Cambridge A Level Mathematics 9709 — 2021 Feb/March Paper 6 · Variant 2

1 A construction company notes the time, t days, that it takes to build each house of a certain design. The results for a random sample of 60 such houses are summarised as follows. Σ t = 4820 Σ t2 = 392 050 (a) Calculate a 98% confidence interval for the population mean time. [6] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ (b) Explain why it was necessary to use the Central Limit theorem in part (a). [1] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................

Mark scheme: 1(a) Est(μ) = 60 4820 or 241 3 or 80.3 (3 sf) B1 Est(σ2) = ) 2 ) 60 4820 ( 60 050 392 ( 59 60 − M1 Use of biased (80.72) score M0 A0. 82.0904 14530 177       to 82.635 or SD = 9.0604 to 9.0904 (3sf) A1 z = 2.326 B1 60 4820 ± z× 60 ' 0904 . 82 ' M1 Expression of the correct form – must be z value. 77.6 to 83.1 (3 sf) A1 CWO Use of biased 77.6 to 83.0(3) can score B1M1A1 (max 4/6). 6 1(b) Population distribution of times unknown B1 Accept ‘not normal’. 1

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3 An architect wishes to investigate whether the buildings in a certain city are higher, on average, than buildings in other cities. He takes a large random sample of buildings from the city and finds the mean height of the buildings in the sample. He calculates the value of the test statistic, z, and finds that z = 2.41. (a) Explain briefly whether he should use a one-tail test or a two-tail test. [1] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ (b) Carry out the test at the 1% significance level. [3] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................

Mark scheme: 3(a) One-tail because investigating whether "higher" B1 OE. Must have both parts. 1 3(b) H0: Population mean (or μ) in city same as for others H1: Population mean (or μ) in city greater than for others B1 FT If (a) two-tail: H0: Pop mean (or μ) in city same as for others. H1: Pop mean (or μ) in region different from others. 2.41 > 2.326 or 0.008 < 0.01 or 0.992 > 0.99 M1 If (a) two-tail: 2.41 < 2.576 or 0.992 < 0.995. There is evidence that buildings are higher [on average]. A1 FT In context, not definite. No contradictions. If (a) two-tail: There is no evidence that the [average] height of buildings is different. 3

More questions on Hypothesis tests

5 The volumes, in litres, of juice in large and small bottles have the distributions N 5.10, 0.0102 and N 2.51, 0.0036 respectively. (a) Find the probability that the total volume of juice in 3 randomly chosen large bottles and 4 randomly chosen small bottles is less than 25.5 litres. [5] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ (b) Find the probability that the volume of juice in a randomly chosen large bottle is at least twice the volume of juice in a randomly chosen small bottle. [5] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................

Mark scheme: 5(a) E(L1+L2+L3+S1+S2+S3+S4) = 3 5.10 4 2.51 [ 25.34] × + × = B1 OE (E(3L + 4S – 25.5) = -0.16) Var(L1+L2+L3+S1+S2+S3+S4) = 3 0.0102 4 0.0036 [ 0.045] × + × = B1 or 3 2 SD 0.2121 20 = = . ' 045 .0' ' 34 . 25 ' 5. 25 − [= 0.754] M1 No SD/variance mix. Standardising with their values (must be from a combination attempt). Φ(‘0.754’) M1 For the correct area consistent with their working. 0.775 (3 sf) A1 5 5(b) E(L – 2S) =5.10 2 2.51 [ 0.08] −× = B1 OE Var(L – 2S) = 2 0.0102 2 0.0036 [ 0.0246] + × = B1 Or SD 0.1568 = . 0 0 08 0 0246 ' . ' ' . ' − [= –0.510] M1 No SD/variance mix. Standardising with their values (must be from a combination attempt). P(Z > '–0.510') = ɸ(‘0.510’) M1 For the correct area consistent with their working. 0.695 (3 sf) A1 5

More questions on Discrete random variables

6 It is known that 8% of adults in a certain town own a Chantor car. After an advertising campaign, a car dealer wishes to investigate whether this proportion has increased. He chooses a random sample of 25 adults from the town and notes how many of them own a Chantor car. (a) He finds that 4 of the 25 adults own a Chantor car. Carry out a hypothesis test at the 5% significance level. [5] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ (b) Explain which of the errors, Type I or Type II, might have been made in carrying out the test in part (a). [2] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ Later, the car dealer takes another random sample of 25 adults from the town and carries out a similar hypothesis test at the 5% significance level. (c) Find the probability of a Type I error. [3] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................

Mark scheme: 6(a) H0: population proportion = 0.08 OE H1: population proportion > 0.08 OE B1 Allow ‘p = 0.08’ etc. P(X > 4) = 1 – P(X < 3) = 1 – (0.9225 + 25×0.9224×0.08 +25C2×0.9223×0.082 + 25C3×0.9222×0.083) M1 Allow 1 – (one term omitted or extra or wrong). 0.135 (3 sf) A1 0.135 > 0.05 M1 Valid comparison. Note: ‘0.865'<0.95 can score M1 A1 and can recover previous M1 A1 for 0.865. There is no evidence that proportion owning Chantor has increased A1 FT In context. Not definite, e.g. not ‘Proportion not increased’. No contradictions. 5 6(b) H0 was not rejected. *B1 FT H0 was rejected (consistent with (a)). Hence Type II might have been made. DB1 FT Type I error. 2 6(c) P(X > 5) = 1 – P(X < 4) = ( ) ( ) 25 21 4 4 1 1 0.1351 C 0.92 0.08 − − + × × [= 0.0451] *M1 Attempted. Note: If critical region method used in (a) marks can be awarded here. 0.0451 < 0.05 A1 Comparison of 0.045[1] with 0.05. Note: If critical region method used in (a) marks can be awarded here. P(Type I error) = 0.0451 or 0.0452 A1 Dependent on M1* only. SC Unsupported answers score: B1 for 0.0451<0.05 and B1 for final answer 0.0451 only. 3

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