This topic covers unbiased estimates, the Central Limit Theorem, confidence intervals, and the reasons a sample may be unsuitable. The reports show that students know the formulae, and the marks are lost in the details of the interval and in the precision of the explanations. This guide collects what Cambridge examiners say students get wrong, based on recent reports.
Summarised in our own words from a reading of the examiner reports.
Mistakes that cost marks
- Unbiased estimates. Confusion between the two standard formulae for the unbiased estimate of the variance, finding the biased estimate, confusing Σx² with (Σx)². Another report records an estimate for the mean of x of 0.3 instead of 2.3.
- Accuracy. Rounding early, such as 14.44 to 14.4, giving the final answer to two significant figures (including confusion between decimal places and significant figures, and which zeros are significant).
- Confidence intervals: the formula. An incorrect z value, omitting √n (√100, √60 and √15 are examples), using σ in the formula, using (p − 1) in place of (1 − p), and factor-of-10 errors when forming equations.
- Confidence intervals: the centre. Centring the interval around 45 instead of 45/60, or around 46 instead of the sample proportion. In another question, 245 + 263 was divided by 50 instead of 2 to find the midpoint.
- The width. Omitting the factor of 2 when setting up an equation for the width, and not giving the values of x as integers, or giving p when x was asked for.
- The final form. The final answer must be written as an interval.
- The confidence level. After finding 1.288 (2.576 ÷ 2), giving 90.12%, which is a one-tail area. Some thought that a change to a lower confidence level meant the interval should increase.
- Inequalities. Errors with signs of inequalities, the wrong sign for the z-value, and not deducing a whole number from a result such as n > 62.9. Also the algebraic error of expanding (10 + a)² as 100 + a².
- The Central Limit Theorem. Statements such as "it should be Normal" earned no credit. Many candidates said that "it" was normally distributed without saying whether "it" was T or the population distribution. Some thought the Central Limit Theorem was needed when it was not.
- Explaining why a sample is small or unsuitable. Many candidates did not identify that the sample size was small and wrote comments such as "the data items were too similar". When asked for two different reasons a sample was unsuitable, many gave only one, or a limited explanation such as stating "it was not representative" without saying why.
- Hypotheses in this topic. Using X or x̄ in the hypotheses, omitting the parameter (H₀ = 41.5), attempting a one-tail test, and a conclusion that implied a two-tailed test, such as "there is insufficient evidence that the mean length is not 41.5".
What examiners advise
- Write the formula for the confidence interval with the numbers in it, and take care with the z value for the stated confidence level.
- State precisely what is normally distributed (the population, or the sample mean) and why.
- Give the final interval in the form (a, b).
- For a sample size, solve the inequality and then give the smallest whole number that satisfies it.
- Give reasons in the context of the question, and two different reasons when two are asked for.
- Keep working to at least 4 significant figures.
Habits that help
- Keep a line in your notes for each of the two variance formulae, and use the one that matches the data you are given (the sums or the raw values).
- For a proportion interval, write p̂ before you start, as the centre.
- To go from a z value to a confidence level, sketch a Normal curve and shade the middle.
Questions students ask
Which formula do I use for an unbiased estimate of variance?
There are two standard formulae, one for sums (Σx and Σx²) and one for raw values. Use the one that matches the data. Students confused them, found the biased estimate, or confused Σx² with (Σx)².
How do I write a confidence interval?
Write the formula with the numbers in, take care with the z value for the confidence level, include √n, and give the answer as an interval (a, b). For a proportion, write p̂ first as the centre.
How do I find the sample size from a width?
Set up the equation for the width, including the factor of 2, solve the inequality, then give the smallest whole number that satisfies it. Students left a result such as n > 62.9 without deducing the integer.
What should I say about the Central Limit Theorem?
State precisely what is normally distributed, the sample mean or the population, and why. "It should be Normal" earned no credit.
How do I explain why a sample is unsuitable?
Give reasons in the context of the question, and two different reasons when two are asked for. Many candidates did not say the sample size was small.
Practice this topic
Ready to try some? Work through past paper questions on this topic one at a time. Attempt each question fully first, and open the mark scheme only afterwards.