Cambridge A Level Mathematics 9709 — 2018 Feb/March Paper 7 · Variant 2

*6786904181* Cambridge International Examinations Cambridge International Advanced Level CANDIDATE NAME CENTRE NUMBER CANDIDATE NUMBER MATHEMATICS 9709/72 Paper 7 Probability & Statistics 2 (S2) February/March 2018 1 hour 15 minutes Candidates answer on the Question Paper. Additional Materials: List of Formulae (MF9) READ THESE INSTRUCTIONS FIRST Write your Centre number, candidate number and name in the spaces at the top of this page. Write in dark blue or black pen. You may use an HB pencil for any diagrams or graphs. Do not use staples, paper clips, glue or correction fluid. DO NOT WRITE IN ANY BARCODES. Answer all the questions in the space provided. If additional space is required, you should use the lined page at the end of this booklet. The question number(s) must be clearly shown. Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place in the case of angles in degrees, unless a different level of accuracy is specified in the question. The use of an electronic calculator is expected, where appropriate. You are reminded of the need for clear presentation in your answers. At the end of the examination, fasten all your work securely together. The number of marks is given in brackets [ ] at the end of each question or part question. The total number of marks for this paper is 50. This document consists of 13 printed pages and 3 blank pages. JC18 03_9709_72/4R © UCLES 2018 [Turn over

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3 1 X is a random variable with mean 4.9 and standard deviation 2.21. A random sample of 75 values of X is taken. Find the probability that the sample mean is greater than 5.0. [3] … … … … … … … … … … … … … … … … … … … … … … … … … © UCLES 2018 9709/72/F/M/18 [Turn over

4 2 The number of phone calls arriving in a 10-minute period at a switchboard is modelled by the random variable X which has the distribution Po4.1. Use an approximating distribution to find the probability that more than 90 calls arrive in a 4-hour period. [5] … … … … … … … … … … … … … … … … … … … … … … … … © UCLES 2018 9709/72/F/M/18

5 3 The heights of plants of type A have mean 1.2 m and standard deviation 0.03 m. A random sample of 5 plants of type A is selected. The sum of the heights of these 5 plants is denoted by HA m. (i) Find the mean and variance of HA. [3] … … … … … … … The heights of plants of type B have mean 0.6 m and standard deviation 0.02 m. A random sample of 5 plants of type B is selected. The sum of the heights of these 5 plants is denoted by HB m. (ii) Find the mean and variance of HA −2HB. [4] … … … … … … … … … … … … … … © UCLES 2018 9709/72/F/M/18 [Turn over

6 4 A store sells two types of computer, laptops and tablets. The number of laptops sold per hour is modelled by a random variable with distribution Po0.9. The number of tablets sold per hour is modelled by an independent random variable with distribution Po1.5. (i) Find the probability that, during a randomly chosen hour, the total number of laptops and tablets sold in the store is less than 4. [3] … … … … … … … … … … … … … … … … … … … … … … … © UCLES 2018 9709/72/F/M/18

7 (ii) The manager claims that on sunny Saturdays fewer laptops than usual are sold. In order to test this claim, an employee notes the number of laptops sold during a 4-hour period on a randomly chosen sunny Saturday. In fact only 1 laptop is sold during this period. Test the manager’s claim at the 10% significance level. [5] … … … … … … … … … … … … … … … … … … … … … … … … © UCLES 2018 9709/72/F/M/18 [Turn over

8 5 Packets of Frugums contain 30 sweets. The manufacturer claims that, on average, 17% of the sweets are orange flavoured. Angela suspects that the average is actually less than 17%. In order to test the manufacturer’s claim, she buys a packet of Frugums. If there are fewer than 3 orange flavoured sweets in the packet, she will conclude that the claim is false. (i) State appropriate null and alternative hypotheses. [1] … … … (ii) Explain what is meant by a Type I error in this situation. [1] … … … … (iii) Calculate the probability of a Type I error. [3] … … … … … … … … … … … … … © UCLES 2018 9709/72/F/M/18

9 (iv) Given that the true percentage of orange flavoured sweets is 5%, calculate the probability of a Type II error. [3] … … … … … … … … … … … … … … … … … … … … … … … … … © UCLES 2018 9709/72/F/M/18 [Turn over

10 6 A random variable X has probability density function given by fx = D 6x1 −x 0 ≤x ≤1, 0 otherwise. (i) Find the probability that X does not lie between 0.3 and 0.7. [4] … … … … … … … … … … … … … … … … … … … … … … … © UCLES 2018 9709/72/F/M/18

11 (ii) Sketch the graph of the probability density function and hence state the value of EX. [2] … (iii) Find VarX. [3] … … … … … … … … … … … … … … … … … … © UCLES 2018 9709/72/F/M/18 [Turn over

12 7 A nutritionist wishes to investigate the mean sugar content in some cereal bars. He takes a random sample of 10 of the bars and measures the mass, in grams, of sugar in each bar. His results are shown below. 11.9 11.7 11.8 11.9 11.6 12.1 11.7 11.9 11.8 11.9 Assume that the mass, in grams, of sugar in bars of this type has the distribution N-, 0.01. (i) Calculate a 99% confidence interval for -. [4] … … … … … … … … … … … … … … … … (ii) Explain whether it was necessary to use the Central Limit theorem in the calculation in part (i). [1] … … … © UCLES 2018 9709/72/F/M/18

13 (iii) The manufacturer claims that the mean mass of sugar in bars of this type is 11.7 g. Explain why your answer to part (i) does not support this claim. [1] … … … (iv) The manufacturer suggests that a 95% confidence interval would be more likely to support his claim than a 99% confidence interval. Without doing a calculation, explain whether this suggestion is correct. [1] … … … (v) It is thought that the value of 0.01 for the population variance may not be correct. Use the values in the sample to calculate an unbiased estimate of the population variance. [3] … … … … … … … … … … … … … … © UCLES 2018 9709/72/F/M/18 [Turn over

14 Additional Page If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown. … … … … … … … … … … … … … … … … … … … … … … … … © UCLES 2018 9709/72/F/M/18

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16 BLANK PAGE Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable effort has been made by the publisher (UCLES) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will be pleased to make amends at the earliest possible opportunity. To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced online in the Cambridge International Examinations Copyright Acknowledgements Booklet. This is produced for each series of examinations and is freely available to download at www.cie.org.uk after the live examination series. Cambridge International Examinations is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge. © UCLES 2018 9709/72/F/M/18

® IGCSE is a registered trademark. This document consists of 8 printed pages. © UCLES 2018 [Turn over Cambridge Assessment International Education Cambridge International Advanced Level MATHEMATICS 9709/72 Paper 7 Probability and Statistics March 2018 MARK SCHEME Maximum Mark: 50 Published This mark scheme is published as an aid to teachers and candidates, to indicate the requirements of the examination. It shows the basis on which Examiners were instructed to award marks. It does not indicate the details of the discussions that took place at an Examiners’ meeting before marking began, which would have considered the acceptability of alternative answers. Mark schemes should be read in conjunction with the question paper and the Principal Examiner Report for Teachers. Cambridge International will not enter into discussions about these mark schemes. Cambridge International is publishing the mark schemes for the March 2018 series for most Cambridge IGCSE®, Cambridge International A and AS Level components and some Cambridge O Level components.

9709/72 Cambridge International A Level – Mark Scheme PUBLISHED March 2018 © UCLES 2018 Page 2 of 8 Generic Marking Principles These general marking principles must be applied by all examiners when marking candidate answers. They should be applied alongside the specific content of the mark scheme or generic level descriptors for a question. Each question paper and mark scheme will also comply with these marking principles. GENERIC MARKING PRINCIPLE 1: Marks must be awarded in line with: • the specific content of the mark scheme or the generic level descriptors for the question • the specific skills defined in the mark scheme or in the generic level descriptors for the question • the standard of response required by a candidate as exemplified by the standardisation scripts. GENERIC MARKING PRINCIPLE 2: Marks awarded are always whole marks (not half marks, or other fractions). GENERIC MARKING PRINCIPLE 3: Marks must be awarded positively: • marks are awarded for correct/valid answers, as defined in the mark scheme. However, credit is given for valid answers which go beyond the scope of the syllabus and mark scheme, referring to your Team Leader as appropriate • marks are awarded when candidates clearly demonstrate what they know and can do • marks are not deducted for errors • marks are not deducted for omissions • answers should only be judged on the quality of spelling, punctuation and grammar when these features are specifically assessed by the question as indicated by the mark scheme. The meaning, however, should be unambiguous. GENERIC MARKING PRINCIPLE 4: Rules must be applied consistently e.g. in situations where candidates have not followed instructions or in the application of generic level descriptors. GENERIC MARKING PRINCIPLE 5: Marks should be awarded using the full range of marks defined in the mark scheme for the question (however; the use of the full mark range may be limited according to the quality of the candidate responses seen). GENERIC MARKING PRINCIPLE 6: Marks awarded are based solely on the requirements as defined in the mark scheme. Marks should not be awarded with grade thresholds or grade descriptors in mind.

9709/72 Cambridge International A Level – Mark Scheme PUBLISHED March 2018 © UCLES 2018 Page 3 of 8 Mark Scheme Notes Marks are of the following three types: M Method mark, awarded for a valid method applied to the problem. Method marks are not lost for numerical errors, algebraic slips or errors in units. However, it is not usually sufficient for a candidate just to indicate an intention of using some method or just to quote a formula; the formula or idea must be applied to the specific problem in hand, e.g. by substituting the relevant quantities into the formula. Correct application of a formula without the formula being quoted obviously earns the M mark and in some cases an M mark can be implied from a correct answer. A Accuracy mark, awarded for a correct answer or intermediate step correctly obtained. Accuracy marks cannot be given unless the associated method mark is earned (or implied). B Mark for a correct result or statement independent of method marks. • When a part of a question has two or more “method” steps, the M marks are generally independent unless the scheme specifically says otherwise; and similarly when there are several B marks allocated. The notation DM or DB (or dep*) is used to indicate that a particular M or B mark is dependent on an earlier M or B (asterisked) mark in the scheme. When two or more steps are run together by the candidate, the earlier marks are implied and full credit is given. • The symbol FT implies that the A or B mark indicated is allowed for work correctly following on from previously incorrect results. Otherwise, A or B marks are given for correct work only. A and B marks are not given for fortuitously “correct” answers or results obtained from incorrect working. • Note: B2 or A2 means that the candidate can earn 2 or 0. B2/1/0 means that the candidate can earn anything from 0 to 2. The marks indicated in the scheme may not be subdivided. If there is genuine doubt whether a candidate has earned a mark, allow the candidate the benefit of the doubt. Unless otherwise indicated, marks once gained cannot subsequently be lost, e.g. wrong working following a correct form of answer is ignored. • Wrong or missing units in an answer should not lead to the loss of a mark unless the scheme specifically indicates otherwise. • For a numerical answer, allow the A or B mark if a value is obtained which is correct to 3 s.f., or which would be correct to 3 s.f. if rounded (1 d.p. in the case of an angle). As stated above, an A or B mark is not given if a correct numerical answer arises fortuitously from incorrect working. For Mechanics questions, allow A or B marks for correct answers which arise from taking g equal to 9.8 or 9.81 instead of 10.

9709/72 Cambridge International A Level – Mark Scheme PUBLISHED March 2018 © UCLES 2018 Page 4 of 8 The following abbreviations may be used in a mark scheme or used on the scripts: AEF/OE Any Equivalent Form (of answer is equally acceptable) / Or Equivalent AG Answer Given on the question paper (so extra checking is needed to ensure that the detailed working leading to the result is valid) CAO Correct Answer Only (emphasising that no “follow through” from a previous error is allowed) CWO Correct Working Only – often written by a ‘fortuitous’ answer ISW Ignore Subsequent Working SOI Seen or implied SR Special Ruling (detailing the mark to be given for a specific wrong solution, or a case where some standard marking practice is to be varied in the light of a particular circumstance) Penalties MR –1 A penalty of MR –1 is deducted from A or B marks when the data of a question or part question are genuinely misread and the object and difficulty of the question remain unaltered. In this case all A and B marks then become “follow through” marks. MR is not applied when the candidate misreads his own figures – this is regarded as an error in accuracy. An MR –2 penalty may be applied in particular cases if agreed at the coordination meeting. PA –1 This is deducted from A or B marks in the case of premature approximation. The PA –1 penalty is usually discussed at the meeting.

9709/72 Cambridge International A Level – Mark Scheme PUBLISHED March 2018 © UCLES 2018 Page 5 of 8 Question Answer Marks Guidance 1 5 4.9 2.21 75 − (= 0.392) M1 Correct stand'n. Must have √75 1 – Φ(“0.392”) M1 Correct area consistent with working = 0.348 (3 sfs) A1 3 Question Answer Marks Guidance 2 λ = 98.4 B1 N(98.4, 98.4) seen or implied B1 90.5 "98.4" "98.4" − (= –0.796) M1 allow with wrong or no cc. No sd/var mix. ɸ("0.796") M1 Correct area consistent with working = 0.787 (3 sf) A1 5 Question Answer Marks Guidance 3(i) E(HA) = 6 B1 Var(HA) = 5 × 0.032 M1 = 0.0045 or 9/2000 A1 3 3(ii) E(HA – 2HB) = 0 B1 From 6–6 Var(HA – 2HB) = '0.0045' + 4×5×0.022 M2 Allow M1 for '0.0045' – 4×5×0.022 or '0.0045' + 2×5×0.022 or '0.0045' + 4×0.022 or '0.0045' + 4×52×0.022 = 0.0125 (3 sf) or 1/80 A1 4

9709/72 Cambridge International A Level – Mark Scheme PUBLISHED March 2018 © UCLES 2018 Page 6 of 8 Question Answer Marks Guidance 4(i) (Po)(2.4) B1 seen or implied e–2.4 (1 + 2.4 + 2 3 2.4 2.4 2 3! + ) M1 allow + P(4)/one end error. Allow wrong λ = 0.779 (3 sfs) A1 Final answer (Note: accept combination method) 3 4(ii) H0: λ (or mean) = 3.6 (or 0.9) H1: λ (or mean) < 3.6 (or 0.9) B1 Accept µ for both e–3.6 (1 + 3.6) M1 Allow any λ = 0.126 A1 0.126 > 0.1 M1 Valid comparison. (Comparison with 0.9 could recover previous M1A1) No evidence that fewer than usual sold A1FT Correct conclusion. No contradictions 5 Question Answer Marks Guidance 5(i) H0: P(Orange) = 0.17 H1: P(Orange) < 0.17 B1 or H0: p = 0.17 H1:p < 0.17 5(ii) Wrongly concluding that % age is less than 17% B1 OE in context allow "fewer than 3 orange in packet even though average 17% is correct" 1 5(iii) B(30, 0.17) stated or implied M1 eg by 0.17p × 0.83q (p + q = 30) or 30Cr (r < 30) (1 – 0.17)30 + 30(1 – 0.17)29 × 0.17 + 30C2(1– 0.17)28 × 0.172 M1 correct, but allow + 30C3(1–0.17)27× 0.173 = 0.0949 (3 sf) A1 (SR: use of N(5.1,4.233) M1 standardising (with or without cc) M1 max 2/3) 3

9709/72 Cambridge International A Level – Mark Scheme PUBLISHED March 2018 © UCLES 2018 Page 7 of 8 Question Answer Marks Guidance 5(iv) P(⩾ 3 orange | p = 0.05) M1 stated or attempted; can be implied = 1– [(0.95)30 + 30(0.95)29 × 0.05 + 30C2(0.95)28 × 0.052] M1 allow + 30C3(0.95)27 × 0.053 in bracket, or ans 0.0608 = 0.188 (3 sfs) A1 3 Question Answer Marks Guidance 6(i) 1 – ( ) 0.7 2 0.3 6 d − ∫x x x M1 or 2 × ( ) 0.3 2 0 6 d − ∫x x x or similar correct expression before integration 1 – 2 3 2 3 6( )   −   x x 0.7 0.3 A1 or similar correct expression after integration 1 – 6 2 3 2 3 0.7 0.7 0.3 0.3 2 3 2 3   − − +     M1 Attempt subst correct limits in this or other correct expression = 0.432 (or 54/125) A1 (SR1 Omission of ‘1–‘ scores B2 for 0.568 or 71/125) (SR2 Omission of ‘2x’ scores B2 for 0.216 or 27/125) 4 6(ii) Correct shape between x = 0 and 1 B1 No curve outside this range. E(X) = 0.5 B1 2 6(iii) ( ) 1 3 4 0 6 d − ∫ x x x = 4 5 6 4 5     −           x x 1 0 M1 attempt int x2f(x), ignore limits 6[ 4 5 1 1 4 5 − ] (= 0.3) M1 attempt subst correct limits in correct integ Var(X) = '0.3' – '0.5'2 = 0.05 A1FT FT their mean, dep their Var(X) > 0 3

9709/72 Cambridge International A Level – Mark Scheme PUBLISHED March 2018 © UCLES 2018 Page 8 of 8 Question Answer Marks Guidance 7(i) x = 11.83 B1 11.83 ± z 0.1 10 M1 any z z = 2.576 B1 accept 2.574 to 2.579 [11.75 to 11.91] A1 or equiv. Accept 11.7 to 11.9 4 7(ii) No because pop normal (so X normally distr) B1 1 7(iii) 11.7 not within CI B1FT 1 7(iv) No because 95% CI is narrower than 99% CI B1 OE 1 7(v) Σx2 (= 1399.67) M1 attempted Est(σ2) = 2 10 "1399.67" "118.3" 9 10 10     −           OE M1 correct sub of their Σs into correct formula = 0.0201 (3 sf) or 181/9000 A1 3