Sampling and estimation: past paper questions

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• Question 4May/June 2026, paper 61Structured11 marks

  The score on one spin of a 5-sided spinner is denoted by the random variable $X$ with probability distribution as shown in the table.

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• Question 2Oct/Nov 2025, paper 61Structured8 marks

  The mean mass of packets of Trueleaf tea is supposed to be 500 grams. An inspector wishes to test whether this value is correct. He weighs 60 randomly chosen packets and notes the mass, $x$ grams, of each packet.

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• Question 1Oct/Nov 2024, paper 61Structured4 marks

  The heights of a certain species of deer are known to have standard deviation 0.35 m. A zoologist takes a random sample of 150 of these deer and finds that the mean height of the deer in the sample is 1.42 m. Calculate a 96% confidence interval for the population mean height.

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• Question 1Oct/Nov 2023, paper 61Structured3 marks

  A random variable $X$ has the distribution $\mathrm{N}(410,400)$. Find the probability that the mean of a random sample of 36 values of $X$ is less than 405.

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• Question 1May/June 2026, paper 62Structured3 marks

  Ten random values of a variable $X$ are given below. $$\begin{array}{llllllllll}8.1& 8.3& 9.5& 10.1& 7.9& 8.7& 9.1& 8.8& 9.4& 9.0\end{array}$$ Calculate unbiased estimates of the population mean and variance of $X$.

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• Question 4Oct/Nov 2025, paper 61Structured4 marks

  The masses of a certain species of animal are known to be normally distributed with standard deviation $\sigma \text{ kg}$.

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• Question 3Oct/Nov 2024, paper 61Structured6 marks

  The times, $T$ minutes, taken by a random sample of 75 students to complete a test were noted.

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• Question 2Oct/Nov 2023, paper 61Structured4 marks

  In a survey of 300 randomly chosen adults in Rickton, 134 said that they exercised regularly. This information was used to calculate an $\alpha \mathrm{\%}$ confidence interval for the proportion of adults in Rickton who exercise regularly.

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• Question 5May/June 2026, paper 62Structured7 marks

  Eric throws darts at a target. The probability that he will hit the centre of the target on one throw is $p$. Eric takes 100 throws and hits the centre $x$ times.

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• Question 3Oct/Nov 2025, paper 62Structured8 marks

  The times, in minutes, taken by students to complete a test have mean $\mu$ and standard deviation $\sigma$.

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• Question 2Oct/Nov 2024, paper 62Structured4 marks

  The lengths of a random sample of 50 roads in a certain region were measured. Using the results, a 95% confidence interval for the mean length, in metres, of all roads in this region was found to be $[245,263]$. Find the mean length of the 50 roads in the sample.

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• Question 7Oct/Nov 2023, paper 61Structured12 marksHas a diagram

  A biologist wishes to test whether the mean concentration $\mu$, in suitable units, of a certain pollutant in a river is below the permitted level of 0.5. She measures the concentration, $x$, of the pollutant at 50 randomly chosen locations in the river.

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• Question 6May/June 2026, paper 62Structured9 marks

  The length of time, $T$ minutes, that customers spend in a certain shop has the distribution $\mathrm{N}(\mu ,9.2)$. Last year the value of $\mu$ was 12.4. The manager of the shop wants to test whether the value of $\mu$ has decreased this year.

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• Question 4Oct/Nov 2024, paper 62Structured6 marks

  A population is normally distributed with mean 35 and standard deviation 8.1. A random sample of size 140 is chosen from this population and the sample mean is denoted by $\bar{X}$. Find $\mathrm{P}(\bar{X}>36)$.

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• Question 2Oct/Nov 2023, paper 62Structured5 marks

  The length, in minutes, of mathematics lectures at a certain college has mean $\mu$ and standard deviation 8.3. The total length of a random sample of 85 lectures was 4590 minutes. Calculate a 95% confidence interval for $\mu$.

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• Question 1May/June 2026, paper 63Structured3 marks

  A five-sided spinner has edges marked 1, 2, 3, 4, 5. Ayesha believes that the spinner is biased so that it lands on five more often than it should. She spins the spinner 200 times, and it lands on five on 60 spins.

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• Question 7Oct/Nov 2024, paper 62Structured9 marks

  The heights of one-year-old trees of a certain variety are known to have mean $2.3\text{ m}$. A scientist believes that, on average, trees of this age and variety in her region are slightly taller than in other places.

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• Question 4Oct/Nov 2023, paper 62Structured5 marks

  The height $H$, in metres, of mature trees of a certain variety is normally distributed with standard deviation 0.67. In order to test whether the population mean of $H$ is greater than 4.23, the heights of a random sample of 200 trees are measured.

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• Question 5May/June 2026, paper 63Structured14 marks

  The lengths, $X\text{ cm}$, of Wamba snakes have mean $\mu \text{ cm}$. Research has shown that, for the whole population of Wamba snakes, $\mu =41.5$. A researcher claims that, for the Wamba snakes in a particular region, the mean length is less than 41.5.

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• Question 2May/June 2025, paper 61Structured7 marks

  The random variable $X$ has the distribution $\mathrm{B}(8,\frac{3}{4})$. A random sample of 100 values of $X$ is chosen, and the sample mean, $\bar{X}$, is found. Find $\mathrm{P}(\bar{X}>6.2)$.

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• Question 3May/June 2025, paper 61Structured8 marks

  The time, $T$ minutes, for a certain daily bus journey is normally distributed. The bus company claims that the mean of $T$ is 45. A passenger believes that the mean of $T$ is actually greater than 45.

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• Question 6May/June 2025, paper 61Structured7 marks

  A manufacturer of cell phones claims that 25% of students own a Pumpkin phone. Jeyeraj thinks that the proportion of students at his large college who own a Pumpkin phone is less than 25%. He plans to test the manufacturer's claim.

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• Question 3May/June 2024, paper 61Structured6 marks

  The time taken in minutes for a certain daily train journey has a normal distribution with standard deviation 5.8 . For a random sample of 20 days the journey times were noted and the mean journey time was found to be 81.5 minutes.

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• Question 1May/June 2025, paper 62Structured6 marks

  One of a group of three students is to be chosen at random. Explain how a single throw of a fair six-sided dice could be used to make the choice.

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• Question 4May/June 2024, paper 61Structured9 marks

  A random sample of 8 boxes of cereal from a certain supplier was taken. Each box was weighed and the masses in grams were as follows. 261 249 259 252 255 256 258 254 Find unbiased estimates of the population mean and variance.

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• Question 3May/June 2023, paper 61Structured6 marks

  In the past, the annual amount of wheat produced per farm by a large number of similar sized farms in a certain region had mean 24.0 tonnes and standard deviation 5.2 tonnes.

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• Question 2May/June 2025, paper 62Structured3 marks

  The height of a certain species of plant is denoted by $H\text{ cm}$. The heights of a random sample of 100 plants were measured, and the following results were found.

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• Question 2May/June 2024, paper 62Structured6 marks

  Henri wants to choose a random sample from the 804 students at his college. He numbers the students from 1 to 804 and then uses random numbers generated by his calculator. The first 20 random digits produced by his calculator are as follows.

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• Question 4May/June 2023, paper 61Structured6 marks

  A certain train journey takes place every day throughout the year. The time taken, in minutes, for the journey is normally distributed with variance 11.2. The mean time for a random sample of $n$ of these journeys was found.

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• Question 4May/June 2025, paper 62Structured4 marks

  A biased spinner has four sides. Each side is of a different colour: yellow, red, green or black. The probability, $p$, that the spinner will land on red is unknown.

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