A-LevelMathematicsProbability and statistics9709/61 · May/June 2023 · Q63 Marks

A sample of 5 randomly selected values of a variable X is as follows: 1 2 6 1 a where a > 0. Given that an unbiased estimate of the variance of X calculated from this sample is 11/2, find the value of a.

Given that an unbiased estimate of the variance of X calculated from this sample is 11/2, find the value of a.[3]

Mark scheme

Marking points from the official mark scheme (May/June 2023). The mark or mark type is shown in brackets.

[M1*] ( (1^2 + 2^2 + 6^2 + 1^2 + a^2) / 5 - ((1+2+6+1+a)/5)^2 ) = 11/2 (OE attempted or e.g., (42+a^2)/5 - ((10+a)/5)^2 = 11/2. Allow use of biased i.e., without n/(n-1).)
[DM1] 4a^2 - 20a + 0 = 0 or a^2 – 5a + 0 = 0 (Two- or three-term quadratic equation in a, with at least two terms correct.)
[A1] a = 5 (Ignore a = 0, if seen.)

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Topic: Probability and statistics, A-Level Mathematics (syllabus 9709). Structured question, 3 marks.
Source: A-Level Mathematics, May/June 2023 Paper 6 Variant 1, question 6.

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