IB DP Mathematics: Applications and Interpretation – Data collection, non-linear regression, hypothesis tests and Markov chains (HL) 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.

This practice set covers an HL unit of IB Diploma Programme Mathematics: Applications and Interpretation, aligned to the IB Mathematics: applications and interpretation guide, first assessment 2021, syllabus sections 4.12, 4.13, 4.18 and 4.19. Every question is HL only (AHL). It follows the IB guide for first assessment 2021, which remains the examined syllabus until the new course is first assessed in May 2029, so it applies to the May and November 2026, 2027 and 2028 HL sessions.

All three HL papers require a GDC, so every question is labelled “(calculator allowed)”. Give answers exactly or to 3 significant figures.

Learn the methods in the study guide and the revision notes. See also the IB DP Maths AI course hub, the printable syllabus checklist and the SL statistics and probability practice.

Questions

1. (calculator allowed, HL) A survey asks: “Don’t you agree that the library should open for longer at weekends?”

(a) State why this question is biased. [1] (b) Rewrite it as an unbiased, structured question. [1]

2. (calculator allowed, HL) A school writes a new reading test.

(a) The same pupils sit it twice, three weeks apart, and the scores are compared. Name this reliability test. [1] (b) Describe one other reliability test. [1] (c) The scores are compared with scores on an established reading test. Name this type of validity. [1]

3. (calculator allowed, HL) A normal model, with μ and σ estimated from the data, is fitted to 150 lengths in 8 classes. Expected frequencies of the two lowest classes are 2.1 and 6.8; of the two highest, 5.9 and 3.4.

(a) State how many classes should be used in a χ² goodness of fit test, giving a reason. [1] (b) Find the number of degrees of freedom. [1] (c) The test gives χ² = 8.94. The critical value at the 5% level is 7.815. State the conclusion. [2]

4. (calculator allowed, HL) The table shows five data points.

x	0	1	2	3	4
y	2.0	3.1	5.9	11.2	18.8

Model A is y = x² + 2. Model B is y = 1.9 × 1.8ˣ.

(a) Find SSres for model A. [2] (b) Find SSres for model B. [1] (c) State which model fits these data better. [1]

5. (calculator allowed, HL) Fuel use f (litres per 100 km) of a car at speed v km/h:

v	40	50	60	70	80	90	100	110
f	7.4	6.6	6.1	5.9	6.0	6.4	7.0	7.9

(a) Find the quadratic regression model f = av² + bv + c. [2] (b) Write down R² and interpret it. [2] (c) Estimate the fuel use at 75 km/h. [1] (d) Comment on using the model at 130 km/h. [1]

6. (calculator allowed, HL) Scores are N(μ, 12²). A sample of 36 has mean 89.1. Test at the 5% level whether μ differs from 85.

(a) State the hypotheses. [1] (b) Find the z-value and the p-value. [2] (c) State the conclusion. [1] (d) Find the critical region for x̄. [2]

7. (calculator allowed, HL) Eight people time a puzzle (minutes) before and after a training session.

Before	14.2	15.8	13.1	16.4	12.9	15.0	14.7	13.6
After	13.5	15.1	13.4	15.2	12.1	14.6	13.8	13.3

Test at 5% whether the session reduces the mean time. [6]

8. (calculator allowed, HL) 20% of customers return an item. After a website change, 25 customers are sampled to test at 5% whether this proportion has increased.

(a) State the hypotheses. [1] (b) Find the critical region. [3] (c) Write down P(Type I error). [1] (d) Find P(Type II error) if the true proportion is 0.35. [2]

9. (calculator allowed, HL) Spam arrived at a mean rate of 6 per hour. After a filter is installed, 5 spam emails arrive in 2 hours. Test at 5% whether the rate has fallen. [5]

10. (calculator allowed, HL) Each year, of network A’s customers 70% stay, 20% move to B and 10% to C. Of B’s, 30% move to A, 60% stay, 10% move to C. Of C’s, 20% move to A, 30% to B, 50% stay. The initial shares are 50%, 30% and 20%.

(a) Write down the transition matrix T. [2] (b) Find the shares after 3 years. [2] (c) Explain why the chain is regular. [1] (d) Find the exact steady state. [3] (e) In the long run, how many of 3600 customers use A? [1]

11. (calculator allowed, HL) Goals per match in 80 matches:

Goals	0	1	2	3	4	5	6
Matches	17	27	18	11	4	2	1

(a) Show that the mean is 1.6. [1] (b) Using Po(1.6), find the expected frequencies and explain which categories to merge. [3] (c) Test at 5% whether a Poisson model fits. [4] (d) The next 6 matches have 4 goals in total. Using a Poisson model with mean 1.6 per match, test at 5% whether the rate has fallen. [3]

12. (calculator allowed, HL) Revision hours h and test scores s of 8 students:

h	2.1	3.4	1.2	4.0	2.8	3.1	1.7	3.6
s	64	71	60	69	74	70	58	76

(a) Assuming bivariate normality, test at 1% whether ρ > 0. [4] (b) Find R² for the linear model of s on h. Interpret it. [2]

Answers

1. (a) It leads the respondent towards “yes”. [1] (b) For example: “How satisfied are you with weekend opening hours? Very satisfied / Satisfied / Neutral / Dissatisfied”. [1] Examiner insight: The rewrite must be neutral and have fixed answer choices; an open question is not structured.

2. (a) Test-retest. [1] (b) Parallel forms: two equivalent versions given to the same pupils and the scores compared. [1] (c) Criterion-related validity. [1] Examiner insight: A description must say what is compared; a name alone scores nothing in (b).

3. (a) 2.1 < 5 and 3.4 < 5, so merge each with its neighbour: 6 classes. [1] (b) df = 6 − 1 − 2 = 3. [1] (c) 8.94 > 7.815 [1], so reject H₀: the normal model is not a good fit. [1] Examiner insight: The “− 2” for estimating μ and σ earns the df mark; df = 5 loses it.

4. (a) Predictions 2, 3, 6, 11, 18; residuals 0, 0.1, −0.1, 0.2, 0.8 [1]. SSres = 0.7 [1] (b) SSres = 1.50 [1] (c) Model A, since its SSres is smaller. [1] Examiner insight: Residual = observed − predicted; list them so the method mark can be awarded if the sum is wrong.

5. (a) a = 0.00141, b = −0.204 [1], c = 13.3 [1] (b) R² = 0.999 [1]. The model accounts for 99.9% of the variability in fuel use. [1] (c) f(75) = 5.92 L/100 km [1] (d) 130 is outside the data range (extrapolation), so it may be unreliable. [1] Examiner insight: Using the 3 s.f. coefficients gives 5.93; accuracy marks expect the stored full values.

6. (a) H₀: μ = 85, H₁: μ ≠ 85 [1] (b) z = (89.1 − 85)/(12/6) = 2.05 [1], p = 0.0404 [1] (c) 0.0404 < 0.05, so reject H₀: evidence that μ differs from 85. [1] (d) 85 ± 1.96 × 2 [1]: x̄ < 81.1 or x̄ > 88.9 [1] Examiner insight: A two-tailed test needs both tails in the critical region; one inequality earns the method mark only.

7. d = before − after: 0.7, 0.7, −0.3, 1.2, 0.8, 0.4, 0.9, 0.3 [1] H₀: μ_d = 0, H₁: μ_d > 0 [1] σ unknown, so a one-sample t-test on d [1] t = 3.65 [1], p = 0.00407 [1] 0.00407 < 0.05: reject H₀; the session reduces mean time. [1] Examiner insight: A two-sample t-test on paired data is the wrong method and loses the later method marks.

8. (a) H₀: p = 0.2, H₁: p > 0.2 [1] (b) X ~ B(25, 0.2) [1]; P(X ≥ 9) = 0.0468 < 0.05 and P(X ≥ 8) = 0.109 > 0.05 [1]; critical region X ≥ 9 [1] (c) 0.0468 [1] (d) P(X ≤ 8 | p = 0.35) [1] = 0.467 [1] Examiner insight: Quote the probabilities either side of the boundary; a bare “X ≥ 9” loses marks.

9. H₀: λ = 6, H₁: λ < 6 (per hour) [1] Under H₀, X ~ Po(12) for 2 hours [1] P(X ≤ 5) [1] = 0.0203 [1] 0.0203 < 0.05: reject H₀; evidence the rate has fallen. [1] Examiner insight: Using Po(6) for a 2-hour count loses the distribution mark and the accuracy mark.

10. (a) Columns are “from”, rows “to” [1]:

      A    B    C
T = [0.7  0.3  0.2]   A
    [0.2  0.6  0.3]   B
    [0.1  0.1  0.5]   C

All entries correct [1] (b) s₃ = T³(0.5, 0.3, 0.2)ᵀ [1] = (0.472, 0.359, 0.169)ᵀ [1] (c) Every entry of T itself is positive. [1] (d) −0.3a + 0.3b + 0.2c = 0 and 0.2a − 0.4b + 0.3c = 0 [1], with a + b + c = 1 [1] s = (17/36, 13/36, 1/6)ᵀ [1] (e) 3600 × 17/36 = 1700 [1] Examiner insight: A decimal steady state such as 0.472 loses the final mark when “exact” is asked.

11. (a) (27 + 36 + 33 + 16 + 10 + 6)/80 = 128/80 = 1.6 [1] (b) Expected: 16.15, 25.84, 20.67, 11.03 [1]; E(4) = 4.41, E(≥ 5) = 1.89 [1]. Both are below 5, so merge into “≥ 4”: expected 6.31, observed 7. [1] (c) H₀: the goals follow a Poisson distribution [1] df = 5 − 1 − 1 = 3 [1] χ² = 0.519, p = 0.915 [1] 0.915 > 0.05: do not reject H₀; a Poisson model is suitable. [1] (d) H₀: λ = 1.6, H₁: λ < 1.6; under H₀, Y ~ Po(9.6) [1] P(Y ≤ 4) = 0.0378 [1] 0.0378 < 0.05: reject H₀; evidence the scoring rate has fallen. [1] Examiner insight: Enter df = 3 in the GDC’s χ² goodness of fit test; df = 4 (categories − 1) gives the wrong p-value.

12. (a) H₀: ρ = 0, H₁: ρ > 0 [1] r = 0.819 [1], p = 0.00645 [1] 0.00645 < 0.01: reject H₀; evidence of positive correlation. [1] (b) R² = r² = 0.671 [1]; 67.1% of the variability in score is accounted for by the linear model. [1] Examiner insight: Use the one-tailed p-value; the two-tailed p (0.0129) gives the wrong conclusion at 1%.

Where marks are usually lost

• Not subtracting estimated parameters from df.
• Describing a reliability test without saying what is compared.
• Predicting with rounded regression coefficients.
• Using a z-test when σ is estimated from the sample.
• Analysing matched pairs as two independent samples.
• Scaling a Poisson rate to the wrong time period.
• Using the H₀ value instead of the true value for P(Type II).
• Giving only one tail of a two-tailed critical region.
• Transposing the transition matrix.

Next steps

• Review the revision notes and the study guide.
• See the IB DP Maths AI course hub and the printable syllabus checklist.
• Read the exam preparation guide.
• 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, syllabus sections AHL 4.12, 4.13, 4.18 and 4.19.