A-LevelMathematicsProbability and statistics9709/62 · Oct/Nov 2025 · Q612 Marks

The weekly profit, in dollars, made by a certain firm has a normal distribution. In the past, the weekly profit had the distribution N(736, 26²). Following a change in management, the mean weekly profit for 35 randomly chosen weeks is $725.

(a)Stating a necessary assumption, test at the 2% significance level whether the mean weekly profit has decreased.[6]
(b)State the probability of a Type I error.[1]
(c)Given that the mean weekly profit is now in fact $718, find the probability of a Type II error.[5]

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2025). The mark or mark type is shown in brackets.

[1] (a) Assume sd is still 26 or is unchanged
[1] (a) Ho: Population mean = 736 H1: Population mean < 736
[1] (a) z = (725-736) / (26/√35)
[1] (a) = -2.503
[1] (a) -2.503 < -2.054(or 5 or 6) or 0.0062 < 0.02
[1] (a) [Reject Ho] There is sufficient evidence that (mean) weekly profit has decreased
[1] (b) 0.02
[1] (c) a-736 / (26/√35) = -2.054 or -2.055
[1] (c) a = 726.973
[1] (c) z = ('726.973'-718) / (26/√35) (= 2.042)
[1] (c) = 1 – Φ('2.042')
[1] (c) = 0.0203 to 0.0207

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