A-LevelMathematicsProbability and statistics9709/61 · Oct/Nov 2025 · Q56 Marks

It is known that 20% of households in a certain country contain more than 4 people. Laxmi believes that, in her town, the percentage is lower than 20%. She chooses a random sample of 40 households in her town and notes the number which contain more than 4 people. She then carries out a test at the 2.5% significance level using a binomial distribution.

(a)Find the probability of a Type I error.[4]
(b)State the rejection region for the test.[1]
(c)Laxmi finds that exactly 2 households in her sample contain more than 4 people. Explain why it is impossible for Laxmi to make a Type II error.[1]

Mark scheme

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

M1 Attempt P(X < 2) or P(X ≤ 3) using B(40, 0.2). Need to see expressions.
A1 SC unjustified X ≤ 2 = 0.00794 scores M0B1.
B1FT Correct term P(X = 3) added to P(X < 2) and one relevant comparison seen. Need to see expressions.
B1 Unjustified final answer 0.00794 scores SCB1B1.
B1 OE.
B1

On Oracle Prep you can write or photograph your own answer and have it marked by AI against this mark scheme, with feedback on what you missed.

Topic: Probability and statistics, A-Level Mathematics (syllabus 9709). Structured question, 6 marks.
Source: A-Level Mathematics, Oct/Nov 2025 Paper 6 Variant 1, question 5.

Questions and mark schemes are from past examination papers and remain the copyright of the exam board. How Oracle Prep builds its content