Cambridge A Level Mathematics 9709 — 2006 Oct/Nov Paper 6 · Variant 1

UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS General Certificate of Education Advanced Subsidiary Level and Advanced Level MATHEMATICS 9709/06 Paper 6 Probability & Statistics 1 (S1) October/November 2006 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 on both sides of the paper. 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. 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. At the end of the examination, fasten all your work securely together. This document consists of 3 printed pages and 1 blank page. © UCLES 2006 [Turn over

2 1 The weights of 30 children in a class, to the nearest kilogram, were as follows. 50 45 61 53 55 47 52 49 46 51 60 52 54 47 57 59 42 46 51 53 56 48 50 51 44 52 49 58 55 45 Construct a grouped frequency table for these data such that there are five equal class intervals with the first class having a lower boundary of 41.5 kg and the fifth class having an upper boundary of 61.5 kg. [4] 2 The discrete random variable X has the following probability distribution. x 0 1 2 3 4 P(X = x) 0.26 q 3q 0.05 0.09 (i) Find the value of q. [2] (ii) Find E(X) and Var(X). [3] 3 In a survey, people were asked how long they took to travel to and from work, on average. The median time was 3 hours 36 minutes, the upper quartile was 4 hours 42 minutes and the interquartile range was 3 hours 48 minutes. The longest time taken was 5 hours 12 minutes and the shortest time was 30 minutes. (i) Find the lower quartile. [2] (ii) Represent the information by a box-and-whisker plot, using a scale of 2 cm to represent 60 minutes. [4] 4 Two fair dice are thrown. (i) Event A is ‘the scores differ by 3 or more’. Find the probability of event A. [3] (ii) Event B is ‘the product of the scores is greater than 8’. Find the probability of event B. [2] (iii) State with a reason whether events A and B are mutually exclusive. [2] 5 (i) Give an example of a variable in real life which could be modelled by a normal distribution. [1] (ii) The random variable X is normally distributed with mean µ and variance 21.0. Given that P(X > 10.0) = 0.7389, find the value of µ. [3] (iii) If 300 observations are taken at random from the distribution in part (ii), estimate how many of these would be greater than 22.0. [4] © UCLES 2006 9709/06/O/N/06

3 6 Six men and three women are standing in a supermarket queue. (i) How many possible arrangements are there if there are no restrictions on order? [2] (ii) How many possible arrangements are there if no two of the women are standing next to each other? [4] (iii) Three of the people in the queue are chosen to take part in a customer survey. How many different choices are possible if at least one woman must be included? [3] 7 A manufacturer makes two sizes of elastic bands: large and small. 40% of the bands produced are large bands and 60% are small bands. Assuming that each pack of these elastic bands contains a random selection, calculate the probability that, in a pack containing 20 bands, there are (i) equal numbers of large and small bands, [2] (ii) more than 17 small bands. [3] An office pack contains 150 elastic bands. (iii) Using a suitable approximation, calculate the probability that the number of small bands in the office pack is between 88 and 97 inclusive. [6] © UCLES 2006 9709/06/O/N/06

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 University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge. 9709/06/O/N/06

UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS GCE Advanced Subsidiary Level and GCE Advanced Level MARK SCHEME for the October/November 2006 question paper 9709 MATHEMATICS 9709 Paper 6, maximum raw mark 50 This mark scheme is published as an aid to teachers and students, 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. All Examiners are instructed that alternative correct answers and unexpected approaches in candidates’ scripts must be given marks that fairly reflect the relevant knowledge and skills demonstrated. Mark schemes must be read in conjunction with the question papers and the report on the examination. The grade thresholds for various grades are published in the report on the examination for most IGCSE, GCE Advanced Level and Advanced Subsidiary Level syllabuses. • CIE will not enter into discussions or correspondence in connection with these mark schemes. CIE is publishing the mark schemes for the October/November 2006 question papers for most IGCSE, GCE Advanced Level and Advanced Subsidiary Level syllabuses and some Ordinary Level syllabuses.

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.

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 Syllabus Paper GCE A/AS LEVEL - OCT/NOV 2006 9709 06 © UCLES 2006 1 Weight freq 41.5-45.5 4 45.5-49.5 7 49.5-53.5 10 53.5-57.5 5 57.5-61.5 4 M1 A1 M1 A1 4 Five groups Correct boundaries, accept 42-45, 46-49 etc Attempt to calculate frequencies Σ 29, 30 or 31. 5 frequencies correct 2 (i) q + 3q + 0.26 + 0.05 + 0.09 = 1 q = 0.15 (ii) E(X) = 1.56 Var (X) = 0.15 + 1.8 + 0.45 + 1.44 – mean2 = 1.41 M1 A1 2 B1ft M1 A1 3 Equation with q in summing probs to 1must be probs Correct answer Correct final answer, ft on wrong q Subst in Σpx2 – mean2 formula Correct final answer 3 (i) LQ = 4 hr 42 min – 3 hr 48 min = 54 min (0.9 hours) (ii) 0 1 2 3 4 5 6 time M1 A1 2 B1 B1 B1ft B1 4 Subtracting IQR from UQ Correct answer Correct whiskers(accept hour decimals or minutes) Correct median line, can be broken or extended Correct UQ and LQ ft on their (i), box ends correct uniform scale label hours or minutes, could be heading or key

Page 5 Mark Scheme Syllabus Paper GCE A/AS LEVEL - OCT/NOV 2006 9709 06 © UCLES 2006 4 (i) list 14,15,16,25,26, 36,and reversed P(scores differ by 3 or more) = 12/36 (1/3)(0.333) (ii) 20/36 (iii) P 0 ) ( ≠ ∩B A implies not mut excl, or equivalent P = ∩ ) ( B A 6/36 so not mut excl M1 A1 A1 3 M1 A1 2 B1 B1 ft 2 For an attempt at listing Selecting at least 6 correct pairs Correct answer Some identification on the list, must include one of 25, 26, 33, 34, 35 Correct answer Correct statement about mut excl events Correct answer using their data 5 (i) heights, weights, times etc of something (ii) z = 0.64 = 21 10 − µ µ = 12.9 (iii) 21 9. 12 22 − = z = 1.986 P(X > 22) = 1 – Φ(1.986) = 1 – 0.9765 = 0.0235 300 × 0.0235 = 7.05 answer = 7 B1 1 B1 M1 A1 3 M1 M1ft M1 A1 4 Any sensible set of data, must be qualified z = ± 0.64 seen equation relating 10, 21 , 21, µ and their z or 1 – their z, (must be a recognisable z value ie not 0.77) correct answer standardising, with or without sq rt, no cc, must be their mean correct area ie < 0.5, ft on their mean >22 mult by 300 correct answer, accept 7 or 8 must be integer 6 (i) 9! = 362880 (363000) (ii) 6! × 7P3 =151200 (iii) 1 woman: 3C1 × 6C2 = 45 2 women: 3C2 × 6C1 = 18 3 women: 3C3 = 1 total = 64 OR no restrictions 9C3 (84) Men only 84 – 20 = 64 B1 B1 2 B1 M1 A1 A1 4 M1 B1 A1 B1 M1 A1 3 9! Or 9P9 only correct answer 6! seen 7P or 7C something or 7 multiplied by something mult by 7P3 correct answer summing cases for 1, 2, 3 women one correct case correct answer 9C3 or 84 or 3 times 8C2 seen attempt at subt of their ‘no women’ case correct answer

Page 6 Mark Scheme Syllabus Paper GCE A/AS LEVEL - OCT/NOV 2006 9709 06 © UCLES 2006 7 (i) (0.6)10 × (0.4)10 × 20C10 = 0.117 (ii) P(18, 19, 20) = (0.6)18 (0.4)2 20C2 + (0.6)19(0.4)1 21C1 +(0.6)20 = 0.003087 + 0.000487 + 0.00003635 = 0.00361 OR using normal approx N(12,4.8) 8.4 12 5. 17 − = z = 2.51 Prob = 1 – 0.9940 = 0.0060 (iii) 90 60 .0 150 = × = µ 36 40 .0 60 .0 150 2 = × × = σ P(88 < X < 97) =       − Φ −       − Φ 6 90 5. 87 6 90 5. 97 = Φ(1.25) – Φ(-0.4166) = 0.8944 – (1 – 0.6616) = 0.556 M1 A1 2 M1 A1 A1 M1 A1 A1 3 B1 M1 M1 A1 M1 A1 6 3 term binomial expression involving 20Csomething and powers summing to 20 Correct final answer Summing three or 4 binomial expressions One correct unsimplified expression allow 0.4 0.6 muddle Correct answer Standardising, cc 16.5 or 17.5, their mean, (their var) 2.51 seen 0.0060 seen must be 0.0060 For seeing 90 and 36 For standardising , with or without cc, must have sq rt on denom one continuity correction 97.5 or 96.5 or 87.5 or 88.5 0.8944 or 0.6616 or 0.3384 or 0.3944 or 0.1616 seen subtracting a probability from their standardised 97 prob correct answer