A-LevelMathematicsProbability and statistics9709/62 · Oct/Nov 2023 · Q45 Marks

The height H, in metres, of mature trees of a certain variety is normally distributed with standard deviation 0.67. In order to test whether the population mean of H is greater than 4.23, the heights of a random sample of 200 trees are measured.

(a)Write down suitable null and alternative hypotheses for the test.[1]
(b)The sample mean height, h metres, of the 200 trees is found and the test is carried out. The result of the test is to reject the null hypothesis at the 5% significance level. Find the set of possible values of h.[3]
(c)Ajit said, ‘In (b) we had to assume that H is normally distributed, so it was necessary to use the Central Limit Theorem.' Explain whether you agree with Ajit.[1]

Mark scheme

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

[B1] Allow μ = 4.23 or population mean of h = 4.23 but NOT h = 4.23 or H = 4.23 or h = 4.23 or H = 4.23. H0: population mean [of H ] = 4.23, H1: population mean [of H ] > 4.23
[M1] For standardising and forming an equation. Must have √200. Allow ± 1.645 or ± 1.96. Accept '>' and '<'. Allow H or any letter instead of h. (h-4.23) / (0.67/√200) = 1.645
[A1] May be implied by h > 4.31. Allow h < 4.31 for this A1 only, condone 4.15 also seen. h = 4.31 (3 sf)
[A1] Condone any letter instead of h. h > 4.31 or h ≥ 4.31 (3 sf)
[B1] Allow h instead of H or just 'The population is normal.' Must use 'population' or 'underlying distribution'. Incorrect, because the population of H is given as normally distributed [with known variance].

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