AS & A Level Further Mathematics (9231) — May–June 2023, Paper 43

6 questions · every question answerable online with instant point-by-point AI marking against the official mark scheme.

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Question 1 · 8 marks · Further Probability & Statistics - continuous random variables (cumulative distribution function)

The continuous random variable X has probability density function f given by f(x) = (1/6)(x^(-1/3) - x^(-2/3)) for 1 <= x <= 27, f(x) = 0 otherwise. Find the cumulative…

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Question 2 · 6 marks · Further Probability & Statistics - confidence intervals (population mean, small sample, t-distribution)

Shane is studying the lengths of the tails of male red kangaroos. He takes a random sample of 14 male red kangaroos and measures the length of the tail, x m, for each kangaroo. He…

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Question 3 · 8 marks · Further Probability & Statistics - non-parametric tests (Wilcoxon matched-pairs signed-rank test)

A large number of students took two test papers in mathematics. The teacher believes that the marks obtained in Paper 1 will be higher than the marks obtained in Paper 2. She…

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Question 4 · 9 marks · Further Probability & Statistics - hypothesis testing (difference of two population means, large samples, z-test)

An inspector is checking the lengths of metal rods produced by two machines, X and Y. These rods should be of the same length, but the inspector suspects that those made by…

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Question 5 · 9 marks · Further Probability & Statistics - probability generating functions (expectation)

The random variable X has probability generating function GX(t) given by GX(t) = k(1 + 3t + 4t^2), where k is a constant. Show that E(X) = 11/8.

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Question 6 · 10 marks · Further Probability & Statistics - chi-squared test (test of independence in a contingency table)

A scientist is investigating whether the ability to remember depends on age. A random sample of 150 students in different age groups is chosen. Each student is shown a set of 20…

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