A-LevelMathematicsProbability and statistics9709/62 · May/June 2024 · Q26 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. 5 6 7 1 0 9 8 4 3 1 0 9 6 6 5 0 2 1 7 6 Henri's first two student numbers are 567 and 109.

(a)Use Henri's digits to find the numbers of the next two students in the sample.[2]
(b)There were 30 students in Henri's sample. He asked each of them how much time, X hours, they spent on social media each week, on average. He summarised the results as follows. n = 30 Σx = 610 Σx² = 12405 Use this information to calculate an unbiased estimate of the mean of X and show that an unbiased estimate of the variance of X is less than 0.1.[3]
(c)Henri's friend claims that Henri has probably made a mistake in his calculation of Σx or Σx². Use your answer to part (b) to comment on this claim.[1]

Mark scheme

Marking points from the official mark scheme (May/June 2024). The mark or mark type is shown in brackets.

[B2] [567, 109], 665, 21 B1 for each. Allow 021. If more than 2 answers given, count first two and ISW.
[B1] Est(μ) = 610/30 or 61/3 or 20.3. OE or 20.3.
[M1] Est(σ²) = 30(12405-(610/30)²) / 29 or (12405-610²/30) / 29 Use of correct formula.
[A1] = 0.0575 (3sf) Accept 17/296.
[B1 FT] Variance is [unrealistically] small so Henri has [probably] made a mistake/claim is [probably] correct Need both parts. Need ‘small' OE, not just < 0.1. FT their < 0.1 variance value (not –ve), e.g. 0.0556 (if omit 30). Accept 's.d. = 0.24 is small, so Henri has probably made a mistake'. Note: 'mean is large/small' scores B0, but 'mean large compared to variance so Henri prob made a mistake' scores B1.

On Oracle Prep you can write or photograph your own answer and have it marked by AI against this mark scheme, with feedback on what you missed.