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

These are original questions written for Marlbridge, for revision and practice on this content. They are not reproduced past-paper questions, and they do not replicate the exam’s exact structure, question count or mark tariffs – examination boards hold copyright in their own papers. Use these alongside the official past papers from your board or school.

These questions cover 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 A Level only. In every question you are given r or a p-value, because calculating the coefficient is excluded, and critical values are supplied as data. Learn the method in the study guide or the revision notes. Other links: the OCR A Level Mathematics hub, the checklist and the free diagnostics.

Questions

1. Four samples P, Q, R and S have coefficients 0.74, −0.97, 0.12 and −0.55 respectively.

(a) Put the samples in order, closest to a straight line first. [2] (b) Describe the trend in sample S. [1]

2. A botanist measures the length and width of 14 leaves and wants to test whether longer leaves tend to be wider.

(a) Write the hypotheses, defining your symbol. [2] (b) Explain why the test has one tail. [1]

3. The age (months) and battery life (hours) of 10 laptops give r = −0.72. Test at the 1% level for negative correlation. The one-tail 1% critical value for n = 10 is 0.7155. [4]

4. For 22 pupils, weekly screen hours and a memory-test score give r = 0.41. Test at the 5% level whether there is any correlation. The two-tail 5% critical value for n = 22 is 0.4227. [4]

5. A test for positive correlation between a river’s depth and its speed has one-tail p-value 0.034.

(a) State the conclusion at the 5% level. [2] (b) State the conclusion at the 1% level. [1] (c) The researcher repeats the analysis as a two-tail test at 5% with the same data. State the decision. [2]

6. For 18 pairs, r = 0.49. The one-tail critical values for n = 18 are 0.4000 (5%), 0.4683 (2.5%), 0.5425 (1%) and 0.5897 (0.5%).

(a) Between which two one-tail levels does the p-value lie? [2] (b) State the decision for H₁: ρ > 0 at 1%. [2] (c) State the decision for H₁: ρ ≠ 0 at 5%. [2]

7. Study X has 9 pairs and study Y has 30 pairs. Both give r = 0.45. For each study, test for positive correlation at 5%. The one-tail 5% critical values are 0.5822 for n = 9 and 0.3061 for n = 30.

(a) Give the decision for study X and for study Y. [4] (b) Explain why the decisions differ. [1]

8. Across 40 towns, the number of libraries and the number of bakeries have a significant positive correlation, r = 0.74. A councillor says that building libraries causes more bakeries to open.

(a) Give two reasons why this claim is not justified. [2] (b) State what the test does show. [1]

9. Seven points have x = −3, −2, −1, 0, 1, 2, 3 and y = x². For these points r = 0.

(a) Explain why r = 0 although y is exactly determined by x. [2] (b) For another sample r = 0.8. Every x is doubled and every y is multiplied by −1. State the new r, with a reason. [2]

10. On 12 school days, a school counts the pupils who cycle in and the cars in the drop-off zone. The coefficient is r = −0.61. The school wants to know whether there is any correlation. The two-tail critical values for n = 12 are 0.5760 (5%) and 0.7079 (1%).

(a) Write the hypotheses, defining your symbol. [2] (b) Carry out the test at the 5% level and state your conclusion. [3] (c) State the conclusion at the 1% level. [2] (d) State an assumption about the data under which the test is valid. [1]

11. On 16 days a wind farm records average wind speed and power output. The coefficient is r = 0.78.

(a) Interpret the value of r in terms of a scatter diagram. [2] (b) Write the hypotheses for a test of positive correlation. [2] (c) The one-tail 1% critical value for n = 16 is 0.5742. Carry out the test at 1%. [3] (d) The one-tail p-value is 0.0002. State how this supports your answer to (c). [1] (e) A colleague says r = 0.78 means the line of best fit has gradient 0.78. Explain the error. [2]

Answers

1. (a) Compare sizes: 0.97, 0.74, 0.55, 0.12 [1]. Order: Q, P, S, R [1]. (b) Negative: as x increases y tends to decrease [1]. [3] Examiner insight: Ordering by closeness to a line uses the size of r, so answers that rank the raw values (putting negatives last) tend to lose the credit.

2. (a) H₀: ρ = 0 and H₁: ρ > 0 [1], where ρ is the population product-moment correlation coefficient between leaf length and leaf width [1]. (b) Only an increase together is of interest, so the alternative is in one direction [1]. [3] Examiner insight: Hypotheses written with r instead of ρ, or with ρ undefined, are a common way to drop the symbol-definition credit.

3. H₀: ρ = 0, H₁: ρ < 0, where ρ is the population correlation between laptop age and battery life [1]. −0.72 < −0.7155 [1], so reject H₀ [1]. There is evidence at the 1% level of negative correlation between laptop age and battery life [1]. [4] Examiner insight: Compare with the negative of the critical value, and finish with a sentence in context that does not claim certainty.

4. H₀: ρ = 0, H₁: ρ ≠ 0, where ρ is the population correlation between screen hours and test score [1]. 0.41 < 0.4227 [1], so do not reject H₀ [1]. There is not enough evidence at the 5% level of correlation between screen hours and memory score [1]. [4] Examiner insight: A result this close to the boundary shows why the decision must come from the comparison, not from a feeling that 0.41 looks large.

5. (a) 0.034 < 0.05 [1], so reject H₀: there is evidence of positive correlation between depth and speed [1]. (b) 0.034 > 0.01, so do not reject H₀ at 1% [1]. (c) The two-tail p-value is 2 × 0.034 = 0.068 [1]. 0.068 > 0.05, so do not reject H₀ [1]. [5] Examiner insight: Say which tail the p-value belongs to before comparing, because the same data can give opposite decisions in one-tail and two-tail tests.

6. (a) 0.4683 < 0.49 < 0.5425 [1], so the one-tail p-value lies between 0.01 and 0.025 [1]. (b) 0.49 < 0.5425 [1], so do not reject H₀ at 1% [1]. (c) Two-tail 5% uses the 2.5% one-tail value, 0.4683 [1]. 0.49 > 0.4683, so reject H₀ [1]. [6] Examiner insight: Show the critical value you chose and the column it came from, so a slip in choosing the column can be seen in the working.

7. (a) Study X: 0.45 < 0.5822 [1], so do not reject H₀. Study Y: 0.45 > 0.3061 [1], so reject H₀ [1]. Both conclusions are stated in words (positive correlation is supported only by Y) [1]. (b) The critical value is lower when n is larger, so the same r is stronger evidence for a larger sample [1]. [5] Examiner insight: Linking the different decisions to sample size, and not just restating the numbers, is how the explanation point is earned.

8. (a) Correlation does not imply causation [1]. A third variable, such as the population of each town, may drive both counts [1]. (b) There is evidence of positive correlation between libraries and bakeries across towns [1]. [3] Examiner insight: Naming a specific alternative explanation in context is stronger than repeating the phrase about causation.

9. (a) The points lie on a parabola, so the relationship is curved [1]. r measures only straight-line association; the downward left half and upward right half cancel [1]. (b) r = −0.8 [1]. Doubling x does not change r, and multiplying y by −1 reverses the sign [1]. [4] Examiner insight: A reason is needed alongside the value, since a bare change of sign could be a guess.

10. (a) H₀: ρ = 0, H₁: ρ ≠ 0 [1], where ρ is the population correlation between pupils cycling and cars in the drop-off zone [1]. (b) The test is two-tail at 5%, with critical value 0.5760. The sample value −0.61 is below −0.5760 [1], so reject H₀ [1]. There is evidence at the 5% level of correlation between the number cycling and the number of cars [1]. (c) −0.61 is not below −0.7079 [1], so do not reject H₀: there is not enough evidence of correlation at the 1% level [1]. (d) The data may be assumed to come from a bivariate normal distribution [1]. [8] Examiner insight: Quote the level beside each conclusion, since the same data are significant at 5% and not at 1%.

11. (a) Positive, so y tends to increase with x [1]. The points lie fairly close to a straight line rising from left to right [1]. (b) H₀: ρ = 0 [1], H₁: ρ > 0, with ρ the population correlation between wind speed and power output [1]. (c) 0.78 > 0.5742 [1], so reject H₀ [1]. There is evidence at the 1% level of positive correlation between wind speed and power output [1]. (d) 0.0002 < 0.01, which agrees with rejecting H₀ [1]. (e) r measures how close the points lie to a line, not its gradient [1]. Lines of very different gradients can both fit data well and give the same r [1]. [10] Examiner insight: Part (e) gains credit for separating closeness from gradient; mentioning only that r lies between −1 and 1 does not answer it.

Where marks are usually lost

• Writing hypotheses about r instead of ρ, or not defining ρ.
• Choosing the tail from the sign of the sample r rather than from the question wording.
• Using the wrong column for a two-tail test.
• Taking n as the number of values instead of the number of pairs.
• Comparing a negative r with a positive critical value without adjusting the inequality.
• Ending with “reject H₀” and no sentence in context.
• Writing “proves” or “accept H₀” in place of cautious wording.
• Mixing up one-tail and two-tail p-values.
• Reading r as a gradient or saying r = 0 means no relationship.

Next steps

• Revise with the revision notes and re-read the study guide.
• Check coverage on the checklist and the course hub.
• Take the free 10-minute diagnostics.
• Book a free trial class.

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.