Sampling and estimation: past paper questions

Real questions on sampling and estimation

The newest 20 of 72 written questions. Open one to see it on the real paper.

1. 9709/62 · October/November 2025 · Q3a[3]
   The times, in minutes, taken by students to complete a test have mean $\mu$ and standard deviation $\sigma$. The times taken by a random sample of 100 students are noted and are used to calculate a 95% confidence interval for $\mu$. (a) Given that the end points of the 95% confidence interval are 31.02 and 33.98, corre...See it on the paper →
2. 9709/62 · October/November 2025 · Q3b[1]
   The times, in minutes, taken by students to complete a test have mean $\mu$ and standard deviation $\sigma$. The times taken by a random sample of 100 students are noted and are used to calculate a 95% confidence interval for $\mu$. (b) The calculation of the confidence interval required the use of the Central Limit th...See it on the paper →
3. 9709/62 · October/November 2025 · Q3c[4]
   The times, in minutes, taken by students to complete a test have mean $\mu$ and standard deviation $\sigma$. The times taken by a random sample of 100 students are noted and are used to calculate a 95% confidence interval for $\mu$. (c) A researcher calculates a number, $r$, of 95% confidence interval for $\mu$. Find t...See it on the paper →
4. 9709/61 · May/June 2025 · Q3a[3]
   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. She notes the times taken for this journey on a random sample of 60 days. The results are summarised below. $n = 60 \...See it on the paper →
5. 9709/61 · May/June 2025 · Q6c[3]
   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. He chooses a random sample of 30 students at his college. If the number of students who...See it on the paper →
6. 9709/62 · May/June 2025 · Q2[3]
   The height of a certain species of plant is denoted by $H$cm. The heights of a random sample of 100 plants were measured, and the following results were found. • The mean, $\bar{h}$, for the sample was 80.2. • An unbiased estimate of the population variance of $H$ was 15.6. Calculate the value of $\Sigma h^2$.See it on the paper →
7. 9709/62 · May/June 2025 · Q4[4]
   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. The spinner was spun 200 times, and the proportion, $a$, of times that it landed on red was noted. This proportion was used to calculate an approximate 90...See it on the paper →
8. 9709/63 · May/June 2025 · Q3a(i)[3]
   A machine dispenses coffee into cups. The volume, $V{\text{cm}}^3$, of coffee in a cup was measured for a random sample of 150 cups. The results were summarised as follows. $\sum v=46350$ $\sum v^2=14410800$ (a) (i) Calculate unbiased estimates of the population mean, $\mu$, and population variance, ${\sigma }^2$.See it on the paper →
9. 9709/63 · May/June 2025 · Q3a(ii)[3]
   A machine dispenses coffee into cups. The volume, $V{\text{ cm}}^3$, of coffee in a cup was measured for a random sample of 150 cups. The results were summarised as follows. $\sum v=46350\sum v^2=14410800$ (ii) Calculate a 95% confidence interval for $\mu$.See it on the paper →
10. 9709/63 · May/June 2025 · Q3b[2]
    A machine dispenses coffee into cups. The volume, $V{\text{ cm}}^3$, of coffee in a cup was measured for a random sample of 150 cups. The results were summarised as follows. $\sum v=46350$ $\sum v^2=14410800$ (b) Without calculation, state whether this confidence interval would be wider or narrower than the confiden...See it on the paper →
11. 9709/62 · February/March 2025 · Q2a[3]
    A researcher records the time, $T$ seconds, taken by adults to complete a questionnaire. The results for a random sample of 60 adults who completed the questionnaire this year are summarised as follows. $n=60\sum t=3678\sum t^2=226313.36$ (a) Find an unbiased estimate of $\text{E}(T)$, and show...See it on the paper →
12. 9709/62 · February/March 2025 · Q2c[1]
    A researcher records the time, $T$ seconds, taken by adults to complete a questionnaire. The results for a random sample of 60 adults who completed the questionnaire this year are summarised as follows. $n=60\sum t=3678\sum t^2=226313.36$ (c) State, with a reason, whether it was necessary to use...See it on the paper →
13. 9709/62 · February/March 2025 · Q3c[5]
    The random variable $X$ has the distribution Po(1.5). (c) The sum of a large number, $n$, of values of $X$ is denoted by $T$. Using a suitable approximation, it was found that $P(T>330)=0.0391$, correct to 3 significant figures. Find the value of $n$.See it on the paper →
14. 9709/62 · February/March 2025 · Q6a[1]
    Nikki is investigating the views of students at her school about the school sports facilities. She plans to give a survey to a sample of students. Nikki’s friend says, “This survey is about sports facilities, so you should choose a sample of students from the school sports teams.” (a) State, with a reason, whether yo...See it on the paper →
15. 9709/62 · February/March 2025 · Q6b[3]
    Nikki is investigating the views of students at her school about the school sports facilities. She plans to give a survey to a sample of students. Nikki's friend says, “This survey is about sports facilities, so you should choose a sample of students from the school sports teams.” (b) Calculate an approximate 95% con...See it on the paper →
16. 9709/62 · February/March 2025 · Q6c[4]
    Nikki is investigating the views of students at her school about the school sports facilities. She plans to give a survey to a sample of students. Nikki’s friend says, “This survey is about sports facilities, so you should choose a sample of students from the school sports teams.” (c) Calculate the value of $x$.See it on the paper →
17. 9709/61 · October/November 2024 · Q3a[3]
    The times, $T$ minutes, taken by a random sample of 75 students to complete a test were noted. The results were summarised by $\sum t=230$ and $\sum t^2=930$. (a) Calculate unbiased estimates of the population mean and variance of $T$.See it on the paper →
18. 9709/61 · May/June 2024 · Q3b[1]
    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. (b) Explain briefly whether this statement is true or not.See it on the paper →
19. 9709/61 · May/June 2024 · Q3c[2]
    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. (c) Given that at least one of these intervals contains the population mean, find t...See it on the paper →
20. 9709/61 · May/June 2024 · Q4a[3]
    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.See it on the paper →

What examiners said

This part was well attempted; candidates knew the formulae to use to find the unbiased estimates of E(T) and Var(T) and there was very little confusion shown between the alternative formulae for the variance.

— Examiner report, February/March 2025, 9709/62 Q2(a) (Cambridge)

This part was not well answered and showed a basic lack of understanding of the Central Limit Theorem by a large number of candidates. There were some candidates who realised that the underlying distribution was not given, but failed to qualify that this was the ‘population’ distribution; the distribution being referred to must be clear, so merely saying ‘the distribution is unknown’ is not precis...

— Examiner report, February/March 2025, 9709/62 Q2(c) (Cambridge)

Many candidates found this question demanding. N(1.5 n, 1.5 n) was required, followed by standardising (with a continuity correction) and equating to 1.761 to form an equation in n. Errors included using an incorrect Normal distribution, equating the standardised expression to 0.0391 rather than 1.761, omitting or using an incorrect continuity correction, and dividing the denominator by √n. The eq...

— Examiner report, February/March 2025, 9709/62 Q3(c) (Cambridge)