A-LevelMathematicsProbability and statistics9709/63 · May/June 2022 · Q26 Marks

Anton believes that 10% of students at his college are left-handed. Aliya believes that this is an under-estimate. She plans to carry out a hypothesis test of the null hypothesis p = 0.1 against the alternative hypothesis p > 0.1, where p is the actual proportion of students at the college that are left-handed. She chooses a random sample of 20 students from the college. She will reject the null hypothesis if at least 5 of these students are left-handed.

(a)Explain what is meant by a Type I error in this context.[1]
(b)Find the probability of a Type I error in the test.[3]
(c)Given that the true value of p is 0.3, find the probability of a Type II error in the test.[2]

Mark scheme

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

[B1] Conclude more than 10% of the students are left handed when this is not true
[M2] 1 – (0.9^20 + 20 × 0.9^19 × 0.1 + 20C2 × 0.9^18 × 0.1^2 + 20C3 × 0.9^17 × 0.1^3 + 20C4 × 0.9^16 × 0.1^4)
[A1] 0.0432 (3 s.f.)
[M1] 0.7^20 + 20 × 0.7^19 × 0.3 + 20C2 × 0.7^18 × 0.3^2 + 20C3 × 0.7^17 × 0.3^3 + 20C4 × 0.7^16 × 0.3^4
[A1] 0.238 or 0.237 (3 s.f.)

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