A-LevelMathematicsProbability and statistics9709/62 · Feb/Mar 2021 · Q610 Marks

It is known that 8% of adults in a certain town own a Chantor car. After an advertising campaign, a car dealer wishes to investigate whether this proportion has increased. He chooses a random sample of 25 adults from the town and notes how many of them own a Chantor car.

(a)He finds that 4 of the 25 adults own a Chantor car. Carry out a hypothesis test at the 5% significance level.[5]
(b)Explain which of the errors, Type I or Type II, might have been made in carrying out the test in part (a).[2]
(c)Find the probability of a Type I error.[3]

Mark scheme

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

[B1] Ho: population proportion = 0.08 OE
[B1] H1: population proportion > 0.08 OE
[M1] P(X > 4) = 1 - P(X ≤ 3) = 1 – (0.92^25 + 25×0.92^24×0.08 +25C2×0.92^23×0.08^2 + 25C3×0.92^22×0.08^3)
[A1] 0.135 (3 sf)
[M1] 0.135 > 0.05
[A1] There is no evidence that proportion owning Chantor has increased
[*B1] Ho was not rejected.
[DB1] Hence Type II might have been made.
[*M1] P(X≥ 5) = 1 - P(X≤4) = 1-((1-0.1351) + 25C4×0.92^21 ×0.08^4) [= 0.0451]
[A1] 0.0451 < 0.05
[A1] P(Type I error) = 0.0451 or 0.0452

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