A-LevelMathematicsProbability and statistics9709/62 · Feb/Mar 2024 · Q76 Marks

The heights, in centimetres, of adult females in Litania have mean μ and standard deviation σ. It is known that in 2004 the values of μ and σ were 163.21 and 6.95 respectively. The government claims that the value of μ this year is greater than it was in 2004. In order to test this claim a researcher plans to carry out a hypothesis test at the 1% significance level. He records the heights of a random sample of 300 adult females in Litania this year and finds the value of the sample mean. You should assume that the value of σ after 2004 remains at 6.95.

(a)State the probability of a Type I error.[1]
(b)Given that the value of μ this year is actually 164.91, find the probability of a Type II error.[5]

Mark scheme

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

[B1] 0.01 or 1%. Note: x < 0.01 scores B0.
[M1] Calculation of critical value for sample mean (h_bar). (2.326 = (h_bar - 163.21) / (6.95 / sqrt(300))). Accept any z (±).
[A1] h_bar = 164.14. Accept 3 sf accuracy here.
[M1] Standardising the critical value with the actual mean μ = 164.91. ((their '164.14' - 164.91) / (6.95 / sqrt(300)) [= -1.919]).
[M1] For area attempt consistent with their values (e.g., Φ(their '-1.919') or 1 – Φ(their '1.919')).
[A1] = 0.0275 or 0.0276 or 0.028[0] (3.s.f). Accept anything in range 0.0275 to 0.028[0].

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