Cambridge A Level Mathematics 9709 — 2022 Feb/March Paper 6 · Variant 2

1 The lengths, in millimetres, of a random sample of 12 rods made by a certain machine are as follows. 200 201 198 202 200 199 199 201 197 202 200 199 (a) Find unbiased estimates of the population mean and variance. [3] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ (b) Give a statistical reason why these estimates may not be reliable. [1] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................

Mark scheme: 1(a) Est (μ) = 1199 6 or 199.833 or 200 or 2398 12 [mm] B1 Accept in any form Est (σ2) = 2 12 479226 '1199' 11 12 6   −     or 2 1 '2398' '479226' 11 6   −     M1 Use of their values in correct formula (may be implied) = 2.33 (3 sf) [mm2] A1 Accept 7 3 3 1(b) Small sample B1 Accept not ‘not representative’ unless qualified. 1

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3 A random sample of 500 households in a certain town was chosen. Using this sample, a confidence interval for the proportion, p, of all households in that town that owned two or more cars was found to be 0.355 < p < 0.445. Find the confidence level of this confidence interval. Give your answer correct to the nearest integer. [5] ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................

Mark scheme: 3 est(p) = 0.4 B1 ( ) [ ] '0.4' 1 '0.4' '0.4' 0.445 500 z × − + = M1 OE Use of their 0.4 in a correct expression z ( ) '0.4' 1 '0.4' 0.045 500   × − = ÷       = 2.054 A1 Condone 2.053 and 2.05 0.98 – (1 – 0.98) M1 96% confidence A1 CWO, must be integer 5

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4 In the past the time, in minutes, taken by students to complete a certain challenge had mean 25.5 and standard deviation 5.2. A new challenge is devised and it is expected that students will take, on average, less than 25.5 minutes to complete this challenge. A random sample of 40 students is chosen and their mean time for the new challenge is found to be 23.7 minutes. (a) Assuming that the standard deviation of the time for the new challenge is 5.2 minutes, test at the 1% significance level whether the population mean time for the new challenge is less than 25.5 minutes. [5] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ (b) State, with a reason, whether it is possible that a Type I error was made in the test in part (a). [1] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................

Mark scheme: 4(a) H0: μ = 25.5 H1: μ < 25.5 B1 23.7 25.5 5.2 40 − ÷ M1 Must have √40 = –2.189 A1 '2.189' < 2.326 M1 For valid comparison For two-tailed test: allow compare 2.576 if H1: μ ≠ 25.5 [Accept H0] No evidence that mean time has decreased A1 FT In context, not definite, no contradictions FT their 2.189 but no FT for two-tailed test N.B. Use of two-tailed test can score max B0 M1 A1 M1 A0 Condone use of critical value method (23.59 M1 A1 and 23.7 > 23.59 M1 A1 correct conclusion or 25.612 M1 A1 and 25.5 < 25.612 M1 A1 with correct conclusion) 5 4(b) No, because H0 was not rejected B1 FT FT their conclusion in (a) 1

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5 The heights of buildings in a large city are normally distributed with mean 18.3m and standard deviation 2.5m. (a) Find the probability that the total height of 5 randomly chosen buildings in the city is more than 95m. [4] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ (b) Find the probability that the difference between the heights of two randomly chosen buildings in the city is less than 1m. [5] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................

Mark scheme: 5(a) B1 SOI 95 '91.5 '31.25' − [= 0.626] M1 FT their mean and variance 1 – Φ(‘0.626’) M1 For finding area consistent with their values 0.266 (3 sf) A1 4 5(b) E(D) = 0 B1 Or E(D ‒ 1) = ‒1 Var(D) = 2.52 × 2 [= 12.5] B1 1 0 '12.5' − [= 0.283] or 1 0 '12.5' −− [= –0.283] M1 FT their E and Var Φ(‘0.283’) – (1 – ɸ(0.283)) [= 0.6115 – 0.3885] M1 For finding area consistent with their values 0.223 (3 sf) A1 5

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