A-LevelMathematicsProbability and statistics9709/61 · Oct/Nov 2022 · Q710 Marks

In the past Laxmi's time, in minutes, for her journey to college had mean 32.5 and standard deviation 3.1. After a change in her route, Laxmi wishes to test whether the mean time has decreased. She notes her journey times for a random sample of 50 journeys and she finds that the sample mean is 31.8 minutes. You should assume that the standard deviation is unchanged.

(a)Carry out a hypothesis test, at the 8% significance level, of whether Laxmi's mean journey time has decreased.[5]
(b)Later Laxmi carries out a similar test with the same hypotheses, at the 8% significance level, using another random sample of size 50. Given that the population mean is now 31.5, find the probability of a Type II error.[5]

Mark scheme

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

[B1] Ho: Population mean time (or μ) = 32.5; H1: Population mean time (or μ) < 32.5
[M1] (31.8-32.5) / (3.1/√50)
[A1] = ±-1.597
[M1] '-1.597' < -1.406 [or ‘1.597' > 1.406]
[A1 FT] [ reject Ho] There is evidence that [population] [mean ] time has decreased
[M1] a-32.5 / (3.1/√50) = -1.406
[A1] a = 31.88 or 31.9
[M1] (their '31.88'-31.5) / (3.1/√50) [= 0.8668 to 0.8760]
[M1] 1 – Φ('0.8668')
[A1] = 0.190 to 0.193 (3 sf)

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