A-LevelMathematicsProbability and statistics9709/63 · May/June 2021 · Q28 Marks

In the past, the time, in hours, for a particular train journey has had mean 1.40 and standard deviation 0.12. Following the introduction of some new signals, it is required to test whether the mean journey time has decreased.

(a)State what is meant by a Type II error in this context.[1]
(b)The mean time for a random sample of 50 journeys is found to be 1.36 hours. Assuming that the standard deviation of journey times is still 0.12 hours, test at the 2.5% significance level whether the population mean journey time has decreased.[5]
(c)State, with a reason, which of the errors, Type I or Type II, might have been made in the test in part (b).[2]

Mark scheme

Marking points from the official mark scheme (May/June 2021). The mark or mark type is shown in brackets.

[B1] Conclude (mean) (journey) time has not decreased when in fact it has. OE in context
[B1] Ho: Pop mean (or μ) = 1.4 H1: Pop mean (or μ) < 1.4 May be seen in (a)
[M1] (1.36-1.4) / (0.12/√50) Accept totals method (68-70)/(0.12/√50 × 50) No mixed methods or no standard deviation/variance mixes
[A1] -2.357 or -2.36 Correct z or correct area if used
[M1] -2.357 < -1.96 or 0.0092 < 0.025 or 0.9908 > 0.975 Or CV method 1.36 < 1.367 valid comparison
[A1 FT] There is evidence that (mean) (journey) times have decreased in context not definite no contradictions NB use of two tail test scores max B0M1A1M1A0 no ft for two tail test
[*B1 FT] Ho was rejected OE FT Ho was accepted OE
[DB1 FT] Type I FT Type II

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