AS & A Level Mathematics (9709) — May–June 2023, Paper 63

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

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Question 1 · 3 marks · Continuous random variables (probability density functions; expectation)

A random variable X has probability density function f, where f(x) = (3/2)(1 - x^2) for 0 <= x <= 1, and f(x) = 0 otherwise. Find E(X).

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Question 2 · 3 marks · Sampling and estimation (random samples)

A club has 264 members, numbered from 1 to 264. Donash wants to choose a random sample of members for a survey. In order to choose the members for the sample he uses his…

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Question 3 · 5 marks · Sampling and estimation (confidence interval for a population proportion)

In a random sample of 100 students at Luciana's college, x students said that they liked exams. Luciana used this result to find an approximate 90% confidence interval for the…

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Question 4 · 6 marks · The normal distribution (linear combinations of independent normal variables)

The mass, in tonnes, of steel produced per day at a factory is normally distributed with mean 65.2 and standard deviation 3.6. It can be assumed that the mass of steel produced…

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Question 5 · 9 marks · Sampling and estimation (unbiased estimates of population mean and variance)

Last year the mean time for pizza deliveries from Pete's Pizza Pit was 32.4 minutes. This year the time, t minutes, for pizza deliveries from Pete's Pizza Pit was recorded for a…

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Question 6 · 5 marks · The Poisson distribution (approximation to the binomial)

It is known that 1 in 5000 people in Atalia have a certain condition. A random sample of 12 500 people from Atalia is chosen for a medical trial. The number having the condition…

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Question 7 · 8 marks · Continuous random variables (probability density functions)

A random variable X has probability density function f, where the graph of y = f(x) is a semicircle with centre (0, 0) and radius sqrt(2/pi), entirely above the x-axis. Elsewhere…

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Question 8 · 11 marks · Hypothesis tests (test for a Poisson mean)

A new light was installed on a certain footpath. A town councillor decided to use a hypothesis test to investigate whether the number of people using the path in the evening had…

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