A-LevelMathematicsProbability and statistics9709/63 · May/June 2023 · Q35 Marks

In a random sample of 100 students at Luciana's college, x students said that they liked exams. Luciana used this result to find an approximate 90% confidence interval for the proportion, p, of all students at her college who liked exams. Her confidence interval had width 0.15792.

(a)Find the two possible values of x.[4]
(b)Suzma independently took another random sample and found another approximate 90% confidence interval for p. Find the probability that neither of the two confidence intervals contains the true value of p.[1]

Mark scheme

Marking points from the official mark scheme (May/June 2023). The mark or mark type is shown in brackets.

[1] (a) z = 1.645 (for 90% confidence interval).
[1] (a) Equation for confidence interval width in correct form, e.g., 2 * z * sqrt(p(1-p)/n) = width. Accept p=x/100. Allow missing factor of 2 initially.
[1] (a) Any correct (likely scalar multiple) three-term quadratic equation in x or p with simplified coefficients, e.g., x² - 100x + 2304 = 0 or p(1-p) = 0.2304.
[1] (a) x = 36 or 64.
[1] (b) 0.1² = 0.01.

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About this question

Topic

Probability and statistics, A-Level Mathematics (syllabus 9709). Structured question, 5 marks.

Source

A-Level Mathematics, May/June 2023 Paper 6 Variant 3, question 3.

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