IB DP Mathematics: Applications and Interpretation – Correlation, regression and hypothesis testing 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 – the IB holds copyright in its own papers. Use these alongside the official past papers available through your school or the IB store.

These practice questions cover correlation, regression and hypothesis testing for IB Diploma Programme Mathematics: Applications and Interpretation. They are aligned to the IB Mathematics: applications and interpretation guide, first assessment 2021, syllabus sections 4.4, 4.10 and 4.11, which are common content for SL and HL. They follow the IB guide for first assessment 2021, which remains the examined syllabus until the new course is first assessed in May 2029, so they apply to the May and November 2026, 2027 and 2028 sessions.

The guide lists every paper in this course as “technology required”, so every question here is labelled (calculator allowed). Give answers exactly or to 3 s.f.

Links: course hub · printable checklist · study guide · revision notes · statistics and probability practice

Questions

1. (calculator allowed) Describe the correlation shown by each value of Pearson’s product-moment correlation coefficient.

(a) r = −0.87 [1] (b) r = 0.08 [1]

2. (calculator allowed) A χ² test for independence is carried out on a contingency table with 4 rows and 3 columns, at the 5% significance level. The critical value is 12.592 and χ²calc = 11.4.

(a) Write down the number of degrees of freedom. [1] (b) State the conclusion of the test, giving a reason. [2]

3. (calculator allowed) Rank the values 7.2, 8.5, 6.9, 8.5, 9.1, 8.5 from smallest (rank 1) to largest, as you would for Spearman’s rank correlation coefficient. [2]

4. (calculator allowed) The fare, y euros, for a taxi trip of x km is modelled by y = 1.85x + 4.20.

(a) Interpret the value 1.85 in context. [1] (b) Interpret the value 4.20 in context. [1] (c) Find the fare for a 12 km trip. [1]

5. (calculator allowed) The age, x years, and value, y thousand euros, of eight cars of the same model are shown.

x	1	2	3	4	5	6	7	8
y	18.5	16.9	15.1	14.2	12.0	11.4	9.6	8.9

(a) Find r and describe the correlation. [2] (b) Find the equation of the regression line of y on x. [2] (c) Estimate the value of a car of this model aged 4.5 years. [1] (d) Explain why the line should not be used for a car aged 15 years. [1]

6. (calculator allowed) Seven students recorded hours spent on a revision app, x, and their quiz score, y.

x	12	15	15	19	22	26	30
y	40	38	45	50	49	61	58

(a) Write down the ranks of the x values. [1] (b) Find Spearman’s rank correlation coefficient, rₛ. [2] (c) Interpret your value of rₛ in context. [1] (d) State what rₛ measures that Pearson’s r does not. [1]

7. (calculator allowed) A spinner has four sectors, claimed to have probabilities 0.4, 0.3, 0.2 and 0.1. It is spun 150 times, with observed frequencies 52, 50, 24 and 24 respectively. A goodness of fit test is carried out at the 5% level.

(a) State the null and alternative hypotheses. [1] (b) Find the expected frequencies. [1] (c) Write down the degrees of freedom. [1] (d) Find the χ² statistic and the p-value. [2] (e) State the conclusion. [1]

8. (calculator allowed) 200 students were asked how they usually travel to school.

	Walk	Bus	Car	Cycle	Total
Year 12	28	26	16	20	90
Year 13	22	34	34	20	110
Total	50	60	50	40	200

A χ² test for independence is carried out at the 5% significance level.

(a) State the null and alternative hypotheses. [1] (b) Show that the expected frequency for Year 12 students who travel by bus is 27. [2] (c) Write down the degrees of freedom. [1] (d) Find the χ² statistic and the p-value. [2] (e) State the conclusion of the test. [1] (f) State, with a reason, whether the conclusion would change at the 10% level. [1]

9. (calculator allowed) Reaction times, in seconds, were recorded for two independent groups.

Group A: 0.42, 0.39, 0.47, 0.44, 0.51, 0.40, 0.46, 0.43
Group B: 0.48, 0.45, 0.52, 0.47, 0.55, 0.50, 0.46

A pooled two-sample t-test is used, at the 5% level, to test whether the mean reaction times differ.

(a) State the null and alternative hypotheses. [2] (b) Find the p-value. [2] (c) State the conclusion of the test in context. [2] (d) State one assumption needed for this test. [1]

10. (calculator allowed) A coach records the number of weeks of training, x, and the number of push-ups an athlete completes, y. In week 9 the athlete was injured.

x	2	3	4	5	6	7	8	9
y	11	14	15	19	20	24	25	5

(a) Find Pearson’s r and Spearman’s rₛ for all eight points. [2] (b) Explain why r is much smaller than rₛ. [2] (c) The week 9 point is removed. Find the regression line of y on x for the other seven points. [2] (d) Use your line to estimate y when x = 5.5. [1] (e) The coach wants to use the line to find how many weeks it takes to reach 30 push-ups. Give two reasons why this is unreliable. [2]

11. (calculator allowed) A two-tailed pooled t-test compares the mean hours of sleep of Year 12 and Year 13 students. The p-value is 0.068.

(a) State the null and alternative hypotheses. [1] (b) State the conclusion at the 5% level. [1] (c) State the conclusion at the 10% level. [1]

Answers

1. (a) Strong negative linear correlation. [1] (b) No (or very weak) linear correlation. [1] Examiner insight: a description needs both strength and direction, and “linear” for r; “negative” alone does not earn the mark.

2. (a) df = (4 − 1)(3 − 1) = 6 [1] (b) 11.4 < 12.592 [1]; do not reject H₀: there is insufficient evidence that the variables are not independent. [1] Examiner insight: the comparison must be written down; a bare “do not reject H₀” without 11.4 < 12.592 loses the first mark.

3. The three 8.5s occupy positions 3, 4 and 5, so each gets (3 + 4 + 5)/3 = 4. [1] Ranks in the given order: 2, 4, 1, 4, 6, 4 [1] [2] Examiner insight: giving the tied values ranks 3, 4 and 5 loses the method mark, and any rₛ found from those ranks will be wrong.

4. (a) The fare increases by €1.85 for each extra kilometre. [1] (b) €4.20 is the fixed charge, the fare before any distance is travelled. [1] (c) 1.85 × 12 + 4.20 = €26.40 [1] Examiner insight: interpretations must use the context (euros, kilometres); “the gradient” or “the y-intercept” alone scores nothing.

5. (a) r = −0.994 [1]; strong negative linear correlation. [1] (b) a = −1.3928…, b = 19.592… [1]; y = −1.39x + 19.6 [1] (c) y = −1.3928… × 4.5 + 19.592… = 13.3 thousand euros [1] (d) 15 years is outside the data range (extrapolation); the line gives a negative value, −1.3. [1] Examiner insight: both coefficients must be to at least 3 s.f.; y = −1.4x + 19.6 loses the accuracy mark under the 3 s.f. rule.

6. (a) 1, 2.5, 2.5, 4, 5, 6, 7 [1] (b) Ranks of y: 2, 1, 3, 5, 4, 7, 6 [1]; rₛ = 0.865 [1] (c) Strong positive monotonic relationship: more app time tends to go with higher scores. [1] (d) rₛ measures any monotonic relationship, not only a linear one. [1] Examiner insight: rₛ is found with technology, but writing the rank lists shows method, so an entry slip can still earn the method mark.

7. (a) H₀: the spinner has probabilities 0.4, 0.3, 0.2, 0.1. H₁: it does not. [1] (b) 150 × each probability: 60, 45, 30, 15 [1] (c) df = 4 − 1 = 3 [1] (d) χ²calc = 8.22 [1]; p-value = 0.0416 [1] (e) 0.0416 < 0.05, so reject H₀: there is evidence the spinner does not have the claimed probabilities. [1] Examiner insight: the conclusion mark needs both the comparison (0.0416 < 0.05) and a statement in context; either alone is not enough.

8. (a) H₀: method of travel is independent of year group. H₁: method of travel is not independent of year group. [1] (b) (row total × column total)/grand total = (90 × 60)/200 [1] = 5400/200 = 27 [1] (c) df = (2 − 1)(4 − 1) = 3 [1] (d) χ²calc = 6.33 [1]; p-value = 0.0966 [1] (e) 0.0966 > 0.05, so do not reject H₀: insufficient evidence that method of travel depends on year group. [1] (f) Yes: 0.0966 < 0.10, so at 10% H₀ would be rejected. [1] Examiner insight: in a “show that” part, writing only “27” earns nothing; the substitution (90 × 60)/200 must appear.

9. (a) H₀: μ_A = μ_B [1]; H₁: μ_A ≠ μ_B [1] (b) Pooled two-sample t-test with the two-tailed (≠) alternative [1]; p-value = 0.0234 [1] (c) 0.0234 < 0.05, so reject H₀ [1]; there is evidence that the mean reaction times of the two groups differ. [1] (d) Reaction times in both populations are normally distributed. [1] Examiner insight: choosing the one-tailed μ_A < μ_B option gives p = 0.0117 instead; the p-value mark is lost, although the conclusion can still earn follow-through.

10. (a) r = 0.177 [1]; rₛ = 0.333 [1] (b) The point (9, 5) is an outlier far below the trend [1]; r is strongly affected by outliers, while rₛ is less sensitive because the outlier only takes the lowest rank. [1] (c) a = 2.3928…, b = 6.3214… [1]; y = 2.39x + 6.32 [1] (d) y = 2.3928… × 5.5 + 6.3214… = 19.5 push-ups (3 s.f.) [1] (e) 30 push-ups is outside the range of the y data, so this is extrapolation. [1] A y on x line should not be used to predict x from a value of y. [1] Examiner insight: in (d), predicting with the rounded 2.39 and 6.32 happens to give the same 3 s.f. answer here, but in general use the stored values or risk the accuracy mark.

11. (a) H₀: the two year groups have equal mean hours of sleep; H₁: the means are different. [1] (b) 0.068 > 0.05: do not reject H₀; insufficient evidence of a difference. [1] (c) 0.068 < 0.10: reject H₀; evidence that the mean hours of sleep differ. [1] Examiner insight: H₁ for a two-tailed test uses ≠ or “different”; writing “>” contradicts the test described and loses the mark.

Where marks are usually lost

• Describing a correlation without both strength and direction, or without “linear” for r.
• Predicting from rounded a and b instead of the stored values.
• Using a regression line outside the data range without a comment.
• Rearranging a y on x line to estimate x.
• Giving tied values consecutive ranks instead of the average rank.
• Counting df as rows × columns, or as n instead of n − 1 in a goodness of fit test.
• Writing “accept H₀”, or stopping at “reject H₀” with no sentence in context.
• Selecting the wrong tail on the GDC for a t-test.

Next steps

• Revision notes for this unit.
• Study guide with full worked examples.
• Course hub.
• Printable checklist.
• Try all free 10-minute diagnostics.
• Book a free trial class.

Official syllabus

International Baccalaureate Organization, Diploma Programme, Mathematics: applications and interpretation guide, first assessment 2021.