AS & A Level Mathematics (9709) — May–June 2024, Paper 63

7 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 · The Poisson distribution (Poisson as an approximation to the binomial)

The random variable X has the distribution B(4000, 0.001). Use a suitable approximating distribution to find P(2 <= X < 5).

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Question 2 · 4 marks · Sampling and estimation (confidence intervals for a population mean; unbiased estimate of population standard deviation)

The widths, w cm, of a random sample of 150 leaves of a certain kind were measured. The sample mean of w was found to be 3.12 cm. Using this sample, an approximate 95% confidence…

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Question 3 · 5 marks · The normal distribution (linear combinations of independent normal random variables)

The masses in kilograms of large and small bags of cement have the independent distributions N(50, 2.4) and N(26, 1.8) respectively. Find the probability that the total mass of 5…

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Question 4 · 5 marks · Hypothesis tests (test for a population proportion using the binomial distribution)

In this question you should not use an approximating distribution. At an election in Menham last year, 24% of voters supported the Today Party. A student wishes to test whether…

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Question 5 · 10 marks · Continuous random variables (probability density functions; total area = 1)

A random variable X has probability density function f given by f(x) = ax - x^3 for 0 <= x <= sqrt(2), and f(x) = 0 otherwise, where a is a constant. Show that a = 2.

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Question 6 · 9 marks · Sampling and estimation (unbiased estimates of population mean and variance from frequency data)

The numbers of green sweets in 200 randomly chosen packets of Frutos are summarised in the table. Number of green sweets 0 1 2 3 3 --- --- --- --- --- --- Number of packets 32 50…

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Question 7 · 13 marks · The Poisson distribution (sum of independent Poisson variables)

The independent random variables X and Y have the distributions Po(1.9) and Po(2.2) respectively. Find P(X + Y < 4).

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