A-LevelMathematicsProbability and statistics9709/62 · Oct/Nov 2022 · Q56 Marks

X is a random variable with distribution B(10, 0.2). A random sample of 160 values of X is taken.

(a)Find the approximate distribution of the sample mean, including the values of the parameters.[3]
(b)Hence find the probability that the sample mean is less than 1.8.[3]

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2022). The mark or mark type is shown in brackets.

[B1] 5(a) For X, μ = 2 σ² = 1.6
[B1] 5(a) Mean = 2
[B1] 5(a) Variance = 1.6/160 or 1/100 or 0.01 Accept Var = 0.1² (accept sd=0.1 if clearly identified). Normal
[M1] 5(b) 1.8-'2'/√'0.01' or ±(287.5 – '320') / √'256' [= -2.03 or -2] Allow with wrong continuity correction. M1 can be implied by correct final answer or for -2.03/-2.0 or 0.9788 / 0.9772 seen.
[M1] 5(b) Φ(‘−2.03’) = 1 – Φ(‘2.03’) Correct area consistent with their values. M1 can be implied by correct final answer.
[A1] 5(b) = 0.0212 or 0.0228 (3 sf)

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Topic: Probability and statistics, A-Level Mathematics (syllabus 9709). Structured question, 6 marks.
Source: A-Level Mathematics, Oct/Nov 2022 Paper 6 Variant 2, question 5.

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