Cambridge A Level Mathematics 9709 — 2020 May/June Paper 6 · Variant 1

1 The lengths, X centimetres, of a random sample of 7 leaves from a certain variety of tree are as follows. 5.2 4.8 5.5 6.1 4.8 3.9 4.4 (a) Calculate unbiased estimates of the population mean and variance of X. [3] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ It is now given that the true value of the population variance of X is 0.55, and that X has a normal distribution. (b) Find a 95% confidence interval for the population mean of X. [3] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................

Mark scheme: 1(a) = = 4.9571 or 4.96 (3 sf) (Σx2 = 175.15) B1 2 7 "175.15" "4.9571" 6 7   −     M1 0.523 (3 sf) A1 3 1(b) 0.523 '4.96' 7 z ± × (FT their mean and standard deviation) M1 z = 1.96 B1 4.42 to 5.49 (3 sf) A1 3 7 x Σ 7 7. 34

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2 In the past the yield of a certain crop, in tonnes per hectare, had mean 0.56 and standard deviation 0.08. Following the introduction of a new fertilizer, the farmer intends to test at the 2.5% significance level whether the mean yield has increased. He finds that the mean yield over 10 years is 0.61 tonnes per hectare. (a) State two assumptions that are necessary for the test. [2] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ (b) Carry out the test. [5] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................

Mark scheme: 2(a) B1 Assume yields normally distributed B1 2 2(b) H0: Population mean yield (or μ) = 0.56 H1: Population mean yield (or μ) > 0.56 B1 0.61 0.56 0.08 10 − M1 1.976 A1 Comp 1.96 M1 There is evidence that mean yield has increased A1 5

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3 The masses, in kilograms, of large sacks of flour and small sacks of flour have the independent distributions N 40, 1.52 and N 12, 0.72 respectively. (a) Find the probability that the total mass of 6 randomly chosen large sacks of flour is more than 245 kg. [4] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ (b) Find the probability that the mass of a randomly chosen large sack of flour is less than 4 times the mass of a randomly chosen small sack of flour. [6] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................

Mark scheme: 3(a) M1 245 "240" "13.5" − (= 1.361) M1 1 – Φ(“1.361”) M1 0.0867 (3 sf) A1 4 3(b) Use of L – 4S or similar M1 E(L – 4S) = –8 B1 Var(L – 4S) = 1.52 + 16 × 0.72 or 10.09 B1 ( )( ) 0 " 8" 2.519 "10.09" − − = M1 Φ(“2.519”) M1 0.994 (3 sf) A1 6

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5 Each week a sports team plays one home match and one away match. In their home matches they score goals at a constant average rate of 2.1 goals per match. In their away matches they score goals at a constant average rate of 0.8 goals per match. You may assume that goals are scored at random times and independently of one another. (a) A week is chosen at random. (i) Find the probability that the team scores a total of 4 goals in their two matches. [2] ................................................................................................................................................ ................................................................................................................................................ ................................................................................................................................................ ................................................................................................................................................ ................................................................................................................................................ ................................................................................................................................................ ................................................................................................................................................ ................................................................................................................................................ ................................................................................................................................................ (ii) Find the probability that the team scores a total of 4 goals, with more goals scored in the home match than in the away match. [3] ................................................................................................................................................ ................................................................................................................................................ ................................................................................................................................................ ................................................................................................................................................ ................................................................................................................................................ ................................................................................................................................................ ................................................................................................................................................ ................................................................................................................................................ ................................................................................................................................................ ................................................................................................................................................ ................................................................................................................................................ (b) Use a suitable approximating distribution to find the probability that the team scores fewer than 25 goals in 10 randomly chosen weeks. [4] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ (c) Justify the use of the approximating distribution used in part (b). [1] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................

Mark scheme: 5(a)(i) 4 2.9 2.9 e 4! − × M1 0.162 (3 sf) A1 2 5(a)(ii) 4 2.1 0.8 2.1 e e 4! − − × × 3 2.1 0.8 2.1 e e 0.8 3! − − + × × × B1 (B1 for either expression correct, M1 for P(4, 0) + P(3, 1)) M1 0.113 (3 sf) A1 3 5(b) N(29, 29) M1 ( ) 24.5 29 0.83563 29 − = − M1 1 – Φ(“0.836”) M1 0.202 (3sf) A1 4 5(c) 29 is large or 29 > 15 B1 1

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