁. Pick the alternative from the wording.
2. Note n (pairs), the significance level and whether the test is one-tail or two-tail.
3. Read c from the table. For a two-tail test at α%, use the one-tail column for (α ÷ 2)%, unless the table is labelled by two-tail level.
4. Compare r with c.
5. Decide and write a contextual conclusion.

H₁	Reject H₀ if
ρ > 0	r > c
ρ < 0	r < −c
ρ ≠ 0	r > c or r < −c

Method in steps: p-value

1. Write the hypotheses as above.
2. Check that the p-value belongs to your tails. Double a one-tail p-value for a two-tail test.
3. Reject H₀ if p is less than the significance level.
4. With a table only, bracket p: if r lies between the 5% and 2.5% one-tail critical values, then 0.025 < p < 0.05.
5. Write the conclusion in context.

Reading r from a scatter diagram

What you see	What it suggests about r
Narrow band rising from left to right	r positive and close to 1
Narrow band falling from left to right	r negative and close to −1
Wide, shapeless cloud	r close to 0
Clear curve, such as a U shape	r can be near 0 even though the relationship is strong
One point far from the rest	r may be pulled up or down by that point, so check it

Only the first two rows show a strong linear pattern. For the curve, a test on r would be misleading, because the test assesses straight-line correlation. Always look at the diagram when one is given, and say what you see before you quote the test result.

Second worked reminder: negative correlation with a p-value

Eighteen pairs give r = −0.49 and a one-tail p-value of 0.0195 for H₁: ρ < 0.

H₀: ρ = 0,  H₁: ρ < 0
p = 0.0195 < 0.05, reject H₀ at 5%
p = 0.0195 > 0.01, do not reject H₀ at 1%

Conclusion at 5%: there is evidence of negative correlation in the population. At 1% there is not enough evidence. Quote the level you used each time.

Sentence frames

• Hypotheses: “H₀: ρ = 0 and H₁: ρ < 0 (or > 0, or ≠ 0). Here ρ stands for the population product-moment correlation coefficient linking [x] with [y].”
• Reject: “Reject H₀ at the 5% level: there is evidence of correlation (state its direction) between [x] and [y].”
• Do not reject: “Do not reject H₀ at the 5% level: the sample gives insufficient evidence of correlation between [x] and [y].”

Worked reminder

Twelve pairs of measurements give r = 0.58. Test at 5% for positive correlation, given the one-tail 5% critical value for n = 12 is 0.4973.

H₀: ρ = 0,  H₁: ρ > 0
0.58 > 0.4973, so reject H₀

There is evidence at the 5% level of positive correlation. If the question asked about correlation of either sign, the 2.5% one-tail value for n = 12 is 0.5760. Since 0.58 > 0.5760, you would still reject H₀.

Must-know distinctions

Pair	Difference
r and ρ	Sample value you are given, and population value you test
Strong and significant	Strong describes the size of r; significant describes the test result
One-tail and two-tail	Wording such as “positive” or “negative” gives one tail; “correlation” or “relationship” gives two
No correlation and no relationship	r near zero rules out only a linear pattern
Do not reject and accept	A failed test means a lack of evidence, not proof that ρ = 0
Correlation and causation	A significant r does not show one variable causes the other

Sample size effect: the same r can fail with a small n and pass with a large n, because the critical value falls as n rises.

Quick self-test

Critical values below are supplied as data and computed to 4 decimal places.

1. Samples have r = 0.2, −0.9 and 0.6. Which is closest to a straight line?
2. Write the hypotheses for a test of whether there is negative correlation between altitude and temperature.
3. A study has 8 pairs of readings. What value of n do you use?
4. For n = 8, the 5% one-tail critical value is 0.6215. Test H₁: ρ < 0 when r = −0.70.
5. For n = 11, the 5% one-tail value is 0.5214. A two-tail test at 10% gives r = 0.55. What is the decision?
6. A two-tail test at 5% is given a two-tail p-value of 0.07. What is the decision?
7. Points plotted on a diagram form a clear U shape, yet r = 0.02. Can we say the two variables are unrelated?
8. For n = 16, the 1% one-tail critical value is 0.5742. Test H₁: ρ > 0 at 1% when r = 0.45.
9. For n = 40, r = 0.28. The 5% one-tail value is 0.2638 and the 2.5% one-tail value is 0.3120. Give the decision for a one-tail and a two-tail test at 5%.
10. All the x-values are converted from centimetres to metres. What happens to r?
11. A test finds significant positive correlation between ice-skating attendance and woollen hat sales. Is it right to say attendance causes the sales?
12. Why is r = 0.5 convincing for n = 30 but not for n = 6?

Answers

1. r = −0.9, because its size is the largest.
2. H₀: ρ = 0, H₁: ρ < 0, where ρ is the population product-moment correlation coefficient between altitude and temperature.
3. n = 8, the number of pairs.
4. −0.70 < −0.6215, so reject H₀. At the 5% level the data support negative correlation.
5. Two-tail 10% uses the 5% one-tail column: c = 0.5214. Since 0.55 > 0.5214, reject H₀.
6. 0.07 > 0.05, so do not reject H₀. There is not enough evidence of correlation.
7. No. A value near zero rules out only a straight-line pattern, and the U shape is a strong curved relationship.
8. 0.45 < 0.5742, so do not reject H₀. There is not enough evidence at the 1% level of positive correlation.
9. One-tail: 0.28 > 0.2638, reject H₀. Two-tail: 0.28 < 0.3120, do not reject H₀.
10. r is unchanged, because the scaling is by a positive number.
11. No. A third variable such as cold weather could drive both; the test shows association only.
12. The critical value is lower for n = 30 (0.3061 at 5% one-tail) than for n = 6 (0.7293), so 0.5 passes the first and fails the second.

Where marks are usually lost

• Writing hypotheses about r instead of ρ, or leaving ρ undefined.
• Choosing the tail from the sign of r instead of from the wording.
• Using n as the number of values rather than pairs.
• Using the 5% column for a two-tail 5% test, instead of the 2.5% one-tail column.
• Comparing a negative r with a positive critical value without changing the inequality.
• Stating “reject H₀” with no conclusion in context.
• Wording the answer as certain: “proves”, “shows that H₀ is true”, “accept H₀”.
• Treating a significant result as proof of causation.
• Forgetting that a one-tail p-value must be doubled for a two-tail test.

Official syllabus

OCR, A Level Mathematics A (H240) specification, Version 3.1 (August 2026), for first assessment in 2018, section 2.05 Statistical Hypothesis Testing, references 2.05f and 2.05g.