Cambridge A Level Mathematics 9709 — 2011 May/June Paper 7 · Variant 2

*4301505300* UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS General Certificate of Education Advanced Level MATHEMATICS 9709/72 Paper 7 Probability & Statistics 2 (S2) May/June 2011 1 hour 15 minutes Additional Materials: Answer Booklet/Paper Graph Paper List of Formulae (MF9) READ THESE INSTRUCTIONS FIRST If you have been given an Answer Booklet, follow the instructions on the front cover of the Booklet. Write your Centre number, candidate number and name on all the work you hand in. Write in dark blue or black pen. You may use a soft pencil for any diagrams or graphs. Do not use staples, paper clips, highlighters, glue or correction fluid. Answer all the questions. 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. Questions carrying smaller numbers of marks are printed earlier in the paper, and questions carrying larger numbers of marks later in the paper. This document consists of 3 printed pages and 1 blank page. JC11 06_9709_72/RP © UCLES 2011 [Turn over

2 1 The weights of bags of fuel have mean 3.2 kg and standard deviation 0.04 kg. The total weight of a random sample of three bags is denoted by T kg. Find the mean and standard deviation of T. [4] 2 X is a random variable having the distribution B12, 1 4. A random sample of 60 values of X is taken. Find the probability that the sample mean is less than 2.8. [5] 3 The number of goals scored per match by Everly Rovers is represented by the random variable X which has mean 1.8. (i) State two conditions for X to be modelled by a Poisson distribution. [2] Assume now that X ∼Po(1.8). (ii) Find P(2 < X < 6). [2] (iii) The manager promises the team a bonus if they score at least 1 goal in each of the next 10 matches. Find the probability that they win the bonus. [3] 4 A doctor wishes to investigate the mean fat content in low-fat burgers. He takes a random sample of 15 burgers and sends them to a laboratory where the mass, in grams, of fat in each burger is determined. The results are as follows. 9 7 8 9 6 11 7 9 8 9 8 10 7 9 9 Assume that the mass, in grams, of fat in low-fat burgers is normally distributed with mean µ and that the population standard deviation is 1.3. (i) Calculate a 99% confidence interval for µ. [4] (ii) Explain whether it was necessary to use the Central Limit theorem in the calculation in part (i). [2] (iii) The manufacturer claims that the mean mass of fat in burgers of this type is 8 g. Use your answer to part (i) to comment on this claim. [2] 5 The number of adult customers arriving in a shop during a 5-minute period is modelled by a random variable with distribution Po(6). The number of child customers arriving in the same shop during a 10-minute period is modelled by an independent random variable with distribution Po(4.5). (i) Find the probability that during a randomly chosen 2-minute period, the total number of adult and child customers who arrive in the shop is less than 3. [3] (ii) During a sale, the manager claims that more adult customers are arriving than usual. In a randomly selected 30-minute period during the sale, 49 adult customers arrive. Test the manager’s claim at the 2.5% significance level. [6] © UCLES 2011 9709/72/M/J/11

3 6 Jeevan thinks that a six-sided die is biased in favour of six. In order to test this, Jeevan throws the die 10 times. If the die shows a six on at least 4 throws out of 10, she will conclude that she is correct. (i) State appropriate null and alternative hypotheses. [1] (ii) Calculate the probability of a Type I error. [3] (iii) Explain what is meant by a Type II error in this situation. [1] (iv) If the die is actually biased so that the probability of throwing a six is 1 2, calculate the probability of a Type II error. [3] 7 A random variable X has probability density function given by f(x) = k(1 −x) −1 ≤x ≤1, 0 otherwise, where k is a constant. (i) Show that k = 1 2. [2] (ii) Find PX > 1 2. [1] (iii) Find the mean of X. [3] (iv) Find a such that P(X < a) = 1 4. [3] © UCLES 2011 9709/72/M/J/11

4 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. University of 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. 9709/72/M/J/11

UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS GCE Advanced Subsidiary Level and GCE Advanced Level MARK SCHEME for the May/June 2011 question paper for the guidance of teachers 9709 MATHEMATICS 9709/72 Paper 7, maximum raw mark 50 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 must be read in conjunction with the question papers and the report on the examination. • Cambridge will not enter into discussions or correspondence in connection with these mark schemes. Cambridge is publishing the mark schemes for the May/June 2011 question papers for most IGCSE, GCE Advanced Level and Advanced Subsidiary Level syllabuses and some Ordinary Level syllabuses.

Page 2 Mark Scheme: Teachers’ version Syllabus Paper GCE AS/A LEVEL – May/June 2011 9709 72 © University of Cambridge International Examinations 2011 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 √ 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.

Page 3 Mark Scheme: Teachers’ version Syllabus Paper GCE AS/A LEVEL – May/June 2011 9709 72 © University of Cambridge International Examinations 2011 The following abbreviations may be used in a mark scheme or used on the scripts: AEF Any Equivalent Form (of answer is equally acceptable) AG Answer Given on the question paper (so extra checking is needed to ensure that the detailed working leading to the result is valid) BOD Benefit of Doubt (allowed when the validity of a solution may not be absolutely clear) 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 MR Misread PA Premature Approximation (resulting in basically correct work that is insufficiently accurate) SOS See Other Solution (the candidate makes a better attempt at the same question) 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.

Page 4 Mark Scheme: Teachers’ version Syllabus Paper GCE AS/A LEVEL – May/June 2011 9709 72 © University of Cambridge International Examinations 2011 1 E(T) = 9.6 Var(wt of one bag) = 0.0016 Var(T) = 3 × 0.0016 sd of T = √(3 × 0.0016) = 0.0693 B1 M1 M1 A1 [4] May be impl. by Var(T) = 0.0048 or 0.0144 [Total: 4] 2 X ~N(3, 60 4 9 ) 60 4 9 3 8.2 − (= –1.033) Φ(“–1.033”) = 1 – Φ(“1.033”) = 0.151 B2 M1 M1 A1 [5] B1 for N & µ = 3; (oe) B1 for 9/4/60 or 3/80 or 0.0375 (oe) (oe working with totals or proportions) With or without c.c. With cc of –1/120, Φ(–1.076) = 1 – Φ(1.076) = 0.141 [Total: 5] 3 (i) Constant average rate of goals scored Goals random Goals indep B1 B1 [2] Any two given in context (SR score B1 for any two not in context) Not Goals scored singly (because this is inherent in the context so it’s not a condition) (ii) e–1.8 ) ( !5 8.1 !4 8.1 !3 8.1 5 4 3 + + = 0.259 M1 A1 [2] Poisson probs, λ = 1.8. Allow 2, 6 included (iii) 1 – e–1.8 (1 – e–1.8)10 = 0.164 M1 M1 A1 [3] Any λ. Allow end errors. [Total: 7] 4 (i) x = 8.4 8.4 ± z 15 3.1 z = 2.576 [7.54, 9.26] B1 M1 B1 A1 [4] Accept 2.574 to 2.579 or equiv. Accept 7.53. Accept 9.27 (ii) No because pop normal so X normally distr B1 B1 [2] SR If ‘Yes’ or no conclusion, but 2 correct statements score B1 (iii) 8 within CI Claim justified B1√ B1√ [2] ft (i) [Total: 8]

Page 5 Mark Scheme: Teachers’ version Syllabus Paper GCE AS/A LEVEL – May/June 2011 9709 72 © University of Cambridge International Examinations 2011 5 (i) Po(3.3) e–3.3(1 + 3.3 + ) 2 3.3 2 = 0.359 B1 M1 A1 [3] seen or implied Poisson P(0) + P(1) + P(2). Allow + P(3) Allow wrong λ. Accept equiv method. (ii) X~Po(36) X~N(36, 36) 36 36 5. 48 − = 2.08(3) comp with 1.96 Evidence to support claim B1 B1 M1 A1 M1 A1√ [6] Allow with no or wrong cc or no √ 2.08(3) or 0.0186/0.0187 if area comparison Valid comparison Correct conclusion (ft their z) [Total: 9] 6 (i) H0: P(6) = 6 1 H1: P(6) > 6 1 B1 [1] Condone undefined p (ii) 3 7 3 10 2 8 2 10 9 10 6 1 6 5 6 1 6 5 6 1 6 5 10 6 5     ×     ×     + ×     ×     + ×     × +     M1 (1 –) P(0,1,2,3) o.e. using B(10,1/6) allow end errors 1 – ( 2 8 2 10 9 10 6 1 6 5 6 1 6 5 10 6 5     ×     ×     + ×     × +     3 7 6 1 6 5 3 10     ×     ×     + ) M1 Attempt at fully correct expression for 1 – P(0,1,2,3) o.e. = 0.0697 (3 sfs) A1 [3] Accept 0.0698 (iii) Die biased towards a six but result < 4 so no evidence of bias B1 [1] or equiv. Must be in context (iv) P(0, 1, 2 or 3 sixes) ) ( 3 7 2 8 9 10 2 1 2 1 3 10 2 1 2 1 2 10 2 1 2 1 10 2 1       ×       ×     +       ×       ×     + ×       × +       = 0.172 or 11/64 B1 M1 A1 [3] Stated or attempted. Can be implied Attempt at P(0,1,2,3) with p = 1/2, allow end errors. [Total: 8]

Page 6 Mark Scheme: Teachers’ version Syllabus Paper GCE AS/A LEVEL – May/June 2011 9709 72 © University of Cambridge International Examinations 2011 7 (i) ∫− − 1 1 d ) 1( x x k = 1 (k[ 2 2 x x − ]1 1 − = 1) 2k = 1 (k = 2 1 AG) M1 A1 [2] Attempt integ f(x) = 1 with correct limits (ii) (∫ − 1 5.0 2 1 d ) 1( x x = 2 1 [ 2 2 x x − ]1 5.0 ) = 16 1 or 0.0625 B1 [1] (iii) ∫− − 1 1 2 2 1 d ) ( x x x = 2 1 [ 2 2 x – 3 3 x ] 1 1 − = – 3 1 or –0.333 M1 A1 A1 [3] ∫ x x x d ) f( ignore limits Correct integrand and limits (iv) ∫− − a x x 1 2 1 d ) 1( = 0.25 ( 2 1 [ 2 2 x x − ] a 1 − = 0.25) ( 2 1 (a – 2 2 a – (–1 – 2 1 ) = 0.25) a2 – 2a – 2 = 0 a = 1 – √3 or –0.732 M1 A1 A1 [3] Correct limits (or integral from a to 1 = 0.75) any correct QE with “= 0”(or in completed square form (a – 1)2 = 3) Not a = 1 ± √3; Not –0.732 or 2.732 [Total: 9]