Statistics Unit Statistics 1B

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1 Centre Number Candidate Number F Examiner s Use Surname Other Names Candidate Signature Examiner s Initials Mathematics Unit Statistics 1B Statistics Unit Statistics 1B General Certificate of Education Advanced Subsidiary Examination MS/SS1B Specimen paper f examinations in June 2010 onwards Question TOTAL Mark F this paper you must have: the blue AQA booklet of fmulae and statistical tables. You may use a graphics calculat. Time allowed 1 hour 30 minutes Instructions Use black ink black ball-point pen. Pencil should only be used f drawing. Fill in the boxes at the top of this page. Answer all questions. Write the question part reference (eg (a), (b)(i) etc) in the left-hand margin. You must answer the questions in the space provided. Do not write outside the box around each page. Show all necessary wking; otherwise marks f method may be lost. Do all rough wk in this book. Cross through any wk that you do not want to be marked. The final answer to questions requiring the use of tables calculats should nmally be given to three significant figures. Infmation The marks f questions are shown in brackets. The maximum mark f this paper is 75. Unit Statistics 1B has a written paper only. Advice Unless stated otherwise, you may quote fmulae, without proof, from the booklet. MS/SS1B

2 2 Do not write outside the box Answer all questions in the spaces provided. 1 A large bookcase contains two types of book: hardback and paperback. The number of books of each type in each of four subject categies is shown in the table. Subject categy Crime Romance Science fiction Thriller Total Type Hardback Paperback Total (a) A book is selected at random from the bookcase. Calculate the probability that the book is: (i) a paperback; (1 mark) (ii) not science fiction; (2 marks) (iii) science fiction a hardback; (iv) a thriller, given that it is a paperback. (2 marks) (2 marks) (b) Three books are selected at random, without replacement, from the bookcase. Calculate, to three decimal places, the probability that one is crime, one is romance and one is science fiction. (4 marks) QUESTION PART REFERENCE (02)

3 4 Do not write outside the box 2 Hermione, who is studying reptiles, measures the length, x cm, and the weight, y grams, of a sample of 11 adult snakes of the same type. Her results are shown in the table. Snake A B C D E F G H I J K x y (a) Calculate the value of the product moment crelation coefficient, r, between x and y. (3 marks) (b) Interpret your value in context. (2 marks) (c) Complete the scatter diagram, opposite, f these data. (2 marks) (d) Subsequently it is found that, of the 11 adult snakes, 9 are male and 2 are female. (i) (ii) Given that female adult snakes are generally larger than male adult snakes, identify the 2 snakes which are most likely to be female. (1 mark) Hence, without further calculation, estimate the value of r f the 9 male snakes and revise, as necessary, your interpretation in part (b). (2 marks) QUESTION PART REFERENCE (04)

4 5 Do not write outside the box QUESTION PART REFERENCE (c) y ~ 90 D Weight (grams) 60 A E C F 50 B Length (cm) ~ x Turn over (05) s

5 6 Do not write outside the box 3 The weight, X grams, of talcum powder in a tin may be modelled by a nmal distribution with mean 253 and standard deviation s. (a) Given that s ¼ 5, determine: (i) PðX<250Þ ; (3 marks) (ii) Pð245 < X < 250Þ ; (2 marks) (iii) PðX ¼ 245Þ. (1 mark) (b) Assuming that the value of the mean remains unchanged, determine the value of s necessary to ensure that 98% of tins contain me than 245 grams of talcum powder. (4 marks) QUESTION PART REFERENCE (06)

6 8 Do not write outside the box 4 As part of an investigation, a chline block is immersed in a large tank of water held at a constant temperature. The block slowly dissolves, and its weight, y grams, is noted x days after immersion. The results are shown in the table. x days y grams (a) Calculate the equation of the least squares regression line of y on x. (4 marks) (b) Hence estimate, to the nearest gram, the initial weight of the block. (1 mark) (c) A company which markets the chline blocks claims that a block will usually dissolve completely after about 13 weeks. Comment, with justification, on this claim. (3 marks) QUESTION PART REFERENCE (08)

7 10 Do not write outside the box 5 A survey of all the households on an estate is undertaken to provide infmation on the number of children per household. The results, f the 99 households with children, are shown in the table. Number of children (x) Number of households ( f ) (a) F these 99 households, calculate values f: (i) the median and the interquartile range; (3 marks) (ii) the mean and the standard deviation. (3 marks) (b) In fact, 163 households were surveyed, of which 64 contained no children. (i) (ii) F all 163 households, calculate the value f the mean number of children per household. (2 marks) State whether the value f the standard deviation, when calculated f all 163 households, will be smaller than, the same as, greater than that calculated in part (a)(ii). (1 mark) (iii) It is claimed that, f all 163 households on the estate, the number of children per household may be modelled approximately by a nmal distribution. Comment, with justification, on this claim. Your comment should refer to a fact other than the discrete nature of the data. (2 marks) QUESTION PART REFERENCE (10)

8 12 Do not write outside the box 6 (a) The time taken, in minutes, by Domesat to install a domestic satellite system may be modelled by a nmal distribution with unknown mean, m, and standard deviation 15. The times taken, in minutes, f a random sample of 10 installations are as follows Construct a 98% confidence interval f m. (5 marks) (b) The time taken, Y minutes, by Teleair to erect a TV aerial and then connect it to a TV is known to have a mean of 108 and a standard deviation of 28. Estimate the probability that the mean of a random sample of 40 observations of Y is me than 120. (4 marks) (c) Indicate, with a reason, where, if at all, in this question you made use of the Central Limit Theem. (2 marks) QUESTION PART REFERENCE (12)

9 14 Do not write outside the box 7 Mr Alott and Miss Fewer wk in a postal sting office. (a) The number of letters per batch, R, sted increctly by Mr Alott when sting batches of 50 letters may be modelled by the distribution Bð50, 0:15Þ. Determine: (i) PðR<10Þ ; (ii) Pð5 4 R 4 10Þ. (4 marks) (b) It is assumed that the probability that Miss Fewer sts a letter increctly is 0.06, and that her sting of a letter increctly is independent from letter to letter. (i) (ii) Calculate the probability that, when sting a batch of 22 letters, Miss Fewer sts exactly 2 letters increctly. (3 marks) Calculate the probability that, when sting a batch of 35 letters, Miss Fewer sts at least 1 letter increctly. (2 marks) (iii) Calculate the mean and the variance f the number of letters sted crectly by Miss Fewer when she sts a batch of 120 letters. (2 marks) (iv) Miss Fewer sts a random sample of 20 batches, each containing 120 letters. The number of letters sted crectly per batch has a mean of and a variance of Comment on the assumptions that the probability that Miss Fewer sts a letter increctly is 0.06, and that her sting of a letter increctly is independent from letter to letter. (3 marks) QUESTION PART REFERENCE (14)

10 Surname Other Names Centre Number Candidate Number Candidate Signature General Certificate of Education June 2009 Advanced Subsidiary Examination MATHEMATICS Unit Statistics 1B MS/SS1B STATISTICS Unit Statistics 1B Insert Insert f use in Question 2. Fill in the boxes at the top of this page. Fasten this insert securely to your answer book. Turn over f Figure 1 P15525/Jun09/MS/SS1B 6/6/6/ Turn over s

11 2 Figure 1 (f use in Question 2) Lengths and Weights of Snakes y ~ 90 D Weight (grams) 60 E A C F 50 B Length (cm) ~ x Copyright Ó 2009 AQA and its licenss. All rights reserved. P15525/Jun09/MS/SS1B

12 Version : klm General Certificate of Education Mathematics 6360 Statistics 6380 MS/SS1B/W Statistics 1B Mark Scheme 2009 examination - June series

13 Mark schemes are prepared by the Principal Examiner and considered, together with the relevant questions, by a panel of subject teachers. This mark scheme includes any amendments made at the standardisation meeting attended by all examiners and is the scheme which was used by them in this examination. The standardisation meeting ensures that the mark scheme covers the candidates responses to questions and that every examiner understands and applies it in the same crect way. As preparation f the standardisation meeting each examiner analyses a number of candidates scripts: alternative answers not already covered by the mark scheme are discussed at the meeting and legislated f. If, after this meeting, examiners encounter unusual answers which have not been discussed at the meeting they are required to refer these to the Principal Examiner. It must be stressed that a mark scheme is a wking document, in many cases further developed and expanded on the basis of candidates reactions to a particular paper. Assumptions about future mark schemes on the basis of one year s document should be avoided; whilst the guiding principles of assessment remain constant, details will change, depending on the content of a particular examination paper. Further copies of this Mark Scheme are available to download from the AQA Website: Copyright 2009 AQA and its licenss. All rights reserved. COPYRIGHT AQA retains the copyright on all its publications. However, registered centres f AQA are permitted to copy material from this booklet f their own internal use, with the following imptant exception: AQA cannot give permission to centres to photocopy any material that is acknowledged to a third party even f internal use within the centre. Set and published by the Assessment and Qualifications Alliance. The Assessment and Qualifications Alliance (AQA) is a company limited by guarantee registered in England and Wales (company number ) and a registered charity (registered charity number ). Registered address: AQA, Devas Street, Manchester M15 6EX Dr Michael Cresswell Direct General

14 MS/SS1B/W - AQA GCE Mark Scheme 2009 June series Key to mark scheme and abbreviations used in marking M m dm A B E mark is f method mark is dependent on one me M marks and is f method mark is dependent on M m marks and is f accuracy mark is independent of M m marks and is f method and accuracy mark is f explanation ft F follow through from previous increct result MC mis-copy CAO crect answer only MR mis-read CSO crect solution only RA required accuracy AWFW anything which falls within FW further wk AWRT anything which rounds to ISW igne subsequent wk ACF any crect fm FIW from increct wk AG answer given BOD given benefit of doubt SC special case WR wk replaced by candidate OE equivalent FB fmulae book A2,1 2 1 ( 0) accuracy marks NOS not on scheme x EE deduct x marks f each err G graph NMS no method shown c candidate PI possibly implied sf significant figure(s) SCA substantially crect approach dp decimal place(s) No Method Shown Where the question specifically requires a particular method to be used, we must usually see evidence of use of this method f any marks to be awarded. However, there are situations in some units where part marks would be appropriate, particularly when similar techniques are involved. Your Principal Examiner will alert you to these and details will be provided on the mark scheme. Where the answer can be reasonably obtained without showing wking and it is very unlikely that the crect answer can be obtained by using an increct method, we must award full marks. However, the obvious penalty to candidates showing no wking is that increct answers, however close, earn no marks. Where a question asks the candidate to state write down a result, no method need be shown f full marks. Where the permitted calculat has functions which reasonably allow the solution of the question directly, the crect answer without wking earns full marks, unless it is given to less than the degree of accuracy accepted in the mark scheme, when it gains no marks. Otherwise we require evidence of a crect method f any marks to be awarded. 3

15 MS/SS1B/W - AQA GCE Mark Scheme 2009 June series MS/SS1B/W Q Solution Marks Total Comments 1(a) In (a), ratios (eg 100:160) are only penalised by 1 mark at first crect answer (i) P(P) = 100/160 = 50/80 = 25/40 = 10/16 = 5/8 = B1 1 CAO (ii) P(S ) = 32 1 P(S) = M1 Or equivalent Igne labels of S & S Can be implied by crect answer = 128/160 = 64/80 = 32/40 = 16/20 = 8/10 = 4/5 = 0.8 A1 2 CAO (iii) P(S H) = P(S H) = Or equivalent M1 Can be implied by crect answer = 74/160 = 37/80 = to A1 2 CAO/AWFW (0.4625) (iv) P(T P) = (i) M1 Or equivalent Can be implied by crect answer But watch f (b) P(1C & 1R & 1S) = = 3/100 = 3/10 = 0.3 A1 2 CAO M1 M1 Multiplication of any 3 different given subject totals Multiplication of 160, 159 & 158 ( ) Accept 3dp accuracy 6 M1 Award f 3 multiplier 6 = to A1 AWFW (0.0642) Do not accept a fraction as answer A crect answer can imply 4 marks Special Case: (Any given subject total) 160 seen anywhere in (b) (M1) 4 Total 11 Can award if no marks sced in (b) Accept a decimal equivalent 4

16 MS/SS1B/W - AQA GCE Mark Scheme 2009 June series MS/SS1B/W (cont) Q Solution Marks Total Comments 2(a) r = to B3 AWFW ( ) r = 0.89 to (B2) AWFW r = 0.8 to 0.95 (B1) AWFW Attempt at Attempt at x xy 2 x y S xx S yy & S xy 2 y & (M1) & (all 5 attempted) & 1661 (all 3 attempted) Attempt at crect cresponding fmula f r (m1) r = to (A1) 3 AWFW (b) Fairly strong / strong / very strong positive (linear) crelation / relationship / association / link (but not trend) B1dep Or equivalent; must qualify strength and indicate positive Dependant on 0.8 r 0.95 B0 f some/average/medium/etc between length and weight of adult snakes B1 2 Context; providing 0 < r < 1 (c) Figure 1: 5 crect labelled points 4 3 crect labelled points B2 (B1) 2 Deduct 1 mark if points not labelled (d)(i) D and G B1 1 Both CAO (ii) r = 0.25 to 0.75 B1 AWFW ( ) No penalty f calculation Accept a range only if whole of it falls within 0.25 to 0.75 Fairly weak / weak / some / moderate positive (linear) crelation / relationship / association / link Do not accept comparison with value in (a) statement in (b) B1dep 2 Total 10 Or equivalent; must qualify strength and indicate positive Dependant on 0.25 r 0.75 B0 f very weak/little/slight/hardly any/fair/average/medium/anything involving strong/etc 5

17 MS/SS1B/W - AQA GCE Mark Scheme 2009 June series MS/SS1B/W (cont) Q Solution Marks Total Comments 3(a) X ~ N(253, 5 2 ) (i) P( X < 250) = P Z < 5 = M1 ( Z <. ) = 1 P( Z < 0. 6) P 0 6 Standardising (249.5, ) with 253 and ( 5, ) and/ (253 x) = m1 Area change; may be implied (ii) (iii) ( < X < ) = [ C's(a)(i) ] P( 245) P = (i) P( Z 1. 6) = to A1 3 < = X < M1 = to 0.22(0) A1 2 P( X = 245) = 0 zero impossible B1 1 AWFW ( ) (1 answer) M1 max Or equivalent; must be clear crect method if answer increct and answer > 0 AWFW ( ) M1 A0 f [1 (i)] = M0 A0 f [(i)] = M1 A1 f [1 (i)] = Igne any wking B0 f f impossible to calculate (b) 98% (0.98) z = 2.05 to 2.06 B1 AWFW; igne sign (2.0537) z = σ M1 Standardising 245 with 253 and σ ; allow ( ) = A1 Only allow: ±2.05 to ±2.06 ±2.32 to ±2.33 Note: σ σ = 3.88 to 3.9(0) A1 AWFW (3.8954) = σ = B1 M1 A1 A0 4 Or equivalent inconsistent signs Total 10 6

18 MS/SS1B/W - AQA GCE Mark Scheme 2009 June series MS/SS1B/W (cont) Q Solution Marks Total Comments 4(a) b (gradient) = to B2 AWFW ( ) b (gradient) = 0.55 to 0.54 (B1) AWFW Omission of ve sign (B0) (b) Attempt at Attempt at a (intercept) = 49.7 to 49.9 B2 AWFW ( ) a (intercept) = 49 to 50 (B1) AWFW x 2 x y & S xx & S xy 2 xy ( y ) (M1) Attempt at crect fmula f b (gradient) (m1) b (gradient) = to (A1) AWFW a (intercept) = 49.7 to 49.9 (A1) 4 AWFW Accept a & b interchanged only if identified crectly by a clearly shown equation (stated answers are not sufficient) in (b) (c) C s value of intercept from (a) providing > 47 Value 50 stated even if (a) increct not attempted B1F & 6980 (10173) (all 4 attempted) & (both attempted) If a and b not identified anywhere in question, then: to B to 49.9 B1 Accept value rounded to nearest integer (50) (c) 13 weeks 91 days B1 Stated used Accept a descriptive answer that includes 91 and a value in range y = 1.1 to B1 AWFW ( ) y = 0 x = 89 to 93 (B1) AWFW ( ) Accept a descriptive answer that includes a value in range and weeks (approximately) (B1) Stated Note: B1 B1 (B1) (B1) are available even if (a) not attempted Thus claim appears justified Thus tablet likely to have dissolved B1 dep Or equivalent; igne reasoning unless contradicty Dependent upon 2 nd B1 in (c) 2 nd (B1) in (c) Extrapolation required so cannot comment (B1) Not dependent Note: If (B1) f extrapolation maximum mark is 2; other mark available is f

19 MS/SS1B/W - AQA GCE Mark Scheme 2009 June series Total 8 8

20 MS/SS1B/W - AQA GCE Mark Scheme 2009 June series MS/SS1B/W (cont) Q Solution Marks Total Comments 5(a) (i) Median (50) = 3 B1 CAO If not identified, then assume der is median then IQR IQR (75 25) = 4 2 = 2 B2 Do not award marks if crect answers are based on shown increct method; eg accept use of 99/2, etc but not 276/2, etc CAO; but 25 th value IQR = 2 B0 Special Cases: Identification that LQ = 2 and UQ = 4 (B1) Both CAO Statement of 4 cumulative frequencies F: (M1) 3 Can award if no marks sced in (i) even if then applied to continuous data (ii) Mean = fx f = = 2.77 to 2.78 B1 AWFW (2.778) If not identified, assume der is x then s Treat rounding to integers as ISW fx 2 = 933 = 1.3(0) to 1.32 B2 AWFW (1.307 & 1.314) SD ( ) Special Case: fx (M1) 3 Can award if no marks sced in (ii) Evidence of 99 (b)(i) 99 Mean99 fx from(a)(ii) Mean163 = M1 Or equivalent; may be implied by an answer within range = 1.68 to 1.69 A1 2 AWFW (1.687) (ii) Increase (iii) Data is (positively/negatively) skewed / not symmetric / bimodal / not bell-shaped from frequency distribution / given table [C s mean in (b)(i)] 2 [C s SD in (a)(ii)] < 0 [C s mean in (b)(i)] 2 [1.69 to 1.71] < 0 B1 B1 Thus claim appears not valid B1 dep 2 1 CAO; equivalent (1.696) Igne any wking (1.702) Or equivalent (1.75 to 0.90) Or equivalent Dependent upon previous B1 Total 11 9

21 MS/SS1B/W - AQA GCE Mark Scheme 2009 June series MS/SS1B/W (cont) Q Solution Marks Total Comments 6(a) Mean = = 47 B1 CAO 98% (0.98) z = 2.32 to 2.33 B1 AWFW (2.3263) σ CI f μ is x ± z M1 n Used Must have n with n > 1 Thus ± A1F F on x and z only 10 Hence 47 ± 11.0 to 11.1 Or (35.9 to 36.0, 58.0 to 58.1) (b) Y ~ N(108, 28 2 ) Variance of Y 40 = 28 2 /40 = 19.6 SD of Y 40 = = to 4.43 A1 5 B1 CAO & AWRT (accept 11) AWRT (accept 36 & 58) CAO Stated used AWFW ( 40 ) P Y > 120 = P Z > M1 Standardising 120 with 108 and (4.425 to 4.43) equivalent; allow ( ) = P( Z > 2. 71) = 1 P( Z 271. ) < m1 Area change; may be implied = = to A1 4 AWFW ( ) (1 answer) B1 M1 max (c) Part (b) Teleair times B1 Or equivalent; igne reasoning Distribution of Y not known Note: To sce B1 B1 there must be both a clear indication of where in question and a valid reason B1 2 Total 11 Or equivalent; must be clear reference to Y population B0 f n > 30 Any reference to part (a) B0 B0 10

22 MS/SS1B/W - AQA GCE Mark Scheme 2009 June series MS/SS1B/W (cont) Q Solution Marks Total Comments 7(a) R ~ B(50, 0.15) (i) P(R < 10) = B1 AWRT (0.7911) (ii) P(5 R 10) = (p 1 ) M1 Accept 3 dp accuracy (1 p 2 ) p 1 M0 M0 A0 p 1 (1 p 2 ) M1 M0 A0 only providing result > 0 minus (p 2 ) M1 Accept 3 dp accuracy = A1 AWRT (0.7680) B(50, 0.15) expressions stated f at least 3 terms within 4 R 10 gives probability = (M1) (A2) 4 Can be implied by crect answer AWRT (b) Confusion of 22, 35, 120 and/ 0.15, 0.06 Do not treat as misreads (i) S ~ B(22, 0.06) M1 Used in (b)(i) as evidenced by any crect binominal term f S > Can be implied by crect answer A1 2 Igne any additional terms = 0.24 to A1 3 AWFW ( ) P(S = 2) = (. ) (. ) (ii) P(S 1) = 1 M1 (B1) 35 q where q Can be implied by crect answer Award f (0.94) 35 seen in an expression but not if accompanied by a multiplier 1 = to 0.89 A1 2 AWFW ( ) (iii) Mean = np = = B1 Either 2 If not identified, assume der is μ then σ Variance = np(1 p) Must clearly state variance value = = 6.76 to 6.78 B1 2 AWFW (6.768) (iv) Means are (approximately) the same stated Variances are (very) different stated B1 Must have sced 1 st B1 in (iii) Must have sced 2 nd B1 in (iii) Agree with P(sts letter increctly) = 0.06 B1 dep Dependent on means same stated Disagree with independent from letter to letter B1 dep 3 Total 14 Paper 75 Dependent on variances different stated 11

23 12 MS/SS1B/W - AQA GCE Mark Scheme 2009 June series

24 Scaled mark component grade boundaries - June 2009 exams klm GCE (legacy) Component Maximum Scaled Mark Grade Boundaries Code Component Title Scaled Mark A B C D E MS03 MATHEMATICS UNIT MS MS04 MATHEMATICS UNIT MS MS/SS1A/W MATHEMATICS UNIT MS1A WRITTEN MS/SS1A/C MATHEMATICS UNIT MS1A CWK MS1B MATHEMATICS UNIT MS1B MS2B MATHEMATICS UNIT MS2B XMCA2 MATHEMATICS UNIT XMCA XMCAS MATHEMATICS UNIT XMCAS MED1 MEDIA STUDIES UNIT MED2 MEDIA STUDIES UNIT MED3 MEDIA STUDIES UNIT MED4 MEDIA STUDIES UNIT MED5 MEDIA STUDIES UNIT MED6 MEDIA STUDIES UNIT HEB1 MODERN HEBREW UNIT HEB2 MODERN HEBREW UNIT MUS1 MUSIC UNIT MUS2 MUSIC UNIT MUS3 MUSIC UNIT MUS4 MUSIC UNIT MUS5 MUSIC UNIT MUS6 MUSIC UNIT PAN1 PANJABI UNIT PAN2 PANJABI UNIT