A-LevelMathematicsProbability and statistics9709/63 · May/June 2025 · Q710 Marks

In the past, one quarter of job applicants at a certain firm had first-class degrees. A change is made in the job description and a director of the firm believes that, on average, the proportion of job applicants with first class degrees has decreased. In the month following the change, there were 35 job applicants, and r of these had first-class degrees. The firm carried out a hypothesis test at the 4% significance level to test the director's belief.

(a)Use a binomial distribution to find the largest value of r that would provide sufficient evidence that the director's belief is correct.[6]
(b)Explain the meaning of a Type I error in this context, and state the probability of a Type I error.[2]
(c)Given that the proportion of job applicants with first class degrees this year is actually 0.05, find the probability of a Type II error.[2]

Mark scheme

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

[M1] SOI.
[M1] Attempt P(r ≤ 3) Expressions or terms.
[A1] SC unsupported answer 0.0136 B1.
[M1] Attempt P(r ≤ 4) Expressions or terms
[A1] SC unsupported answer 0.041 B1.
[A1] Dep on above two correct probabilities seen and at least one comparison with 0.04.
[B1] Or concluding that the director's belief is correct when in fact it isn't.
[B1] FT their P(r ≤ 3), dep < 0.04 OE.
[M1] Attempt 1 – P(r ≤ 3) with B(35, 0.05). Expressions or terms.
[A1] SC unsupported answer 0.0958 B1.

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