AS & A Level Further Mathematics (9231) — May–June 2023, Paper 41

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

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Question 1 · 4 marks · Estimation (confidence interval for a normal mean, small sample, t-distribution)

The lengths of the leaves of a particular type of tree are normally distributed with mean mu cm. The lengths, x cm, of a random sample of 12 leaves of this type are recorded. The…

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Question 2 · 8 marks · Hypothesis tests (difference of two population means, pooled two-sample t-test, equal variances)

The children at two large schools, P and Q, are all given the same puzzle to solve. A random sample of size 10 is taken from the children at school P. Their individual times to…

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Question 3 · 9 marks · Chi-squared tests (goodness of fit — expected frequencies from a pdf)

A random sample of 50 values of the continuous random variable X was taken. These values are summarised in the following table. Interval 1 <= x < 1.5 1.5 <= x < 2 2 <= x < 2.5 2.5…

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Question 4 · 9 marks · Non-parametric tests (Wilcoxon matched-pairs signed-rank test)

A random sample of 13 technology companies is chosen and the numbers of employees in 2018 and in 2022 are recorded. Company A B C D E F G H I J K L M --- --- --- --- --- --- ---…

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Question 5 · 9 marks · Probability generating functions (constructing a PGF)

Harry has three coins. - One coin is biased so that, when it is thrown, the probability of obtaining a head is 1/3. - The second coin is biased so that, when it is thrown, the…

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Question 6 · 11 marks · Continuous random variables (cumulative distribution function)

The continuous random variable X has probability density function f given by f(x) = (3/28)(e^(x/2) + 4e^(-x/2)) for 0 <= x <= 2 ln 3, f(x) = 0 otherwise. (a) Find the cumulative…

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