A-LevelMathematicsProbability and statistics9709/61 · Oct/Nov 2023 · Q712 Marks

A biologist wishes to test whether the mean concentration u, in suitable units, of a certain pollutant in a river is below the permitted level of 0.5. She measures the concentration, x, of the pollutant at 50 randomly chosen locations in the river. The results are summarised below. n = 50 Σx = 23.0 Σx² = 13.02

(a)Carry out a test at the 5% significance level of the null hypothesis µ = 0.5 against the alternative hypothesis μ < 0.5.[7]
(b)Given that, in fact, the value of u is 0.4, find the probability of a Type II error.[5]

Mark scheme

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

[B1]
[M1] For an expression of the correct form for unbiased standard deviation or variance.
[A1]
[M1] Standardising with their values.
[A1]
[M1] For a valid comparison.
[A1 FT] In context, not definite. E.g., not 'Mean concentration is not less than 0.5'. No contradictions.
[M1]
[A1]
[M1]
[M1] For area consistent with their working.
[A1]

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Topic: Probability and statistics, A-Level Mathematics (syllabus 9709). Structured question, 12 marks.
Source: A-Level Mathematics, Oct/Nov 2023 Paper 6 Variant 1, question 7.

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