Cambridge A Level Mathematics 9709 — 2005 May/June Paper 6 · Variant 1

UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS General Certificate of Education Advanced Subsidiary Level and Advanced Level Advanced International Certificate of Education MATHEMATICS STATISTICS 9709/06 0390/06 Paper 6 Probability & Statistics 1 (S1) May/June 2005 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. 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. The use of an electronic calculator is expected, where appropriate. You are reminded of the need for clear presentation in your answers. This document consists of 4 printed pages. © UCLES 2005 [Turn over

2 1 It is known that, on average, 2 people in 5 in a certain country are overweight. A random sample of 400 people is chosen. Using a suitable approximation, find the probability that fewer than 165 people in the sample are overweight. [5] 2 The following table shows the results of a survey to find the average daily time, in minutes, that a group of schoolchildren spent in internet chat rooms. Time per day Frequency (t minutes) 0 ≤t < 10 2 10 ≤t < 20 f 20 ≤t < 40 11 40 ≤t < 80 4 The mean time was calculated to be 27.5 minutes. (i) Form an equation involving f and hence show that the total number of children in the survey was 26. [4] (ii) Find the standard deviation of these times. [2] 3 A fair dice has four faces. One face is coloured pink, one is coloured orange, one is coloured green and one is coloured black. Five such dice are thrown and the number that fall on a green face are counted. The random variable X is the number of dice that fall on a green face. (i) Show that the probability of 4 dice landing on a green face is 0.0146, correct to 4 decimal places. [2] (ii) Draw up a table for the probability distribution of X, giving your answers correct to 4 decimal places. [5] © UCLES 2005 9709/06/M/J/05

3 4 The following back-to-back stem-and-leaf diagram shows the cholesterol count for a group of 45 people who exercise daily and for another group of 63 who do not exercise. The figures in brackets show the number of people corresponding to each set of leaves. People who exercise People who do not exercise (9) 9 8 7 6 4 3 2 2 1 3 1 5 7 7 (4) (12) 9 8 8 8 7 6 6 5 3 3 2 2 4 2 3 4 4 5 8 (6) (9) 8 7 7 7 6 5 3 3 1 5 1 2 2 2 3 4 4 5 6 7 8 8 9 (13) (7) 6 6 6 6 4 3 2 6 1 2 3 3 3 4 5 5 5 7 7 8 9 9 (14) (3) 8 4 1 7 2 4 5 5 6 6 7 8 8 (9) (4) 9 5 5 2 8 1 3 3 4 6 7 9 9 9 (9) (1) 4 9 1 4 5 5 8 (5) (0) 10 3 3 6 (3) Key: 2 | 8 | 1 represents a cholesterol count of 8.2 in the group who exercise and 8.1 in the group who do not exercise. (i) Give one useful feature of a stem-and-leaf diagram. [1] (ii) Find the median and the quartiles of the cholesterol count for the group who do not exercise. [3] You are given that the lower quartile, median and upper quartile of the cholesterol count for the group who exercise are 4.25, 5.3 and 6.6 respectively. (iii) On a single diagram on graph paper, draw two box-and-whisker plots to illustrate the data. [4] 5 Data about employment for males and females in a small rural area are shown in the table. Unemployed Employed Male 206 412 Female 358 305 A person from this area is chosen at random. Let M be the event that the person is male and let E be the event that the person is employed. (i) Find P(M). [2] (ii) Find P(M and E). [1] (iii) Are M and E independent events? Justify your answer. [3] (iv) Given that the person chosen is unemployed, find the probability that the person is female. [2] 6 Tyre pressures on a certain type of car independently follow a normal distribution with mean 1.9 bars and standard deviation 0.15 bars. (i) Find the probability that all four tyres on a car of this type have pressures between 1.82 bars and 1.92 bars. [5] (ii) Safety regulations state that the pressures must be between 1.9 −b bars and 1.9 + b bars. It is known that 80% of tyres are within these safety limits. Find the safety limits. [3] © UCLES 2005 9709/06/M/J/05 [Turn over

4 7 (a) A football team consists of 3 players who play in a defence position, 3 players who play in a midfield position and 5 players who play in a forward position. Three players are chosen to collect a gold medal for the team. Find in how many ways this can be done (i) if the captain, who is a midfield player, must be included, together with one defence and one forward player, [2] (ii) if exactly one forward player must be included, together with any two others. [2] (b) Find how many different arrangements there are of the nine letters in the words GOLD MEDAL (i) if there are no restrictions on the order of the letters, [2] (ii) if the two letters D come first and the two letters L come last. [2] 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. © UCLES 2005 9709/06/M/J/05

UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS GCE Advanced Subsidiary, Advanced Level and AICE MARK SCHEME for the June 2005 question paper 9709/0390 MATHEMATICS 9709/06, 0390/06 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 initially instructed to award marks. It does not indicate the details of the discussions that took place at an Examiners’ meeting before marking began. Any substantial changes to the mark scheme that arose from these discussions will be recorded in the published Report on the Examination. 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. • CIE will not enter into discussion or correspondence in connection with these mark schemes. CIE is publishing the mark schemes for the June 2005 question papers for most IGCSE and GCE Advanced Level and Advanced Subsidiary Level syllabuses and some Ordinary Level syllabuses’.

Grade thresholds for Syllabus 9709/0390 (Mathematics) in the June 2005 examination. minimum mark required for grade: maximum mark available A B E Component 6 50 39 35 20 The thresholds (minimum marks) for Grades C and D are normally set by dividing the mark range between the B and the E thresholds into three. For example, if the difference between the B and the E threshold is 24 marks, the C threshold is set 8 marks below the B threshold and the D threshold is set another 8 marks down. If dividing the interval by three results in a fraction of a mark, then the threshold is normally rounded down.

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.

JUNE 2005 GCE A, AS LEVEL and AICE MARK SCHEME MAXIMUM MARK: 50 SYLLABUS/COMPONENT: 9709/06, 0390/06 MATHEMATICS (Probability and Statistics 1)

Page 1 Mark Scheme Syllabus Paper A AND AS LEVEL, AICE – JUNE 2005 9709/0390 6 © University of Cambridge International Examinations 2005 1 µ = 160, σ 2 = 96 P(≤165) = Φ       − 96 160 5. 164 = Φ (0.4593) = 0.677 B1 M1 M1 M1 A1 [5] For 160 and 96 seen or implied by 9.798 For standardising, must have square root For continuity correction, either 165.5 or 164.5 For using tables and finding correct area (i.e.> 0.5) For correct answer 2 (i) 5 × 2 + 15f + 30 × 11 + 60 × 4 = 27.5(17 + f) f = 9 total = 26 AG (ii) σ = 16.1 M1 M1 A1 A1 [4] M1 A1 [2] For attempt at LHS, accept end points or cl width For attempt at RHS, must have 17+ f For correct f For correct answer given, ft if previous answer rounds to 9 For method including sq rt and mean squared (can be implied if using calculator, must be x2f on mid-points) or ∑ − 26 ) ( 2 x x f For correct answer 3 (i) P(G, G, G, G, NG) = (0.25)4×(0.75)1 × 5C4 = 0.0146 AG (ii) X 0 1 2 P(X = x) 0.2373 0.3955 0.2637 (cont) X 3 4 5 P(X = x) 0.0879 0.0146 0.0010 M1 A1 [2] B1 B1 B1 B1 B1 [5] For relevant binomial calculation, need 5Cr or 5 or all 5 options For correct answer. AG For all correct X values For one correct prob excluding P( X = 4) For 2 correct probs excluding P( X = 4) For 3 correct probs excluding P( X = 4) All correct and in decimals

Page 2 Mark Scheme Syllabus Paper A AND AS LEVEL, AICE – JUNE 2005 9709/0390 6 © University of Cambridge International Examinations 2005 4 (i) shows all the data (ii) Not exercise LQ = 5.4 Median = 6.5 UQ = 8.3 (iii) B1 [1] B1 B1ft B1ft [3] B1 B1ft B1 B1 [4] Or other suitable advantage e.g. can see the shape, mode etc. ft on first answer missing the decimal point For one linear numbered scale from 3 to 9.5, or two identically positioned scales For not exercise all correct on linear scale For exercise correct on linear scale For two labels and cholesterol and scale labelled SR non linear scale max B0 B0 B0 B1 SR no graph paper lose one mark 5 (i) 618/1281 (0.482) (ii) 412/1281 (0.322) or tree diagram options (iii) P(E) = 717/1281 Their (i) × their P(E) ≠ their (ii) Not independent (iv) 358/564 (0.635) or (0.279/0.440) B1 B1 [2] B1ft [1] M1 M1dep A1ft [3] B1 B1 [2] For correct numerator For correct denominator Follow through on their denominator if p <1 or 2/3 × their (i) For attempting to find P(E) For showing they know what independence means, mathematically ft on their (i) × their P(E) ≠ their (ii) For correct numerator, 0.28 gets B0 with PA For correct denominator 3 4 5 6 7 8 9 10 not ex ex

Page 3 Mark Scheme Syllabus Paper A AND AS LEVEL, AICE – JUNE 2005 9709/0390 6 © University of Cambridge International Examinations 2005 6 (i) z1 = 0.02/0.15 = 0.1333 z2 = - 0.08/0.15 = − 0.5333 area= Φ (0.1333) − Φ (−0.533) = Φ (0.1333) − [1 - Φ (0.5333)] = 0.5529 + 0.7029 − 1 = 0.256 Prob all 4 = (0.256)4 (0.00428 to 0.00430) (ii) z = ± 1.282 or 1.28 or 1.281 15 .0 282 .1 b = ± limits between 1.71 and 2.09 M1 M1 M1 A1 A1ft [5] B1 M1 A1ft [3] For standardising one value, no cc For standardising the other value, no cc. SR ft on no sq rt For finding correct area (i.e. two Φ s - 1) For correct answer For correct answer, ft from their (i), if p<1, allow 0.0043 For correct z, + or - or both For seeing an equation involving + or - of their z, b, 0.15 (their z can only be 0.842 or 0.84 or 0.841) both limits needed, ft 1.77 to 2.03 on 0.842 only 7 (a) (i) 3C1 × 5C1 = 15 (ii) 5C1 × 6C2 = 75 (b) (i) 9!/2!2! = 90720 (ii) 5! Or 5P5 = 120 M1 B1 [2] M1 A1 [2] M1 A1 [2] B1 B1 [2] For multiplying two combinations together For correct answer For seeing 6C2, or separating it into three alternatives either added or multiplied For correct answer For dividing by 2! twice For correct answer 5! seen in a numerator For correct final answer