A-LevelMathematicsProbability and statistics9709/61 · Oct/Nov 2022 · Q28 Marks

A spinner has five sectors, each printed with a different colour. Susma and Sanjay both wish to test whether the spinner is biased so that it lands on red on fewer spins than it would if it were fair. Susma spins the spinner 40 times. She finds that it lands on red exactly 4 times.

(a)Use a binomial distribution to carry out the test at the 5% significance level.[5]
(b)Sanjay also spins the spinner 40 times. He finds that it lands on red r times. Use a binomial distribution to find the largest value of r that lies in the rejection region for the test at the 5% significance level.[3]

Mark scheme

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

[B1] Ho: P(red) = 0.2 H1: P(red) < 0.2
[M1] P(X ≤ 4) = 0.8^40 + 40×0.8^39×0.2 + 40C2×0.8^38×0.2^2 +40C3×0.8^37×0.2^3 + 40C4×0.8^36×0.2^4
[A1] 0.0759
[M1] their '0.0759' > 0.05
[A1 FT] [Do not reject Ho ]. Not enough evidence that it lands on red fewer times than if it were fair or not enough evidence to suggest that the spinner is biased
[M1] P(X < 3) = ' 0.0759 ' – 40C4×0.8^36×0.2^4
[*A1] = 0.0285 or 0.0284
[DA1] Largest value of r is 3

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