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A-Level Further Maths Paper 4 Past Papers & Drilling: Statistics with Answers
TUTORZONE ORIGINAL DRILLING MOCK · CAMBRIDGE 9231 PAPER 4 (FURTHER PROBABILITY & STATISTICS)
Written to the Cambridge 9231 syllabus (2026-2027): Paper 4 Further Probability & Statistics is 50 marks, 1 h 30 min and 5-7 structured questions, worth 40% of AS and 20% of A Level. The AS Level needs Paper 1 plus Paper 3 or Paper 4; this is a 5-question, 50-mark drilling mock with answers and techniques.
What is A-Level Further Maths (9231) Paper 4 worth?
Direct answer: Paper 4 Further Probability & Statistics is 50 marks, 1 h 30 min and 5-7 structured questions, worth 40% of AS and 20% of A Level (Cambridge 9231, 2026-2027).
Assessment at a glance (2027)
| Component | Content | Weighting | Duration |
|---|---|---|---|
| Paper 1 (Further Pure Mathematics 1) | 6-8 structured questions, 75 marks | 60% AS / 30% A Level | 2 hours |
| Paper 2 (Further Pure Mathematics 2) | 7-9 structured questions, 75 marks | 30% A Level | 2 hours |
| Paper 3 (Further Mechanics) | 5-7 structured questions, 50 marks | 40% AS / 20% A Level | 1 h 30 min |
| Paper 4 (Further Probability & Statistics) | 5-7 structured questions, 50 marks | 40% AS / 20% A Level | 1 h 30 min |
Paper 4 technique: 1 h 30 min for 5 questions, about 18 minutes each. Take the PGF and sign-test questions first, then the chi-squared test and the t interval, and leave the continuous random variable integral until last. Remember: the pdf integrates to 1; E(X) is the integral of x f(x); Var(X) = G”(1) + G'(1) – [G'(1)]^2; and the confidence interval is the mean plus or minus t times the standard error.
How this paper was built
This is a TutorZone original drilling mock: the questions follow the Cambridge 9231 Further Mathematics structure (Paper 4 Further Probability & Statistics: 50 marks, 1 h 30 min, 5-7 structured questions, 20% of A Level) and the answers are our own, not Cambridge past papers or marking references.
Every numeric answer was recomputed programmatically, so you can use the model answers as a self-check after drilling.
Pacing
| Section | Marks | Suggested time | What to do first |
|---|---|---|---|
| Paper 1 (Further Pure Mathematics 1) | 60% AS / 30% A Level | about 18 min per question | roots, series, matrices, polar coordinates, vectors, induction |
| Paper 2 (Further Pure Mathematics 2) | 30% A Level | about 16 min per question | hyperbolic functions, eigenvalues, series, complex numbers, ODEs |
| Paper 3 (Further Mechanics) | 40% AS / 20% A Level | about 15 min per question | projectiles, rigid bodies, circular motion, Hooke’s law, momentum |
| Paper 4 (Further Probability & Statistics) | 40% AS / 20% A Level | about 15 min per question | continuous random variables, t-tests, chi-squared, non-parametric, PGFs |
What do the five questions test?
Direct answer: a continuous random variable’s density, mean and variance; a t confidence interval for a normal mean; a chi-squared goodness-of-fit test; the non-parametric sign test; and a probability generating function.
Paper 4: Further Probability & Statistics drilling (5 questions, 50 marks)
The five questions cover the Further Probability & Statistics topics: continuous random variables, inference with the normal and t distributions, chi-squared tests, the non-parametric sign test and probability generating functions.
Question 1 (12 marks)
A continuous random variable X has probability density function f(x) = k x^2 for 0 at most x at most 3, and f(x) = 0 otherwise. (a) Find k. (3) (b) Find E(X) and Var(X). (6) (c) Find P(X > 2). (3)
Question 2 (10 marks)
A random sample of 12 observations from a normal distribution has mean 24.6 and sample standard deviation 1.8. The t-value for 11 degrees of freedom at 95% is 2.201. (a) Find the 95% confidence interval for the population mean. (7) (b) Explain what the interval means. (3)
Question 3 (10 marks)
A die is rolled 120 times; the observed counts are 14, 27, 18, 22, 19 and 20. Test at the 5% level whether the die is fair, using a chi-squared goodness-of-fit test (5 degrees of freedom, 5% critical value 11.07). (a) State the expected frequency and find the test statistic. (7) (b) State your conclusion. (3)
Question 4 (9 marks)
Among 12 paired observations, the second measurement is higher in 8 pairs, the first is higher in 3 pairs, and 1 pair ties. Use a sign test at the 5% level to test whether the median has changed. (a) State the test statistic and the effective sample size. (3) (b) Find the two-tailed p-value. (4) (c) State your conclusion. (2)
Question 5 (9 marks)
The probability generating function of a discrete random variable X is G(t) = (1/3)t + (1/2)t^2 + (1/6)t^3. (a) Find P(X = 2). (2) (b) Find E(X). (3) (c) Use Var(X) = G”(1) + G'(1) – [G'(1)]^2 to find Var(X). (4)
What does a chi-squared statistic of 4.7 mean at the 5% level?
Direct answer: the 5% critical value with 5 degrees of freedom is 11.07; since 4.7 is below it, we do not reject fairness – there is no evidence that the die is biased.
Paper 4 model answers
Question 1
(a) The integral of k x^2 from 0 to 3 equals 9k = 1, so k = 1/9. (b) E(X) = (1/9) times the integral of x^3 = 9/4 = 2.25; E(X^2) = (1/9) times the integral of x^4 = 27/5 = 5.4; so Var(X) = 5.4 – 2.25^2 = 27/80 = 0.3375. (c) P(X > 2) = 1 – (1/9)(8/3) = 19/27, about 0.704.
Question 2
(a) Standard error = 1.8/root 12 = 0.5196 and the half-width = 2.201 x 0.5196 = 1.1437, so the interval is 24.6 plus or minus 1.14, that is 23.46 to 25.74 (23.5 to 25.7 to 3 significant figures). (b) The interval is a statement about the method: if sampling were repeated with the same method, about 95% of the intervals would contain the true mean. It is not a 95% probability that the mean lies between 23.46 and 25.74.
Question 3
(a) If the die is fair each face has expected frequency 120/6 = 20, and the statistic is (14-20)^2/20 + (27-20)^2/20 + (18-20)^2/20 + (22-20)^2/20 + (19-20)^2/20 + 0 = 4.7. (b) Since 4.7 is below 11.07 we do not reject fairness: there is no evidence that the die is biased.
Question 4
(a) Discard the tied pair, so n = 11 and the smaller count is X = 3. (b) P(X at most 3) = (1 + 11 + 55 + 165)/2^11 = 232/2048 = 29/256, about 0.113, so the two-tailed p-value is about 0.227. (c) Since 0.227 exceeds 0.05 we do not reject the null hypothesis: there is no evidence that the median changed.
Question 5
(a) The coefficient of t^2 is P(X = 2) = 1/2. (b) G'(t) = 1/3 + t + (1/2)t^2, so E(X) = G'(1) = 11/6, about 1.833. (c) G”(t) = 1 + t gives G”(1) = 2, so Var(X) = 2 + 11/6 – (11/6)^2 = 17/36, about 0.472.
How many questions and marks does this paper have?
Direct answer: 5 questions for 50 marks, matching Paper 4; allow the full 1 h 30 min.
Where these marks are usually lost
| Mistake | What it looks like | What it costs | Instead |
|---|---|---|---|
| Unnormalised density | Integrating before finding k | Use the fact that the density integrates to 1 | k = 1/9 |
| t and z confused | Using 1.96 for a small sample | Small sample with unknown sigma needs a t-value | 23.46 to 25.74 |
| Wrong chi-squared degrees of freedom | Using 6 for six classes | Degrees of freedom = classes – 1 = 5 | 4.7 |
| Tied pairs kept in a sign test | Counting the tied pair | Discard tied pairs, so n = 11 | p about 0.227 |
| Variance term missing in the PGF | Using G”(1) alone | Var(X) = G”(1) + G'(1) – [G'(1)]^2 | 17/36 |
How to use this paper
First drill: the full 1 h 30 min on all 5 questions, with MF19 and the statistical tables.
Mark it: check the degrees of freedom, the significance level and the conclusion; compare p-values with critical values.
Second drill: redo three to seven days later, aiming to finish inside 70 minutes.
What should I do after drilling Paper 4?
Direct answer: log every degrees-of-freedom and conclusion error, redo the paper in three to seven days, then revise Papers 1 and 3 to complete the 9231 cycle.
FAQ
Is this a Cambridge past paper?
No. This is not a Cambridge past paper: the questions and answers are TutorZone originals, not marking references.
What is in A-Level Further Maths 9231 Paper 4?
Further Probability & Statistics: continuous random variables, inference with the normal and t distributions, chi-squared tests, non-parametric tests and probability generating functions; 50 marks, 1 h 30 min, 5-7 structured questions.
How much is Paper 4 worth?
Paper 4 is 40% of the AS Level and 20% of the A Level; the AS Level is Paper 1 plus Paper 3 or Paper 4.
Do I need the statistical tables?
Yes in a real exam; every critical or t value used here is printed in the question so you can still self-mark.
Official references
| Resource | What to check |
|---|---|
| Cambridge A-Level Further Mathematics 9231 syllabus (2026-2027) | Confirm the marks, times and weightings for Papers 1-4 |
| https://www.tutorzone.com.hk/en-alevel-further-mathematics-9231-paper-3-mock-2027/ | A-Level Further Mathematics Paper 3 drilling mock |
Related: practice area · A-Level Further Mathematics Paper 2 drilling mock · A-Level Further Mathematics Paper 3 drilling mock
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