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2 Version 1.1: 0607 abc General Certificate of Education Mathematics 6360 Statistics 6380 MS/SS1B Statistics 1B Mark Scheme 006 examination - January series 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. Copyright 006 AQA and its licenss. All rights reserved.

3 MS/SS1B AQA GCE Mark Scheme, 006 January 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 A,1 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.

4 AQA GCE Mark Scheme, 006 January series MS/SS1B MS/SS1B 1(a) Gradient, b = to B AWFW b = 0.88 to 0.89 () AWFW Intercept, a =.31 to.33 B AWFW a =.3 () AWRT Attempt at Σx Σx Σy Σxy () 7, 64, 87, 70 Attempt at S xx S xy 105.6, 93.6 Attempt at a crect fmula f b (m1) b = to (A1) AWFW a =.31 to.33 (A1) AWFW Accept a & b interchanged only if y = ax+ b stated subsequently used crectly in either (b) (c) 4 (b) a: average waiting time of.3 minutes (139 seconds) when entering empty restaurant OE; accept minimum waiting time b: average increase in waiting time of minutes (53 seconds) f each customer in restaurant on entry OE (c) Use of y = a + 5b y = a + 5b (i) (ii) F x = 5 y = 6.6 to 6.8 F x = 5 y = 4.3 to 4.6 A1 Both; AWFW (d)(i) Reliable as interpolation and small residuals Reliable as interpolation but large percentage residuals so inconclusive Large percentage residuals so unreliable () () () Within range OE OE (ii) Unreliable as extrapolation 3 Outside range OE Total 11 3

5 MS/SS1B AQA GCE Mark Scheme, 006 January series MS/SS1B (cont) (a) P(X) = 0.3 P(Y) = 0.4 P(Z) = 0. (i) P(X Y Z) = = (ii) P(X' Y' Z') = At least crect terms = A1 CAO (iii) P(X' Y' Z) = Crect numerical expression (b) P(W Z) = 0.9 P(W Z') = 0.5 = A1 CAO (i) P(Z W) = Crect numerical expression = 0.18 A1 CAO (ii) P((Z W') (Z W)) 1 [P((Z W) (Z W'))] = 0. (1 0.9) (b)(i) + (1 0.) 0.5 (1 0.) (1 0.5) Cannot sce an in both methods = ( ) = 0. A1 3 CAO Total 11 4

6 AQA GCE Mark Scheme, 006 January series MS/SS1B MS/SS1B (cont) 3(a) Mean = 86.5 = 5.73 CAO Standard deviation = = 0.95 to AWFW (b) 99% z =.57 to.58 AWFW.5758 CI f µ is x ( σ s) ± z n Use of Must have ( n ) with n > 1 Thus 5.73 ±.5758 ( 0.95 to 0.961) 50 A1 on z and s > 0 but not on x Accept only f n 5.73 ± (0.34 to 0.36)! Dependent 5.37 to 5.39, 6.07 to 6.09) A1 4 AWFW (c) CI excludes both values of 5 and 6½ on (b); OE so! Dependent Neither claim appears valid on (b); OE CI excludes 5 so claim not valid ( ) on (b); OE and CI excludes 6½ so claim not valid ( ) on (b); OE Total 8 5

7 MS/SS1B AQA GCE Mark Scheme, 006 January series MS/SS1B (cont) 4(a) fx = 805 fx = Mean ( x ) = 80. to 80.3 B AWFW 80.5 Standard Deviation (s n, s n-1 ) = 30.9 to 31. B AWFW MPs (x): 5, 35, 50, 70, 90, 110, 135, 165 () At least 4 crect Mean ( ) fx x = 100 () 4 Use of (b)(i) Large (n > 30) sample Central Limit Theem 1 OE (ii) Mean ( Y ) = 80. to 80.3 on (a) Standard err ( Y ) 30.9 to 31. = 36 = 5.1 to 5.5 s > 0 in (a) 36 6 (iii) P( Y < 90) = ( ) ( 5.1 to 5.5) to 80.3 P Z < Standardising 90 Using values from (b)(ii) with s 36 > 0 s 100 > 0 = P(Z < 1.84 to 1.93) = to A1 3 AWFW Total 10 6

8 AQA GCE Mark Scheme, 006 January series MS/SS1B MS/SS1B (cont) 5(a) Scatter Diagram B 4 labelled points plotted () 3 labelled points plotted () 4 unlabelled points plotted (b)(i) Positive/linear crelation/relationship OE except f two unusual values/results OE (ii) CAO; accept 3 rd /final/last value (c) C and D CAO C is likely freestyle champion D is likely backstroke champion Style identified C is likely freestyle champion D is likely backstroke champion () () (d)(i) r = 0.91 to B3 AWFW r = 0.91 to to 0.47 B AWFW r = 0.9 AWRT Attempt at Σx Σx Σy Σy Σxy 70.4, , Attempt at S xx S yy S xy () 48.94, 67.5, 5.45 Attempt at a crect fmula f r (m1) r = 0.91 to A1 3 AWFW (ii) Boys are faster/slower at both strokes Boys are equally good at both strokes 1 Total 11 OE;do not accept freestyle times are proptional to backstroke times 7

9 MS/SS1B AQA GCE Mark Scheme, 006 January series Question 5(a) Swimming Times I H 38.0 C G J F Backstroke ( y seconds) A E B D Freestyle ( x seconds) (a) Scatter Diagram 4 labelled points plotted B 3 labelled points plotted () 4 unlabelled points plotted () Graph = 8

10 AQA GCE Mark Scheme, 006 January series MS/SS1B MS/SS1B (cont) 6(a)(i) B(50, 0.) Use of in (a) P(R 15) = to 0.97 A1 AWFW (ii) P(R = 10) = P(R 10) P(R 9) P(R = 10) = ( 0. ) ( 0.8 ) 10 Stated implied Stated implied = = to A1 AWFW (iii) P(5 < R < 15) = P(R 14 15) = Accept values to 3 dp minus P(R 5 4) = Accept values to 3 dp = 0.89 to A1 AWFW B(50, 0.) expressions stated f at least 3 () of 5 R 15 Answer (A) 3 (b) Mean, µ = np = = 10 Estimate of p, ˆp = 0.1 Variance, np( 1 p) σ = = = 8 CAO Or implied by a crect answer Either; CAO Mean Estimate of p is similar to that expected 10.5 and and 0. but Either point Variance (standard deviation) is different from that expected 0.41 and and.8 Reason to doubt validity of Sly s claim 4 Total 11 Must be based on both and 8 on both and.8 crectly 9

11 MS/SS1B AQA GCE Mark Scheme, 006 January series MS/SS1B (cont) 7 (a) Weight, X ~ N(406, 4. ) (i) P(X < 400) = P Z < 4. = P(Z < 1.48 to 1.43) = 1 P(Z < 1.48 to 1.43) m1 Φ( z) = 1 Φ(z) Standardising (399.5, ) with 406 and ( 4., ) and/ (406 x) = to A1 3 AWRT (ii) P(40.5 < X < 407.5) = P(X < 407.5) P(X < 40.5) = Difference OE P(Z < 0.36) P(Z < 0.83) B,1 AWRT; igning signs = ( ) = to 0.44 A1 4 AWFW (b)(i) z = 1.96 Accept explanation in wds 310 µ P(Y < 310) = P Z < σ x = µ + / ± zσ Standardising 310 using µ and σ Accept in wds Thus 310 µ = 1.96 result σ 310 = µ σ result NB: Wking backwards from given equation at most M0 mo m1 3 Equating AG Substitution (ii) 0.86 z = 1.08 AWRT µ = 1.96σ µ = 1.08σ.5 = 0.88σ Attempt at solving equations each of fm x µ = zσ σ =.84 to.84 A1 AWFW.841 µ = to A1 4 AWFW Total 14 TOTAL 75 10

12 Version 1.0: 0606 abc General Certificate of Education Mathematics 6360 MS1B Statistics 1B Mark Scheme 006 examination - June series 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. Copyright 006 AQA and its licenss. All rights reserved.

13 MS1B AQA GCE Mark Scheme, 006 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 A,1 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.

14 AQA GCE Mark Scheme, 006 June series MS1B MS1B 1(a)(i) r = to B3 AWFW r = 0.14 to B AWFW r = 0.1 to 0. AWRT Attempt at Σx Σx Σy Σy Σxy 395, , Attempt at S xx S yy S xy 75994, , 99.5 Attempt at a crect fmula f r m1 r = to A1 3 AWFW (ii) Little/weak/no crelation/relationship/association between number of pages and (retail) price equivalent; but not po context (iii) Size (page, thickness), auth, ranking, publicity/marketing, cover design, recommendations on back, publisher, font, popularity, quality, print-run, etc 1 any sensible variable but not pictures, coloured pictures, age, wds, weight, mass (b) (Very) strong/almost exact positive/perfect crelation/relationship/ association between number of pages and sale/new price equivalent context Sale price appears to be determined by number of pages B equivalent Total 8 3

15 MS1B AQA GCE Mark Scheme, 006 June series MS1B (cont) (a) Height, X ~ N(185, 10 ) (i) P(X < 00) = P Z < 10 standardising (199.5, ) with 185 and ( 10, ) and/ (185 x) = P(Z < 1.5) A1 CAO; igne sign = Φ(1.5) = A1 3 AWRT ( ) (ii) P(X > 175) = P Z > 10 standardising (174.5, ) with 185 and ( 10, ) and/ (185 x) = P(Z > 1) = P(Z < 1) m1 area change = A1 3 AWRT ( ) (iii) P(175 < X < 00) = (i) [1 (ii)] equivalent = [ ] = to A1 AWFW ( ) on (i) and (ii) providing > 0 (b) Mean of X = 185 CAO; may be implied by use in standardising 10 Variance of X = 4 = 5 CAO; equivalent P( X > 190) = P Z > standardising 190 with 185 and 5 and/ 5 ( ) = P(Z > 1) = 1 Φ(1) = A1 4 AWRT ( ) on (a)(ii) if used Total 1 4

16 AQA GCE Mark Scheme, 006 June series MS1B MS1B (cont) 3(a)(i) Gradient, b = 3.4 to 3.6 B AWFW ( 3.5) b = 3. to 3.3 AWFW Intercept, a = 6 to 64 B AWFW ( 6.88! ) a = 60 to 70 AWFW Attempt at Σx Σx Σy Σxy 108, 1836, 015, 45 Attempt at S xx S xy 540, 1755 Attempt at a crect fmula f b m1 b = 3.4 to 3.6 A1 AWRT a = 6 to 64 A1 4 AWFW Accept a & b interchanged only if identified crectly in (b) and (c) (ii) Gradient, b: Decrease in pressure per month B equivalent Change in pressure better (iii) Intercept, a: Initial pressure pressure at x = 0 equivalent; not y-intercept Reference to 65, actual expected value (b)(i) Value f b = [gradient b from accept b; igne sign (a)(i)] = 6.4 to 6.6 A1 AWFW ( 6.5) from (a)(i) but must be < 0 (ii) P 8 = must use 65 and x = 8 and [b (< 0) from (a)(i)] [from (b)(i) (< 0)] = 1 to 14 A1 AWFW (< 0) AG Total 1 5

17 MS1B AQA GCE Mark Scheme, 006 June series MS1B (cont) 4(a)(i) Mean, x = 505. CAO; stated implied 99% z =.57 to.58 99% t = 3.5 (Knowledge of the t-distribution is not required in this unit) σ CI f µ is x ( z t) n AWFW (.5758) AWRT (3.50) ( s) ± use of; must have ( n ) with n > 1 Thus 505. ± A1 on x and z only ± A1 on x only Hence 505. ± 4.9 use of t 505. ± 6.1 (500.3, 510.1) A1 5 AWRT (ii) Weights of packets can be assumed to be nmally distributed 1 accept population of weights ; not sample of weights it (iii) Given sample: 3 in 10/ some of packets have weights below 500 grams equivalent Confidence interval: CI > 500 on CI in (a)(i) Conclusion: Statement does not appear justified dep 3 equivalent dependent on both and (b) % 1 CAO; equivalent Total 10 6

18 AQA GCE Mark Scheme, 006 June series MS1B MS1B (cont) 5(a) B(15, 0.3) use of in (a) (i) P(K = 5) = P(K 5) P(K 4) may be implied P(K = 5) = ( 0.3 ) ( 0.7 ) 5 = = to A1 3 AWFW (0.061) (ii) (Fewer than) half 7 7½ 8 stated implied Thus require P(K 7 < 8) used implied by crect answer = to A1 3 AWFW (0.9500) (iii) P( < K < 7) = minus = to A1 3 AWFW (0.741) B(15, 0.3) expressions stated f at least 3 implied by a crect answer terms within K 7 Answer A (b)(i) Mean, µ = np = = 6 CAO np 1 p Variance, σ = ( ) = = 3.6 use of σ = np( 1 p) Standard deviation = 3.6 = 1.89 to 1.9 A1 3 AWFW; equivalent (ii) Mean, x = 6 CAO (Σx = 60) CSO if evidence of np(1 p) 1.9 Standard deviation, s σ =.8 to.99 AWFW; equivalent. (Σx = 440) (iii) Means are same/equal on means; accept 6 = 0.4 if not 15 contradicted by x in (ii) Standard deviations are different dep dependent on crect SDs Reason to doubt validity of Kirk s claim dep 3 dependent on crect SDs Total 17 7

19 MS1B AQA GCE Mark Scheme, 006 June series MS1B (cont) 6 0(R) 1(S) (T) 3 T_ D(D) SD(D ) T (a)(i) P(D) = CAO; equivalent (ii) P(D R) = CSO; equivalent (iii) (iv) (v) P(D T) = P(D R) = P(R D ) = = P = A1 CAO; equivalent ( D R) ( ii) = = ( ) P( R) P( R) ( ) 4 30 M0 if independence assumed = A1 CAO; equivalent P ( R D ) P( D ) = 40 ( 30 ) ( ) numerat allow independence assumed denominat = A1 3 CAO; equivalent (b)(i) R and S R and T S and T 1 not D and D (ii) P(D) = = P(D R) (i) = (iv) P(D) P(R) = = = P(D R) (ii) P(R D) = P(R) = 0., etc so YES A1 (c)(i) A semi-detached house two children ( both) (ii) A detached house and/with less than two children Total 16 TOTAL 75 CAO equivalent CAO (0 1 must not include both ) 8

20 Version abc General Certificate of Education Mathematics 6360 Statistics 6380 MS/SS1B Statistics 1B Mark Scheme 007 examination - January series

21 MS/SS1B - AQA GCE Mark Scheme 007 January series 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 007 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 5 6EX Dr Michael Cresswell Direct General

22 MS/SS1B - AQA GCE Mark Scheme 007 January 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 A,1 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. Jan 07 3

23 MS/SS1B - AQA GCE Mark Scheme 007 January series MS/SS1B 1(a) Mean ( x ) = 39.3 to 39.4 AWFW (39.35) Standard Deviation (, ) sn sn 1 = 1.3 to 1.7 B 3 AWFW ( ) If neither crect but wking shown, then x = 787 Mean ( x ) x = 0 () Used (b) Median = 4 B CAO Median = () CAO x = 3403 Interquartile Range = = 4 B 4 CAO; allow f identification of 31 and 55; B0 if method shown is increct Interquartile Range = 1 to 7 () AWFW (c)(i) Mode: eg Does not exist If exists, must be > All / too many different values Sparse data OE (ii) Range: eg Maximum value is unknown / > OE; accept slowest but not smallest Total 9 4

24 MS/SS1B - AQA GCE Mark Scheme 007 January series MS/SS1B (cont) (a) Use of binomial in (a), (b) (c) Can be implied Allow p = 0.45, 0.5, = 0.11 A1 3 AWRT (0.113) P(E = 5) = ( p) ( 1 p) (b)(i) B(50, 0.5) Used; can be implied P(C 1) = AWRT (0.5110) (ii) P(10 < B < 0) = Allow 3 dp accuracy minus Allow 3 dp accuracy = A1 3 AWRT (0.8363) B(50, 0.30) expressions stated f at least 3 terms within 10 B 0 () Or implied by a crect answer Answer = (A) AWRT (c) n = 40, p = 0.7 Both used; can be implied Mean µ = np = 8 CAO; on p only Variance σ = np(1 p) = 8.4 Use of np(1 p) even if SD Standard deviation = 8.4 =.89 to.9 A1 4 CAO; AWFW Total 1 5

25 MS/SS1B - AQA GCE Mark Scheme 007 January series MS/SS1B (cont) 3(a) 0.5 Value 0.95 B Value is actually 0.8 Positive value < 1 (and > 0) () (b) 0. Value +0. Value is actually 0.0 (c) 0.95 Value 0.5 B 5 Value is actually 0.7 Negative value > 1 (and < 0) () Total 5 4(a) 90% z = 1.64 to 1.65 AWFW (1.6449) 90% t = 1.66 to 1.67 () AWFW (1.6649) (Knowledge of the t distribution is not required in this unit) CI f µ is x ( zt) Thus ( s s ) n 1 n ± Used; must have n with n > 1 n ( ) ( 3 3.) 184 ± A1 on z t only Hence 184 ± (5.94 to 6.13) 184 ± 6 ( ) ( 178, 190) A1 4 AWRT; igne units (b)(i) Likely to be valid Accept valid equivalent (ii) Different plays have different: programme prices, sales, marketing, etc theatre audience sizes, etc popularity, artists, etc so Dep Unlikely to be valid 3 Accept not valid equivalent Total 7 6

26 MS/SS1B - AQA GCE Mark Scheme 007 January series MS/SS1B (cont) 5(a) P(D E F ) = At least 1 probability crect = 0.04 A1 CAO; OE (b) P(D E F) = At least probabilities crect = A1 CAO; OE (c) P(One) = (b) + P(D E F ) + P(D E F ) Use of 3 possibilities; igne multipliers =(b) + ( ) + ( ) At least 1 new term crect = = A1 3 CAO; OE (d) P(One two) = (c) + (3 terms each of 3 probabilities) = 1 (a) (1 term of 3 probabilities) (c) + P(Two) Used; OE; igne multipliers 1 (a) P(Three) = ( ) + ( ) + ( ) = At least 1 new term crect = ( ) = = 0.64 A1 3 CAO; OE Total 10 7

27 MS/SS1B - AQA GCE Mark Scheme 007 January series MS/SS1B (cont) 6(a)(i) Standardising (44.5, ) with 37 P(X < 45) = P Z < 8 and ( 8, 8 8 ) and/ (37 x) = P(Z < 1) A1 CAO; igne sign = A1 3 AWRT ( ) (ii) P(30 < X < 45) = (i) P(X < 30) Used; OE = (i) P(Z < 0.875) = (i) [1 ( to )] m1 Area change = to 0.65 A1 3 AWFW ( ) (b) 0.1 z = 1.17 to 1.18 AWFW; igne sign (1.1750) z = σ = m1 Standardising 45 with 40 and σ Equating z-term to z-value but not using 0.1, z σ = 4.3 to 4.8 A1 4 AWFW (c) Route A: P(X > 45) = 1 (a)(i) Route B: P(Y > 45) = 0.1 OE; must use 45 so Dep Monica should use Route B (smaller prob) on (a)(i); allow Route Y (d) Mean of W = 18 CAO; can be implied by use in standardising 1 Variance of W = = 4 CAO; OE Standardising 0 with 18 and P( W > 0) = P Z > and/ (18 0) = P(Z > 1) = A1 4 AWRT ( ); on (a)(i) if used (e) In part (d) 1 CAO; OE Total 17 8

28 MS/SS1B - AQA GCE Mark Scheme 007 January series Question 7 (a) and (b) Lawn Areas and Mowing Times Time (y minutes) Area (x m ) (a) 8 7 points plotted accurately B (6 5 points plotted accurately ) (b) Line plotted accurately B (Evidence of crect method f points ) (Graph = 4) 9

29 MS/SS1B - AQA GCE Mark Scheme 007 January series MS/SS1B (cont) 7(a) 8 7 points plotted accurately B (6 5 points plotted accurately) () (b) Gradient, b = to B AWFW ( ) (b = 0.11 to 0.1) () Intercept, a = 15.9 to 16.1 B AWFW ( ) (a = 13 to 19) () Attempt at x, x, y and xy 440, , 635 and () Attempt at S xx and S xy and Attempt at crect fmula f b (m1) b = to (A1) AWFW a = 15.9 to 16.1 (A1) AWFW Accept a and b interchanged only if then identified crectly later in question Line plotted accurately (Evidence of crect method f points) B () 6 At least from x = 00 to 1000 (c) Res H = y H Y H = 70 (a + b 480) Used; implied by crect answer; allow f Y H y H shown = 1.5 to 0.5 A1 AWFW ( 1.06) Point H is (almost) on / just below the line 3 Accept near / close / just above equivalent (d) Y = a + b 560 reading from scatter diagram Used = 79 to 81 A1 AWFW (80.) 1 Y Cost = Y Used 60 5 = 15.8 to 16. A1 4 AWFW; igne units ( 16.05) Total 15 TOTAL 75 10

30 Version abc General Certificate of Education Mathematics 6360 Statistics 6380 MS/SS1B Statistics 1B Mark Scheme 007 examination - June series

31 MS/SS1B - AQA GCE Mark Scheme 007 June series 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 007 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 5 6EX Dr Michael Cresswell Direct General

32 MS/SS1B - AQA GCE Mark Scheme 007 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 A,1 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

33 MS/SS1B - AQA GCE Mark Scheme 007 June series MS/SS1B 1(a) r = 0.56 to 0.55 B3 AWFW r = 0.53 to 0.5 (B) AWFW; igne sign r = 0.6 to 0.4 () AWFW; igne sign OR Attempt at x, x, y, () Attempt at S xx, S yy and S xy 10, 38.1 and 47 y and xy 60, 6970, 143, 083 and 3671 Attempt at a crect fmula f r (m1) r = 0.56 to 0.55 (A1) 3 AWFW (b) Weak/some/moderate negative crelation (relationship/association) between OE; must qualify strength and indicate negative B0 f strong/po/reasonable/average B0 if r > 0 r < 1 B0 if contradicty statements length and (maximum) diameter Context Igne subsequent comments (as below) only if already sced OR Some evidence that large lengths are associated with small diameters OR Longer melons tend to have smaller diameters / be thinner () () () () Total 5 OE; must qualify strength and indicate negative OE; must qualify strength and indicate negative 4

34 MS/SS1B - AQA GCE Mark Scheme 007 June series MS/SS1B (cont) Ratios: Penalise first occurrence only of a crect answer (a)(i) P(Welsh back) = CAO; OE (ii) P(English) = = Crect expression; PI CAO; OE (iii) P(not English) = 1 (ii) = on (ii) if used; 0 < p < 1 (iv) P(Irish back) = P Irish back ( ) P( back) = 6 ( back) = Used; may be implied by values answer to 0.61 A1 CAO/AWFW (6/50 0) (v) P(fward not Scottish) = ( not Scottish) P( not Scottish) P fward = 7 = = Used; OE May be implied by values answer to A1 CAO/AWFW (5/50 0) (b) P(4 English) = = Reducing non-tabulated value 4 times Reducing 50 and multiplying 4 terms (igne multipliers) to 0.03 A1 3 CAO/AWFW Total 11 5

35 MS/SS1B - AQA GCE Mark Scheme 007 June series MS/SS1B (cont) 3(a) 95% z = 1.96 CAO 95% t =.0 to.01 () AWFW (.009) (Knowledge of the t distribution is not required in this unit) CI f µ is x ( zt) ( s s ) 1 ± n n n Used; must have n with n > 1 Note that = = Max of A0 A1 Thus ( ) ( to5.4) 34 ± A1 on z t only Hence 34 ± (6.95 to 7.30) ( ) ie 34 ± 7 (7, 41) A1 4 AWRT (b) Customers are likely to choose large / similar sized potatoes 1 OE; accept any sensible alternative Total 5 6

36 MS/SS1B - AQA GCE Mark Scheme 007 June series MS/SS1B (cont) 4(a)(i) Mode = CAO Range = 15 CAO (ii) CF: x: Median (48 th ) = 3 B CAO; B0 if shown method is increct Interquartile Range (7 nd 4 th ) = 4 = B CAO Allow f identification of 4 and B0 if shown method is increct If neither crect but CF attempted and matched crectly with 5 x values () (A1) 4 Allow f median = + 17 x (iii) Mean ( x ) = 4. B CAO fx = 399 Standard Deviation ( s s ) n, n 1 = 3.88 to 3.91 B fx = 3111 AWFW ( ) If neither crect but mid-points of 7 and 1 seen fx and use of mean ( x ) = (b)(i) Unknown values (16) have no effect on median and IQR median and IQR are exact values but x and s are estimates 95 () () 4 Allow f 4.1 x (ii) Use all available data Enable further analyses 1 Total 1 7

37 MS/SS1B - AQA GCE Mark Scheme 007 June series MS/SS1B (cont) 5(a) Time taken depends upon temperature 1 OE; not x set values (b) b (gradient) = to B AWFW ( !! ) b (gradient) = 0.09 to 0.08 () AWFW; B0 a (intercept) = 5.94 to 5.96 B AWFW ( !! ) a (intercept) = 5.6 to 6.1 () AWFW Attempt at x, x, y and xy 396, 16016, 30.9 and () Attempt at S xx and S xy 1760 and Attempt at crect fmula f b (m1) b = to (A1) AWFW a = 5.94 to 5.96 (A1) 4 AWFW Accept a and b interchanged only if then identified crectly later in question (c)(i) Each 1 ºC rise in temperature results in an (average) decrease of m (5 s) in time taken f pellets to dissolve (ii) a is y value at x = 0 at which water is solid/ice/frozen so pellets cannot dissolve Quantified rise in x (results in) Decrease in y OE Indication that it is y at x = 0 Mention of solid ice frozen (d)(i) When x = 30 y = 3.3 to 3.4 B AWFW ( 3.337!! ) y =.9 to 3.7 () AWFW If B0, use of their equation with x = 30 () (ii) When x = 75 y < 0 negative OE which Dep is impossible OE; not extrapolation Total 13 8

38 MS/SS1B - AQA GCE Mark Scheme 007 June series MS/SS1B (cont) 6(a) Use of binomial in (a) (b)(i) PI (i) P(T 10 3) = 0.38 to AWFW (0.383) (ii) P(10 < T 40 < 0) = Allow 3 dp accuracy minus Allow 3 dp accuracy OR = 0.83 to 0.84 A1 AWFW (0.835) B(40, 0.40) expressions stated f at least 3 terms within 10 T 40 0 () Or implied by a crect answer Answer = 0.83 to 0.84 (A) 3 AWFW (b)(i) n = 5 p = 0.4 Mean, µ = np = CAO Variance, σ = np(1 p) = 1. Use of np(1 p) even if SD Standard deviation = 1. = 1.09 to 1.1 A1 3 CAO AWFW (ii) Mean ( x ) = CAO x = 6 sn sn 1 Standard Deviation (, ) = 1.1 to 1.16 B x = 68 AWFW ( ) If neither crect but use of mean ( x ) = x 13 () 3 (iii) Means are same and SDs are similar/same Means are same but SDs are different so Dep Trina s claims appear valid / invalid Total 13 Must have sced full marks in (b)(i) and (b)(ii) 9

39 MS/SS1B - AQA GCE Mark Scheme 007 June series MS/SS1B (cont) 7(a) Time, X ~ N(48, 0 ) (i) P(X < 60) = P Z < 0 = Standardising (59.5, ) with 48 and ( 0,0 0 ) and/ (48 x) P(Z < 0.6) = 0.75 to 0.73 A1 AWFW (0.7575) (ii) P(30 < X < 60) = P(X < 60) P(X < 30) = (i) P(X < 30) = (i) P(Z < 0.9) = (i) {1 P(Z < +0.9)} = { } = m1 Difference equivalent Standardising other than 60 and 30 max of m1 A0 Area change 0.54 to 0.54 A1 3 AWFW ( ) (iii) 0.9 z = 1.8 to 1.8 AWFW (1.816) k 48 z = 0 Standardising k with 48 and 0 = m1 Equating z-term to z-value; not using 0.9, 0.1, 1 z Φ(0.9) = k = 73.6 to 74 A1 4 AWFW (b) Time, Y ~ N(37, 5 ) (i) Use of µ ( 3) σ = 37 (50 75) Or equivalent justification < 0 likely negative times f (likely) negative times (ii) Central Limit Theem n large / > (iii) Variance of Y = P( Y 40) P > = Z > 5 35 = OE; stated used Standardising 40 with 37 and 5 35 and/ (37 40) P(Z > 0.71) = 1 P(Z < 0.71) = m1 Area change 0.38 to 0.4 A1 4 AWFW ( ) Total 16 TOTAL 75 10

40 Version 1.0: abc General Certificate of Education Mathematics 6360 Statistics 6380 MS/SS1B Statistics 1B Mark Scheme 008 examination - January series

41 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 008 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 5 6EX Dr Michael Cresswell Direct General

42 MS/SS1B - AQA GCE Mark Scheme 008 January 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 A,1 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

43 MS/SS1B - AQA GCE Mark Scheme 008 January series MS/SS1B (a)(i) P(X < 3.5) = P Z < 0.16 = Standardising (3.45, ) with 3.3 & ( 0.16, ) and/ (3.3 x) P(Z < 1.5) = A1 CAO; igne sign to A1 3 AWFW ( ) (ii) P(X > 3.0) = P Z > 0.16 = Standardising (.95, ) with 3.3 & ( 0.16, ) and/ (3.3 x) P(Z > 1.875) = P(Z < 1.875) = m1 Crect area change to 0.97(0) A1 3 AWFW ( ) (iii) P(3.0 < X < 3.5) = (i) [1 (ii)] = OE to A1 AWFW: CSO ( ) (b) 0.05 z = 1.96 CAO; igne sign z 3.1 µ = 0.16 Standardising 3.1 with µ and 0.16; allow (µ 3.1) = 1.96 m1 Equating z-term to z-value; not using 0.05, 0.975, 1 z Φ(0.05) = to 0.51 Hence µ = 3.4(0) to 3.4 A1 4 AWFW; CSO (3.4136) Total 1 4

44 MS/SS1B - AQA GCE Mark Scheme 008 January series MS/SS1B (cont) (a) r = = Allow no 0.69 to 0.7(0) A1 AWFW (0.693) ( ) (b) (Quite fairly) weak / some / moderate (quite fairly) strong positive crelation (relationship / association) between A1 OE; must qualify strength and indicate positive A0 f po / reasonable / average / medium / good A0 f very weak / very strong etc head & body length and tail length Context; accept body and tail even head and tail Igne subsequent alternative comments only if A1 already sced OR Some evidence that mice with large head & body lengths also have long tails (A1) () OE; must qualify strength and indicate positive in context (c) 0.69 to 0.7(0) OR Answer to (a) 1 0 < r < 1 (d) Existence of: Non-linear relationship Outliers Me than one relationship Any one; OE Not reasons identifiable from context (eg spurious) Sensible related sketch SC: Check on calculation B0 Total 7 5

45 MS/SS1B - AQA GCE Mark Scheme 008 January series MS/SS1B (cont) 3(a) 1 elephants are a random sample OR are selected independently OE; eg representative Mean 39.4 x = = 3.7 CAO 1 98% z =.3 to.33 AWFW (.363) CI f µ is x σ ± z Used; must have n with n > 1 n Thus ±.363 A1 on x and z only 1 Hence 3.7 ± Hence 3.7 ± (0.13 to 0.14) OR (3.13 to 3.14, 3.4(0) to 3.41) A1 6 AWFW (b) Value of.90 is below / outside CI on (a); OE Suggests mean height of adult male Asian elephants is less than / different from that / mean height of adult male African elephants dep Total 8 on (a); OE 6

46 MS/SS1B - AQA GCE Mark Scheme 008 January series MS/SS1B (cont) 4(a) 8 points plotted accurately B ( 6 points plotted accurately) () (b) b (gradient) = 1.19 to 1.(0) B AWFW ( ) (b (gradient) = 1.1 to 1.3) () a (intercept) = 3.8 to 4(.0) B 4 AWFW ( ) (a (intercept) =. to 5.4) () Attempt at OR Attempt at x, x, y and xy 160, 758, 30 and () S 198 and S xx and xy Attempt at crect fmula f b (gradient) (m1) b (gradient) = 1.19 to 1.(0) (A1) AWFW a (intercept) = 3.8 to 4(.0) (A1) AWFW Accept a and b interchanged only if then identified crectly later in question (c) Line plotted accurately (Evidence of crect method f points) (d)(i) When x = 15: B () At least from x 7.5 to.0 x = 10 y = 15.5 to 16.5 x = 0 y = 7.0 to 8.5 y = 1.5 to (.0) B AWFW (1.8) (y = 18.5 to 5(.0)) () AWFW If B0, then use of c s equation with x = 15 (ii) Points are quite widely scattered about line () When x = 14 then y = 14.5 When x = 16 then y = 7.5 Hence not very reliable dep B0 B0 f interpolation so reliable Total 1 7

47 MS/SS1B - AQA GCE Mark Scheme 008 January series MS/SS1B (cont) 5(a)(i) P(G ) = = 0.3(0) 1 CAO; OE (ii) P(G S ) = 0.70 ( ) Can be implied only if answer is crect = 0.45 A1 CAO; OE (iii) P(1 only) = ( 0.5) Can be implied only if answer is crect; allow no ( ) but not by implication from answer = 0.75 A1 CAO; OE (b) P(G G G G ) = [(a)(i)] 4 Can be implied by crect answer Igne multiplier of 4 = A1 CAO; OE (c) P(H G ) = P(A G H G ) + P(A G H G ) = ( ) 0.4 Can be implied by crect answer Igne additional terms ( ) 0.03 Can be implied by crect answer = = 0.45 A1 3 CAO; OE (d) P(H O ) = 1 [ (c)] Can be implied by crect answer = 0.(0) A1 CAO; OE Total 1 8

48 MS/SS1B - AQA GCE Mark Scheme 008 January series MS/SS1B (cont) 6 (a)(i) x: F: Median ( th ) = B CAO; B0 if shown method increct Interquartile range ( th 95.5 th ) = 4 1 = 3 B 4 CAO; B0 if shown method increct f identification of 4 and 1 x If neither is crect but F attempted () Allow f median = and matched crectly with 5 x-values (A1) (ii) Mean ( x ) =.56 to.57 B AWFW (.56316) (.5 to.6) () AWFW fx = 974 and fx = 3546 Standard Deviation (s n, s n 1 ) = 1.66 to 1.67 B 4 AWFW ( ) (1.6 to 1.7) () AWFW ( ) If neither is crect but fx attempted and result divided by 380 () () (b)(i) Average: Same/similar/greater in 004/05 dep OE; dep on and.5 to.6 Spread: Similar/greater in 004/05 dep OE; dep on 3 and 1.6 to 1.7 (ii) Rule applies to data that is (approximately) symmetric/nmal/bellshaped OE Data f 005/06 is (positively) skewed/ not symmetric/not nmal/not bell-shaped OE Total 1 9

49 MS/SS1B - AQA GCE Mark Scheme 008 January series MS/SS1B (cont) 7(a) Use of binomial in (a) (b) Can be implied by answers (i) P(X x) = 1 P(X x 1) OR = 1 B(Σx, 50, 0.08) Identified from an answer / 1 answer Can be implied from a crect answer Identified from an answer/expression = = to 0.9(0) A1 AWFW (0.9173) 1 crect (ii) P(X 3) = = 0.77(0) to A1 4 AWFW (0.7740) (b)(i) P(Y = 0) = (1 0.05) 15 = Can be implied from crect answer = 0.68(0) to A1 AWFW (0.6840) (ii) P(Y 1) = 1 (i) Can be implied from answer if ε (0, 1) = to 0.3(0) A1 4 on (i) if ε (0, 1) (0.3160) (c) Probability = [(b)(ii) (a)(i)] ( ) [(b)(i) (a)(ii)] ( ) Igne additional terms = A1 terms added with 1 crect = 0.81 to 0.83 A1 4 AWFW (0.8193) Total 1 TOTAL 75 10

50 Version 1.0: 0608 abc General Certificate of Education Mathematics 6360 Statistics 6380 MS/SS1B Statistics 1B Mark Scheme 008 examination - June series

51 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 008 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 5 6EX Dr Michael Cresswell Direct General

52 MS/SS1B - AQA GCE Mark Scheme 008 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 A,1 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

53 MS/SS1B - AQA GCE Mark Scheme 008 June series MS/SS1B 1(a) b (gradient) = 1.01 to 1(.00) B AWFW ( ) (b (gradient) = 1.05 to 0.95) () a (intercept) = 53(.0) to 53. B 4 AWFW ( ) (a (intercept) = 5(.0) to 54(.0)) () OR Attempt at x, x, y and xy 180, 3986, 97 and () Attempt at S xx and S xy 386 and Attempt at crect fmula f b (gradient) (m1) b (gradient) = 1.01 to 1(.00) (A1) AWFW a (intercept) = 53(.0) to 53. (A1) AWFW Accept a and b interchanged only if then identified crectly in part (b), but B in (b) does not necessarily imply 4 marks in (a) (b) When x = 1, y = 31.7 to 3. B AWFW (3.0) (y = 9.9 to 34.1) () AWFW Evidence of use of 1 in c s equation () Special Cases (if seen): y = = 31.8 to 31.9 () AWFW; equivalent y = without wking () Total 6 4

54 MS/SS1B - AQA GCE Mark Scheme 008 June series MS/SS1B (cont) (a) P(Blue) = = CAO; equivalent In (b) to (e), method marks are f single fractions, equivalents, only (b) P(Marker) = Numerat 90 and Numerat < Denominat 400 = A1 CAO; equivalent (c) P(B M) = P(B M) = = = Numerat 31 and Numerat < Denominat 400 = 0.80 to A1 AWFW/CAO (0.805) (d) P(Green Highlighter) = P(G H) = 4 10 Numerat = 4 and 110 Denominat 10 = A1 CAO; equivalent (e) P(Non-Permanent Red) = P(P R) = 1 90 Numerat = 1 and 80 Denominat 90 = 0.33 to A1 AWFW/CAO (0.333) Total 9 5

55 MS/SS1B - AQA GCE Mark Scheme 008 June series MS/SS1B (cont) 3(a) r = to B3 3 AWFW ( ) (r = 0.8(0) to 0.81) (B) AWFW (r = 0.7 to 0.9) () AWFW OR Attempt at x, x, y, y and xy 859, , 148, and () Attempt at S xx, S yy and S xy 418.5, 410 and 334 Attempt at crect fmula f r (m1) r = to (A1) AWFW (b) Moderate/fairly strong/strong positive crelation (relationship/association) Or equivalent; must qualify strength and indicate positive B0 f some/average/medium/very strong/etc between length and width of plaques Context; providing 0 < r < 1 (c) Figure 1: 6 crect labelled points (5 crect labelled points) (4 crect labelled points) B3 (B) () 3 Deduct 1 mark if not labelled (d) A to F: r = 0. to +0. Accept Zero but not No crelation G to L: r = 0. to +0. Special Cases: AWFW ( 0.075) No penalties f calculations Statements must include a single value within range AWFW ( ) r = 0. to +0. with no sources () AWFW r = 0. to +0. f each/both source(s) (B) AWFW; equivalent identification If B0 B0 but both values of r = 0.4 to +0.4 () AWFW Total 10 6

56 MS/SS1B - AQA GCE Mark Scheme 008 June series MS/SS1B (cont) 4(a) Ordering: a May be implied by 40 and/ 63 and 13 Median (6 th ) = 40 CAO IQR = Q 3 (9 th ) Q 1 (3 rd ) = = 50 (b)(i) Mode: Zero is not representative / sensible reason Wide range of (known) values Small number of values mostly different () B 4 Identification of 63 and 13 CAO Or equivalent (ii) Range: Largest value, a, is unknown Cannot be calculated Or equivalent Total 6 7

57 MS/SS1B - AQA GCE Mark Scheme 008 June series MS/SS1B (cont) 5 Height X ~ N(140,.5 ) (a)(i) P(X < 145) = P Z <.5 = Standardising (144.5, ) with 140 and (.5,.5.5 ) and/ (140 x) P(Z < ) = A1 CAO; igne sign to 0.98(0) A1 3 AWFW (0.9775) (ii) P(138 < X < 14) = P(X < 14) P(X < 138) = Difference (14 138) P(Z < 0.8) P(Z < 0.8) = 0.8 CAO P(Z < 0.8) {1 P(Z < 0.8)} = ( ) ( ) = m1 Crect area change to 0.58(0) A1 4 AWFW (0.5768) (b) 0.85 (85%) z = 1.03 to 1.04 AWFW; igne sign ( ) z x 140 =.5 Standardising x with 140 and.5; allow (140 x) = ±1.03 to ± 1.04 A1 Equating z-term to the z-value Hence x = to A1 4 AWFW; CSO (137.41) (c) Variance of X 4 =.5 4 SD of X 4 =.5 = 1.56(5) = 1.5 CAO; stated used P( X4 139) P > = Z >.5 4 = Standardising 139 with 140 and 1.5; allow ( ) P(Z > 0.8) = P(Z < 0.8) = m1 Crect area change to 0.79(0) A1 4 AWFW ( ) Total 15 8

58 MS/SS1B - AQA GCE Mark Scheme 008 June series MS/SS1B (cont) 6 Binomial distribution Used somewhere in question (a)(i) M ~ B(40, 0.35) A1 Used; may be implied P(M 15) = 0.69(0) to A1 3 AWFW (0.6946) (ii) P(10 < M < 0) = Accept 3 dp accuracy minus Accept 3 dp accuracy = 0.84(0) to A1 3 AWFW (0.84) OR B(40, 0.35) expressions stated f at least 3 terms within 10 M 0 Answer = 0.84(0) to () (A) Or implied by a crect answer AWFW (b) W ~ B(10, 0.9) Used; may be implied P(W = 3) = ( ) ( ) Stated; may be implied = 0.66 to A1 3 AWFW (0.66) Note: B(10, 0.3) (c)(i) n = 0 p = 0.71 Stated used; may be implied by 14. Mean, µ = np = 14. CAO Variance, σ = np(1 p) = 4.11 to AWFW (4.118) (ii) Mean of 16.5 is greater/different 16.5/0 = 0.85 is greater/different to 0.71 Means and variances are different dep Dependent on µ = 14. (B,1 dep) Variance of.50 is smaller/different dep Dependent on σ = 4.11 to 4.1 Suggests claim that groups are not random samples is justified dep 3 Total 15 Dependent on previous marks Or equivalent 9

59 MS/SS1B - AQA GCE Mark Scheme 008 June series MS/SS1B (cont) 7(a)(i) x: f: Mean ( x ) = 1.9 (0.9 to.9) B () CAO (190) AWFW Standard deviation (s n-1 σ n ) = 3.3(0) to 3.3 (3(.00) to 3.5(0)) B () 4 (145) AWFW ( ) AWFW ( ) If no marks sced but fx attempted and result divided by 100 () (ii) Mean = 60 + x = 61.9 A1 on (a)(i) Standard deviation = 3.3(0) to on (a)(i); accept same as only providing answer in (a)(i) (b)(i) 98% z =.3 to.33 ( t =.36 to.37) AWFW (.363) AWFW (.364) CI f µ is x ± z/ t s n n 1 σ n n 1 Used; must have n with n > 1 Thus 3.3 to ±.363 A1 on (a)(ii) and z/t only Hence 61.9 ± (0.7 to 0.8) (61.1 to 61., 6.6 to 6.7) (ii) Mean and SD based upon grouped data SD (not mean) calculated from a sample CLT used / Times (may) not (be) nmal A1 4 1 Accept 1.03 ± (0.01 to 0.013) AWFW Accept (1.01 to 1.0, 1.04 to 1.05) Actual times/values unknown Or equivalent (c) S > 1 hour 60 minutes: Valid as 74/ % > 50% Must use 74 etc Or equivalent S >> 1 hour 60 minutes: Not valid as UCL 1 hour (Accept Both limits 1 hour) dep Total 14 TOTAL 75 Dependent on UCL = 6.6 to 6.7 UCL = 1.04 to

60 Version 1.0: 0109 hij General Certificate of Education Mathematics 6360 Statistics 6380 MS/SS1B Statistics 1B Mark Scheme 009 examination - January series

61 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 009 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 5 6EX Dr Michael Cresswell Direct General

62 MS.SS1B - AQA GCE Mark Scheme 009 January 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 A,1 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

63 MS.SS1B - AQA GCE Mark Scheme 009 January series MS/SS1B 1 (a) 47 fx Mean = x 5 = ¾ B CAO (4.75 = 5 ISW) B 5 If B0 but evidence of Median (6, 6½) = 5 fx 5 If B0 but evidence of cumulative frequencies F: (0) If data assumed continuous so use of x 4 + where 0 < x < 7 () B () () CAO Stated identification of 6 6½ Need to see attempt at 4 F values (4 < median < 4.9) Mode(s) = 4 and 6 5 CAO both (so mode = 5 B0) (b) Mode(s) CAO Me than one mode/value Two modes/values No unique mode/value dep Or equivalent; eg not unique Dep only on previous sced Notes: If data treated as two separate sets, then Modes = 1 and 13 B0 in (a) but only marks available are dep in (b) dep available in (b) If averages confused then mark (a) as stated eg median = 4 and 6 B0 in (a) and in (b) median, as two values B0 B0 Total 7 4

64 MS.SS1B - AQA GCE Mark Scheme 009 January series MS/SS1B (cont) (a)(i) r = 0.0 to 0.03 B3 AWFW (0.0557) r = 0.0 to 0.03 (B) AWFW (ii) OR Attempt at Attempt at r = 0.1 to 0.1 x xy x y S xx S yy & S xy y & () () AWFW Attempt at crect fmula f r (m1) r = 0.0 to 0.03 (A1) 3 AWFW (Almost/virtually) no/zero (linear) crelation (relationship/association/link) between length and (maximum) diameter of carrots & (all 5 attempted) & 0.33 (all 3 attempted) Or equivalent qualification of NO strength; do not follow-through from (i) B0 f very weak/weak/some/ little/slight/positive/hardly any/etc unless crect qualification also stated Context; providing 1 < r < 1 (b) Unlikely/wrong/increct/invalid Or equivalent Would expect a positive value Would expect weight to increase with length Would imply shter carrots are heavier Total 7 Or equivalent reason 5

65 MS.SS1B - AQA GCE Mark Scheme 009 January series MS/SS1B (cont) 3 X ~ N(5.08, 0.05 ) (a)(i) (ii) P( X < 5) = = 1 ( ) ( 5< < 510) = P( X 510. ) P X. = P( Z 0. 4) P Z < 005. = Standardising (4.5, 4.95, 5, ) with 5.08 and ( 005., ) P( Z < 1. 6) and/ (5.08 x) P Z < 16. = m1 Area change; may be implied = to A1 AWFW (0.0548) 3 (1 answer) max Or equivalent; must be clear crect < (i) method if answer increct and answer > 0 < (i) = = 0.6 to A1 AWFW (0.6006) (b)(i) Variance of X 4 = 0.05 /4 = SD of X 4 = 0.05/ = P( X 4 > 5. 05) = P Z > CAO; stated used Standardising 5.05 with 5.08 and 0.05; allow ( ) = P( Z > 1. ) = P( Z 1. ) < m1 Area change; may be implied AWFW ( ) = to A1 4 (1 answer) max (ii) Zero 1 CAO; equivalent (igne any wking) (c) 1% (0.01) z =.33 to.3 AWFW; igne sign (.363) 5 µ z = 005. =.363 A1 Standardising 5 with µ and ; allow (µ 5) Only allow: ±.05 to ±.06 ±.3 to ±.33 ±.57 to ±.58 µ = 5.11 to 5.1 Note: 5 µ = A1 A0 A1 AWFW (5.1163) 4 Total 14 Or equivalent inconsistent signs 6

66 MS.SS1B - AQA GCE Mark Scheme 009 January series MS/SS1B (cont) 4 P(C) = 0.6 P(C B) = 0.5 In (a), ratios (eg 4 : 10) are only penalised by 1 mark {P(C only) = 0.35 P(B only) = 0.4} at first crect answer (a) (i) P(C ) = 1 P(C) = = CAO; equivalent (ii) P(C B ) = Can be implied by crect answer = 1 ( ) = 0.35 A1 CAO; equivalent (iii) P(B) = (i) + p with p < 0.6 Can be implied by crect answer = (i) A1 Can be implied by crect answer = 0.65 A1 CAO; equivalent OR P(B) = 1 (ii) (M) Can be implied by crect answer = 0.65 (A1) OR 1 = P(C) + P(B) P(C B) Thus P(B) = 1 ( ) = 0.65 () (A1) (A1) 3 Can be implied by crect answer Can be implied by crect answer CAO; equivalent (b) P(L G C ) = 0.9 P(L G CB ) = 0.7 P(L G B ) = 0.3 (i) P(G L) (a)(ii) 0.9 (0.315) (0.175) [(a)(iii) 0.5] 0.3 (0.1) Note: Each pair of multiplied probabilities must be > 0 to sce the cresponding method mark Follow through crect Follow through crect Igne any multiplying facts Igne any additional terms = 0.61 A1 4 CAO (ii) Probability = {1 (b)(i)} 5 Allow 5 {1 (b)(i)} 5 = = A1 AWRT (0.0090) Total 1 7

67 MS.SS1B - AQA GCE Mark Scheme 009 January series MS/SS1B (cont) 5 (a) Mean = = 54 1 CAO; cannot be gained in (b) (b) 98% (0.98) z =.3 to.33 AWFW (.363) CI f µ is x ± z σ Used n Must have n with n > 1 Thus 54 ± F on x (but not 160) and z only 30 A1F Allow x = 54 even if B0 in (a) Hence 54 ± (3.38 to 3.4) CAO & AWFW (54 & 3.4) A1 4 (50.58 to 50.6, to 57.4) AWFW (50.6, 57.4) Notes: Use of n = 1 in (b) must not be deemed as answer to (c) Use of n = 1 in (b) followed by use of n = 1 in (c) (b), (c) A1 max Use of n = 1 with (b) (c) not identified (b), (c) 0 max (c) Repeat of structure in (b) but with n = 1 and 1.96 z 3.03 Or equivalent Thus 54 ± (18.56 to 18.64) (35.36 to 35.44, 7.56 to 7.64) Note: Accept sensible non-symmetric intervals such as: (0, ) = (0, 70.4 to 70.5) A1F CAO & AWFW (54 & 18.6) If z-value increct, then must use 54 ± 8 z from( b) AWFW (35.4, 7.6) (d) Nowhere No 1 Total 8 CAO; equivalent (igne any reasoning) 8

68 MS.SS1B - AQA GCE Mark Scheme 009 January series MS/SS1B (cont) 6(a) Figure 1: 3 crect labelled points B crect labelled points () Deduct 1 mark if not labelled (b) OR Attempt at b (gradient) = B AWRT (0.6850) b (gradient) = 0.68 to 0.69 () AWFW a (intercept) = B AWRT ( ) a (intercept) = 0.34 to 0.35 () AWFW x x y & xy & 7853 (all 4 attempted) () Attempt at S xx & S xy 654 & 448 (both attempted) Attempt at crect fmula f b (gradient) (m1) b (gradient) = (A1) AWRT a (intercept) = (A1) AWRT Accept a & b interchanged only if then identified crectly by a stated used equation in (c) (d) 4 (c) Figure 1: Crect line (50, 34 to 35) (60, 40½ to 4) (70, 47¼ to 49) (80, 54 to 56) If B0 but evidence of use of line f points within range 50 x 80 (d)(i) Residual = y (a + bx) [ (a + bx) y] H I J.5 to 4(.0).5 to 4(.0) (.0) to 4(.0) Bdep Dep on A1 A0 in (b) At least from x 55 to 70 Any two () Calc n points shown on graph A,1 ( 1 EE) Used implied; equivalent (using graph); 1 residual crect (.98) AWFW; igne signs only (3.19) providing all the same (.70) (ii) y 65 = a + b 65 y 65 = 44 to 45.5 Mean =.3 to 4(.0) A1dep 4 + [(d)(i)] [.95 to.97] m1 Special Cases: Line drawn/calc d on H, I & J linear interp n using I & J = 47 to to 45.5 seen with no evidence AWFW; do not igne sign (.96) Dep on previous A sced Use shown AWFW (44.9) Use shown AWFW; igne sign of mean residual = 46 to 50 A1 AWFW (47.8) (B) 3 Total 15 y M = x 478. OR no evidence of method {from (d)(i) and/ (d)(ii)} Evidence of increct method B0 9

69 MS.SS1B - AQA GCE Mark Scheme 009 January series MS/SS1B (cont) 7 (a)(i) B( , 0.45) Used at least once in (a)(i) to (iii) (ii) May be implied by crect answer A1 3 Igne any additional terms = 0.01 to 0.0 A1 3 AWFW (0.015) Accept 3 dp accuracy from tables P(S < 10) = calculation = 0.4 to 0.43 AWFW (0.44) P(S = 3) = (. ) (. ) (iii) P(15 S 0) Accept 3 dp accuracy = (p 1 ) p p 1 M0 M0 A0 p 1 (1 p ) M0 A0 minus (p ) Accept 3 dp accuracy / truncation = to A1 AWFW (0.6544) OR B(40, 0.45) expressions stated f at least 3 terms within 14 S 0 gives probability = to () (A) 3 Or implied by a crect answer AWFW (iv) Mean, µ = np = =.5 ½ Variance, σ = np(1 p) = = 1.3 to 1.4 CAO (.5 = 3 ISW) Accept AWFW (1.375) (b)(i) Non-independence of seni citizens travel Seni citizens tend to travel in pairs/groups (ii) 7.15 am is outside 9.30 am to am Cannot use SCPs befe 9.30 am Cannot use 7.15 am Cannot use SCPs during mning rush hour Value of p likely to be smaller/different/zero Data not available Seni citizens not out at this time Passengers likely to be wkers/school children 1 1 Total 1 Paper 75 Or equivalent; but must be a clear indication of non-independent events Or equivalent Accept other sensible reasons Distribution of types of passenger different 10

70 Version : klm General Certificate of Education Mathematics 6360 Statistics 6380 MS/SS1B/W Statistics 1B Mark Scheme 009 examination - June series

71 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 009 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 5 6EX Dr Michael Cresswell Direct General

72 MS/SS1B/W - AQA GCE Mark Scheme 009 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 A,1 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

73 MS/SS1B/W - AQA GCE Mark Scheme 009 June series MS/SS1B/W 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 = 5/40 = 10/16 = 5/8 = CAO (ii) P(S ) = 3 1 P(S) = Or equivalent Igne labels of S & S Can be implied by crect answer = 18/160 = 64/80 = 3/40 = 16/0 = 8/10 = 4/5 = 0.8 A1 CAO (iii) P(S H) = P(S H) = Or equivalent Can be implied by crect answer = 74/160 = 37/80 = 0.46 to A1 CAO/AWFW (0.465) (iv) P(T P) = (i) Or equivalent Can be implied by crect answer But watch f (b) P(1C & 1R & 1S) = = 3/100 = 3/10 = 0.3 A1 CAO Multiplication of any 3 different given subject totals Multiplication of 160, 159 & 158 ( ) Accept 3dp accuracy 6 Award f 3 multiplier 6 = to A1 AWFW (0.064) 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) () 4 Total 11 Can award if no marks sced in (b) Accept a decimal equivalent 4

74 MS/SS1B/W - AQA GCE Mark Scheme 009 June series MS/SS1B/W (cont) (a) r = to B3 AWFW ( ) r = 0.89 to (B) AWFW r = 0.8 to 0.95 () AWFW Attempt at Attempt at x xy x y S xx S yy & S xy y & () & 3588 (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) dep 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 Context; providing 0 < r < 1 (c) Figure 1: 5 crect labelled points 4 3 crect labelled points B () Deduct 1 mark if points not labelled (d)(i) D and G 1 Both CAO (ii) r = 0.5 to 0.75 AWFW ( ) No penalty f calculation Accept a range only if whole of it falls within 0.5 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) dep Total 10 Or equivalent; must qualify strength and indicate positive Dependant on 0.5 r 0.75 B0 f very weak/little/slight/hardly any/fair/average/medium/anything involving strong/etc 5

75 MS/SS1B/W - AQA GCE Mark Scheme 009 June series MS/SS1B/W (cont) 3(a) X ~ N(53, 5 ) (i) P( X < 50) = Standardising (49.5, P Z < 5 = 50.5) with 53 and ( 5, 5 5 ) and/ (53 x) P Z < 0. 6 = m1 Area change; may be implied ( Z <. ) = 1 ( ) P 0 6 (ii) (iii) ( < X < ) = [ C's(a)(i) ] P( 45) P = (i) P( Z 1. 6) = 0.74 to 0.75 A1 3 < = X < = 0.19 to 0.(0) A1 P( X = 45) = 0 zero impossible 1 AWFW (0.745) (1 answer) max Or equivalent; must be clear crect method if answer increct and answer > 0 AWFW (0.1945) A0 f [1 (i)] = M0 A0 f [(i)] = A1 f [1 (i)] = Igne any wking B0 f f impossible to calculate (b) 98% (0.98) z =.05 to.06 AWFW; igne sign (.0537) z = σ Standardising 45 with 53 and σ ; allow (53 45) =.0537 A1 Only allow: ±.05 to ±.06 ±.3 to ±.33 Note: σ σ = 3.88 to 3.9(0) A1 AWFW (3.8954) =.0537 σ = A1 A0 4 Or equivalent inconsistent signs Total 10 6

76 MS/SS1B/W - AQA GCE Mark Scheme 009 June series MS/SS1B/W (cont) 4(a) b (gradient) = to B AWFW ( ) b (gradient) = 0.55 to 0.54 () AWFW Omission of ve sign (B0) (b) Attempt at Attempt at a (intercept) = 49.7 to 49.9 B AWFW (49.798) a (intercept) = 49 to 50 () AWFW x x y & S xx & S xy xy ( y ) () 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 F & 6980 (10173) (all 4 attempted) & (both attempted) If a and b not identified anywhere in question, then: to to 49.9 Accept value rounded to nearest integer (50) (c) 13 weeks 91 days Stated used Accept a descriptive answer that includes 91 and a value in range y = 1.1 to AWFW ( ) y = 0 x = 89 to 93 () AWFW ( ) Accept a descriptive answer that includes a value in range and weeks (approximately) () Stated Note: () () are available even if (a) not attempted Thus claim appears justified Thus tablet likely to have dissolved dep Or equivalent; igne reasoning unless contradicty Dependent upon nd in (c) nd () in (c) Extrapolation required so cannot comment () Not dependent Note: If () f extrapolation maximum mark is ; other mark available is f

77 MS/SS1B/W - AQA GCE Mark Scheme 009 June series Total 8 8

78 MS/SS1B/W - AQA GCE Mark Scheme 009 June series MS/SS1B/W (cont) 5(a) (i) Median (50) = 3 CAO If not identified, then assume der is median then IQR IQR (75 5) = 4 = B Do not award marks if crect answers are based on shown increct method; eg accept use of 99/, etc but not 76/, etc CAO; but 5 th value IQR = B0 Special Cases: Identification that LQ = and UQ = 4 () Both CAO Statement of 4 cumulative frequencies F: () 3 Can award if no marks sced in (i) even if then applied to continuous data (ii) Mean = fx f = =.77 to.78 AWFW (.778) If not identified, assume der is x then s Treat rounding to integers as ISW fx = 933 = 1.3(0) to 1.3 B AWFW (1.307 & 1.314) SD ( ) Special Case: fx () 3 Can award if no marks sced in (ii) Evidence of 99 (b)(i) 99 Mean99 fx from(a)(ii) Mean163 = Or equivalent; may be implied by an answer within range = 1.68 to 1.69 A1 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)] [C s SD in (a)(ii)] < 0 [C s mean in (b)(i)] [1.69 to 1.71] < 0 Thus claim appears not valid dep 1 CAO; equivalent (1.696) Igne any wking (1.70) Or equivalent ( 1.75 to 0.90) Or equivalent Dependent upon previous Total 11 9

79 MS/SS1B/W - AQA GCE Mark Scheme 009 June series MS/SS1B/W (cont) 6(a) Mean = = 47 CAO 98% (0.98) z =.3 to.33 AWFW (.363) σ CI f µ is x ± z n Used Must have n with n > 1 Thus ±. 363 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, 8 ) Variance of Y 40 = 8 /40 = 19.6 SD of Y 40 = 8 40 = 4.45 to 4.43 A1 5 CAO & AWRT (accept 11) AWRT (accept 36 & 58) CAO Stated used AWFW ( 40 ) P Y > 10 = P Z > 8 40 Standardising 10 with 108 and (4.45 to 4.43) equivalent; allow (108 10) = P( Z >. 71) = 1 P( Z 71. ) < m1 Area change; may be implied = = to A1 4 AWFW ( ) (1 answer) max (c) Part (b) Teleair times Or equivalent; igne reasoning Distribution of Y not known Note: To sce there must be both a clear indication of where in question and a valid reason Total 11 Or equivalent; must be clear reference to Y population B0 f n > 30 Any reference to part (a) B0 B0 10

80 MS/SS1B/W - AQA GCE Mark Scheme 009 June series MS/SS1B/W (cont) 7(a) R ~ B(50, 0.15) (i) P(R < 10) = AWRT (0.7911) (ii) P(5 R 10) = (p 1 ) Accept 3 dp accuracy (1 p ) p 1 M0 M0 A0 p 1 (1 p ) M0 A0 only providing result > 0 minus (p ) 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 = () (A) 4 Can be implied by crect answer AWRT (b) Confusion of, 35, 10 and/ 0.15, 0.06 Do not treat as misreads (i) S ~ B(, 0.06) Used in (b)(i) as evidenced by any crect binominal term f S > Can be implied by crect answer A1 Igne any additional terms = 0.4 to 0.4 A1 3 AWFW (0.415) P(S = ) = (. ) (. ) (ii) P(S 1) = 1 () 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 AWFW (0.8853) (iii) Mean = np = = Either If not identified, assume der is µ then σ Variance = np(1 p) Must clearly state variance value = = 6.76 to 6.78 AWFW (6.768) (iv) Means are (approximately) the same stated Variances are (very) different stated Must have sced 1 st in (iii) Must have sced nd in (iii) Agree with P(sts letter increctly) = 0.06 dep Dependent on means same stated Disagree with independent from letter to letter dep 3 Total 14 Paper 75 Dependent on variances different stated 11

81 MS/SS1B/W - AQA GCE Mark Scheme 009 June series 1

82 Version 1.0: 0110 klm General Certificate of Education Mathematics 6360 Statistics 6380 MS/SS1B/W Statistics 1B Mark Scheme 010 examination - January series

83 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 010 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 5 6EX Dr Michael Cresswell Direct General

84 MS/SS1B/W - AQA GCE Mark Scheme 010 January 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 A,1 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

85 MS/SS1B/W - AQA GCE Mark Scheme 010 January series MS/SS1B 1(a)(i) X ~ N(10., 0.15 ) ( X. ) P < 10 5 = P Z < 015. Standardising (10.45, ) with 10. and ( 015., ) and/ (10. x) = P(Z < ) A1 CAO; igne inequality and sign May be implied by a crect answer = A1 3 AWRT (0.9775) (ii) P100 (. < X < 105. ) = [C s (a)(i)] P( X < 10. 0) Or equivalent; must be clear crect method if answer increct and answer > 0 = (a)(i) P( Z 133. ) < Method crect using 1.3 gives 0.88 to = (a)(i) (1 p) m1 A0 = ( ) m1 Area change May be implied by a crect answer answer > 0.5 = to A1 3 AWFW ( ) m1 A1 f [1 (a)(i)] = m0 A0 f (a)(i) = M0 mo A0 f answer < 0 (b) P(X > 10) = p[from (a)(ii)] = to F Crect value F on value used implied in (a)(ii) providing > 0.5 Use of 1.3 gives P(6 rolls > 10) = Accept any probability to power to A1 3 AWFW Note: B0F A0 is possible Total 9 4

86 MS/SS1B/W - AQA GCE Mark Scheme 010 January series MS/SS1B (cont) (a) Ordering values gives: (a) (b) May be implied by crect median crect IQR Igne any reference to a and b Median = 6 A1 CAO IQR = = 16 A CAO Special Case: Identification that LQ = 18 and UQ = 34 (A1) 4 Both CAO (b)(i) Two values (5 and 37) of mode No unique value Sparse data Many different values Or equivalent (ii) a and b (two values) unknown Impossible to calculate Cannot be calculated Or equivalent (c) Mean = x n = = 6 CAO If not identified, assume der is x then s = 9.4 to 9.8 SD ( x = 1147) AWFW (9.43 & 9.754) Treat rounding of a crect stated answer to an integer as ISW Special Case: Evidence of x 15 () Total 8 Can only be awarded if no marks sced elsewhere in (c) 5

87 MS/SS1B/W - AQA GCE Mark Scheme 010 January series MS/SS1B (cont) 3(a) b (gradient) = 7.05 B AWRT ( ) b (gradient) = 7(.00) to 7.1(0) () AWFW Treat rounding of crect stated answers as ISW a (intercept) = 500 to 50 B 4 AWFW ( ) a (intercept) = 490 to 510 () AWFW Attempt at x x y & xy Attempt at S xx & xy ( y ) S ( S yy ) () & ( ) (all 4 attempted) 7304 & (147894) (both attempted) Attempt at crect fmula f b (gradient) (m1) b (gradient) = 7.05 (A1) AWRT a (intercept) = 500 to 50 (A1) AWFW Accept a & b interchanged only if identified crectly by a clearly shown equation (stated answers are not sufficient) in (b) If a and b are not identified anywhere in solution, then: to 50 (b) y 00 = a + b 00 Used May be implied by crect answer = 3890 to 3930 A1 AWFW ( ) (c) Large residuals / residual range suggest estimate may be unreliable Largest residuals only small in relation to y-values (10%) so estimate may be reliable (unreliable) dep dep (unreliable) requires (10% equivalent) Special Case: If B0 B0dep then: Involves interpolation Does not involve extrapolation Within observed range () Any one; equivalent Total 8 6

88 MS/SS1B/W - AQA GCE Mark Scheme 010 January series MS/SS1B (cont) Ratios (eg 65:1000) are only penalised by 1 mark at first crect answer 4(a)(i) P(all 3 walk) = Can be implied by crect answer = 65/1000 = 13/00 = A1 CAO; do not confuse with 0.65 (ii) P(Rita by bus) = 0.5 (1 0.15) (1 0.0) Can be implied by crect answer = 17/100 = 0.17 A1 CAO (iii) P( cycle) = ( ) = ( ) 0.55 = ( ) = 0.75 (0.735) CAO at least 1 of these 3 terms equivalent but allow a 3 P(3 cycle) = = CAO P( cycle) = P( cycle) + P(3 cycle) Sum of 4 7 terms each a product of 3 probabilities but not 3 = 0.98 A1 4 CAO P(0 cycle) = = 0.75 () CAO P(1 cycles) = = = () (0.4795) = 0.75 P( cycle) = 1 [P(0 cycle) + P(1 cycles)] () CAO at least 1 of these 3 terms but allow a 3 1 [sum of 4 terms each a product of 3 probabilities but not 3 ] = 0.98 (A1) CAO (b)(i) P(WW) = ( ) = P(CC) = ( ) = CAO either P(WW CC) = Sum of terms each a product of probabilities = A1 3 CAO; equivalent (ii) P(different) = 1 (b)(i) = F 1 F on (b)(i) providing 0 < p < 1 Total 1 7

89 MS/SS1B/W - AQA GCE Mark Scheme 010 January series MS/SS1B (cont) 5(a)(i) Mean = = 1010 CAO 98% (0.98) z =.3 to.33 AWFW (.363) σ CI f µ is x ± z n Used Must have n with n > 1 Thus ± A1F F on x and z only Hence 1010 ± (7(.0) to 7.1) (1003, 1017) A1dep 5 CAO & AWFW (accept 7) Dependent on A1F AWRT Notes: Use of t 11 ( 099. ) = 718. maximum of B0 A0F A0 Use of a crected 10.5 maximum of A0F A0 (ii) Weight of flour in a bag (may be assumed to be) is nmally distributed 1 Or equivalent; must refer to weight (iii) Any number such that 0 number 50 1 Must be a single integer value Igne any reasoning (b) 1 kg 1000 grams is outside / below CI From CI, (population) mean weight is greater than 1kg 1000 grams 3 3/1 5% of bags in sample weigh less than 1kg 1000 grams F Or equivalent F on (a)(i) Any reference to 1010 B0F Or equivalent; but not some Statement appears dubious/increct/invalid dep 3 Dependent on both F and (c) /100 1/ % 1 CAO; not 0.0% Total 11 8

90 MS/SS1B/W - AQA GCE Mark Scheme 010 January series MS/SS1B (cont) 6(a)(i) R ~ B(14, 0.35) Used somewhere in (a); may be implied P(R 7) = 0.94 to 0.95 A1 AWFW (0.9466) (ii) P(R 11) = 1 P(R 10) = 1 ( ) Requires 1 and 4 dp accuracy = A1 AWRT ( ) (iii) P(5 < R < 10) = (p 1 ) Accept 3 dp accuracy p p 1 M0 M0 A0 (1 p ) p 1 M0 M0 A0 p 1 (1 p ) M0 A0 only providing result > 0 minus (p ) Accept 3 dp accuracy = to A1 3 AWFW ( ) B(14, 0.35) expressions stated f at least 3 terms within 4 R 11 gives () Can be implied by crect answer probability = to (A) AWFW ( ) (b) R ~ B(1, 0.35) Implied from crect stated fmula; do not accept misreads Can be implied by a crect answer A1 4 Igne any additional terms = to A1 3 AWFW ( ) P(R = 4) = (. ) (. ) (c)(i) S ~ B(7, 5/7) Mean = np = 7 5/7 = 5 CAO If not identified, assume der is µ then σ Variance = np(1 p) = 7 5/7 /7 = 10/7 1.4 to 1.43 Must clearly state variance value if standard deviation (also) stated CAO / AWFW (ii) Means are the same and (both comparisons clearly stated) Variances/standard deviations are similar Do not accept statements involving crect/increct/exact/etc dep Must have sced in (i) B0 plus 10/7 v / 7 v 1.5 stated Barry s claim appears/is sound/valid/crect/likely dep Must have sced previous dep Total 14 9

91 MS/SS1B/W - AQA GCE Mark Scheme 010 January series MS/SS1B (cont) 7(a) r = to B3 3 AWFW ( ) r = to (B) AWFW r = 0.04 to () AWFW Attempt at Attempt at x xy x y S xx S yy & S xy y & () &38605 (all 5 attempted) & 509 (all 3 attempted) Attempt at substitution into crect cresponding fmula f r (m1) r = to (A1) AWFW (b) Almost/virtually/practically no / zero (linear) crelation / relationship / association / link (but not no trend ) dep Dependent on 0.1 < r < 0.1 Or equivalent; must qualify strength as zero ; B0dep f very weak/weak/etc unless then qualified crectly between purchase and auction prices of antiques Context; providing 1 < r < 1 (c)(i) Figure 1: 6 crect labelled points 5 4 crect labelled points 3 crect labelled points B3 (B) () 3 Deduct 1 mark if > 1 point not labelled labelled increctly (ii) (Two) outlier/anomaly/unusual identification of J and L Or equivalent (Otherwise) a positive/linear crelation Or equivalent; igne any qualification of strength (d)(i) r = Used Award B f a crect answer without/with different method r = to A1 AWFW ( ) (ii) Very strong/strong positive (linear) crelation/relationship/association/link Previous calculation of r was not appropriate (due to outliers) dep () 1 Total 13 TOTAL 75 Dependent on 0.9 < r < 1 Or equivalent; must qualify strength and indicate positive; B0dep f high/etc 10

92 Version 1.0 klm General Certificate of Education June 010 Mathematics Statistics MS1B SS1B Statistics 1B Mark Scheme

93

94 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 010 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 5 6EX

95 MS/SS1B - AQA GCE Mark Scheme 010 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 A,1 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

96 MS/SS1B - AQA GCE Mark Scheme 010 June series MS/SS1B 1(a) r = B3 AWRT ( ) OR Attempt at xy Attempt at x x y S xx S yy and r = 0.91 to 0.9 (B) AWFW r = 0.88 to 0.95 () AWFW S xy y and () and (all 5 attempted) and 3388 (all 3 attempted) Attempt at substitution into crect cresponding fmula f r (m1) r = (A1) 3 AWRT (b) Very strong / strong / fairly strong positive (linear) crelation / relationship / association / link (but not trend ) between dep Dependent on 0.88 < r < 0.95 Or equivalent; must qualify strength and indicate positive B0dep f (almost) perfect / high / average / medium / some / etc weight and (engine) power/bhp of (hatchback) cars Examples: The me weight/heavier the me/greater power B0dep Strong crelation and as weight/kg increases so does engine power / bhp B0dep Context; providing 0 < r < 1 No mention of strength Mention of strength but implied suggestion of positive not sufficient Total 5 4

97 MS/SS1B - AQA GCE Mark Scheme 010 June series MS/SS1B (cont) (a)(i) Mean, d = 1.5 CAO d = 18 Igne notation and units Standard deviation, σ d s ( ) d = 11.7 to 1.3 AWFW d = 1680 (ii) Mean, x = 50 + d = 51.5 F F on (a)(i) crect x: x = 618 x = Igne notation and units Standard deviation, σ x s x = 11.7 to 1.3 F F on (a)(i) providing > 0 crect (b) [Values, mean sd in (a)(i) (a)(ii)] Award if use seen implied by 1 Subsequent crect (crect 100) answer Mean = 0.68 to 0.63 A1 AWFW (0.683) Standard deviation = 0.14 to A1 3 AWFW ( ) Special Cases: At least one answer crect with no stated units increct stated units A1 A1 max At least one answer 100 with its units stated as cents A1 A1 max At least one answer 100 with no units stated units stated as euros / pence / only cents attached to 1 answer 100 Total 7 5

98 MS/SS1B - AQA GCE Mark Scheme 010 June series MS/SS1B (cont) 3 Time, X ~ N(65, 0 ) (a) (i) P(X < 90) = P Z < P Z < = P( Z < 3. 5) = Standardising (89.5, , ) with 65 and ( 0, 0 0 ) and/ (65 x) May be gained in (a)(i) (a)(ii) = P(Z < 1.5) A1 CAO; igne inequality and sign May be implied by a crect answer (ii) P(X > 60) = P(Z > 0.5) = to A1 AWFW ( ) = P(Z < 0.5) Area change May be implied by a crect answer answer > 0.5 = to A1 5 AWFW ( ) (b) (i) P(1 in 6 = 60) = 0 zero impossible 1 Igne any wking B0 f impossible to calculate (ii) P(X < 60) = 1 [(a)(ii)] (0.401 to 0.40) May be implied P(6 in 6 < 60) = p 6 with 0 < p < 1 Any probability to power 6; do not allow multiplying facts = (0.4019) 6 = to A1dep 3 Dependent on ( ) (iii) Variance of X 6 = 0 /6 = 66.6 to 66.7 Sd of X 6 = 0/ 6 = 8.16 to 8.17 ( 6 ) P X < 60 = P Z < 0 6 P(Z < 0.61) = 1 P(Z < 0.61) = m1 = = = 0.7(0) to 0.71 A1 4 Note: Watch f answers to (ii) and (iii) interchanged Total 13 CAO/AWFW Stated used anywhere in (b) CAO/AWFW Standardising 60 with 65 and 0/ 6 equivalent allow (65 60) Area change May be implied by a crect answer answer < 0.5 AWFW (0.7093) (1 answer) max 6

99 MS/SS1B - AQA GCE Mark Scheme 010 June series MS/SS1B (cont) 4(a) M ~ B(50, 0.15) Used somewhere in (a); may be implied (i) P(M 10) = 0.88(0) A1 AWRT (0.8801) (ii) P(M 5) = 1 P(M 4) = 1 ( ) Requires 1 ; accept 3 dp accuracy Implied by but not by = A1 AWRT (0.8879) (iii) P(6 < R < 1) = (p 1 ) Accept 3 dp accuracy rounding truncation p p 1 M0 M0 A0 (1 p ) p 1 M0 M0 A0 p 1 (1 p ) M0 A0 only providing result > 0 minus (p ) Accept 3 dp accuracy = A1 AWRT (0.5759) OR B(50, 0.15) expressions stated f at least 3 terms within 5 R 1 gives () Can be implied by crect answer probability = (A) 3 AWFW (0.5759) (b) F ~ B(35, 0.11) Implied from crect stated fmula; do not accept misreads Can be implied by a crect answer A1 4 Igne any additional terms = 0.06 to 0.08 A1 3 AWFW (0.0685) P(F = 4) = (. ) (. ) (c) P(M and LH) = = ) N(M) = = 1040 ) P(F and LH) = = 0.058) N(F) = = 960 ) N(M and LH) = = = 156 ) N(F and LH) = = = ) P(LH) = = ) A1 A1 1 of these probabilities 1 of these numbers attempted; may be implied probabilities numbers evaluated crectly Evaluation of 1 of these numbers Addition of these probabilities N(LH) = = = 61 to 6 A1 4 6/000 A0 AWFW (61.6) Total 14 7

100 MS/SS1B - AQA GCE Mark Scheme 010 June series MS/SS1B (cont) 5 Ratios (eg 63:100) are only penalised by1 mark at first crect answer F marks can only be awarded if 0 < p < 1 (a) P(J) = 0.9 P(R J) = 0.7 P(R J ) = 0. (i) P(both at trough) = Can be implied by crect answer = 0.63 = 63/100 A1 CAO (ii) P(neither at trough) = (1 0.9) (1 0.) = Can be implied by crect answer = 0.08 = 8/100 = 4/50 = /5 A1 CAO (iii) P(at least one at trough) = (1 (ii)) = 0.9 = 9/100 = 46/50 = 3/5 F 1 F on (ii) crect answer (b)(i) M M Total D D Total Notes: Use of Venn tree diagrams without table completion B0 B0 Table not completed on page 13 but completed on page 10 max of Both row and column totals ie 0.5 and 0.40; CAO Three table values ie 0.35 and 0.0 and 0.05; CAO (ii) Accept answers 1.00 (A) P(neither at gate) = 0.05 F 1 F on table crect answer by otherwise (B) P(only Daisy at gate) = 0.35 F 1 F on table crect answer by otherwise (C) P(exactly one at gate) = P D M + P D M ( ) ( ) Only crect two values from c s table shown and added Can be implied by crect answer = 0.55 A1F F on table crect answer by otherwise Total 11 8

101 MS/SS1B - AQA GCE Mark Scheme 010 June series MS/SS1B (cont) 6 (a) b (gradient) = 3.5 to 3.6 B AWFW (3.547) b (gradient) = 3. to 3.3 () AWFW Treat rounding of crect answers as ISW a (intercept) = 509 to 510 B AWFW ( ) a (intercept) = 507 to 513 () AWFW OR Attempt at ( y ) Attempt at x x y and S xx and xy S ( S yy ) xy () Attempt at crect fmula f b (gradient) (m1) b (gradient) = 3.5 to 3.6 (A1) AWFW a (intercept) = 509 to 510 (A1) 4 AWFW Accept a and b interchanged only if identified crectly by a clearly shown drawn equation and ( ) (all 4 attempted) 17 and 4140 (435100) (both attempted) If a and b are not identified anywhere in question, then: 3.5 to to 510 (b) (i) Crect line drawn on graph (40, 630 to 650) (80, 750 to 790) If B0 but evidence of use of line f points within range 0 x 80 intercept and means (ii) Outliers / at least E and H identified / wide scatter (of points) / large residuals Evidence of a (+ ve) relationship crelation /model is not appropriate Bdep () Dep on A1 A0 in (a) From x 40 to 80 Calc n points shown on graph Allow point ( 0, 500 to 50) Graph Or equivalent wds Or equivalent wds; none of strong/ negative/trend/etc unreliable/invalid (c) (i) Crect two points marked on graph 1 Labels are not required; n is Graph (ii) b (gradient) = 11.6 AWRT ( ) a (intercept) = 3 to 4 AWFW ( ) Crect line on graph (40, 480 to 500) (80, 930 to 970) dep 3 Graph Dependent on (iii) No outliers / less scatter / small residuals Or equivalent wds Strong(er)/me evidence of a positive link/ relationship me rapid increase (of reaction time with age) model is me appropriate Total 14 Or equivalent wds; must indicate change from (b)(ii) in context; not some/weak/etc reliable/valid References to crelation alone B0 9

102 MS/SS1B - AQA GCE Mark Scheme 010 June series MS/SS1B (cont) 7(a)(i) t s = =.48 to.47 AWRT (.4764) Negative value is impossible f a measurement of time Or equivalent; allow if negative value increct not stated (ii) Sample size, n = 80 is large / > 5 Indication that given sample is large Thus sample mean ( T ) ~ approximately nmal due to CLT dep Dependent on previous Requires mean and nmal and CLT (b) CI f µ is 98% (0.98) z =.3 to.33 t () s ± z/t n AWFW (.363) t 79 ( 099. ) =.37 AWRT Used Must have n with n > 1 Thus ± A1F F on z/t only Hence 6.31 ± (1.13 to 1.15) (5.16 to 5.18, 7.44 to 7.46) A1 CAO and AWFW AWFW (5.17, 7.45) Note: Use of t gives 6.31 ± (1.17) (5.14, 7.48) (A1) 4 AWRT (c) µ T < 8 Since CI / UCL < 8 Yes F F on (b); must clearly compare 8 with CI/UCL and state a crect followthrough conclusion P(T 0) > 95% P(T > 0) = 1/80 = 0.01 to P(T 0) = 79/80 = to 0.99 CAO/AWFW; accept eg 1 in 80 B0 f use of nmal distribution CAO/AWFW; accept eg 79 in 80 P(T > 0) < % P(T 0) > % Yes dep 3 Dependent on previous A crect comparison must be clearly stated together with clear conclusion Do not accept use of % 98% OE Total 11 TOTAL 75 10

103 Version 1.0 General Certificate of Education (A-level) January 011 Mathematics MS/SS1B (Specification 6360) Statistics 1B Mark Scheme

104 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 events which all examiners participate in and is the scheme which was used by them in this examination. The standardisation process 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 standardisation each examiner analyses a number of candidates scripts: alternative answers not already covered by the mark scheme are discussed and legislated f. If, after the standardisation process, examiners encounter unusual answers which have not been raised 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 from: aqa.g.uk Copyright 011 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 5 6EX.

105 Mark Scheme General Certificate of Education (A-level) Mathematics Statistics 1B January 011 Key to mark scheme abbreviations M mark is f method m dm mark is dependent on one me M marks and is f method A mark is dependent on M m marks and is f accuracy B mark is independent of M m marks and is f method and accuracy E mark is f explanation ft F follow through from previous increct result CAO crect answer only CSO crect solution only AWFW anything which falls within AWRT anything which rounds to ACF any crect fm AG answer given SC special case OE equivalent A,1 1 ( 0) accuracy marks x EE deduct x marks f each err NMS no method shown PI possibly implied SCA substantially crect approach c candidate sf significant figure(s) 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. 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

106 Mark Scheme General Certificate of Education (A-level) Mathematics Statistics 1B January 011 MS/SS1B r = AWFW ( 0.8) 1(a)(i) 0.6 to 0.98 r = AWFW ( ) If answers are not labelled, assume der is (a)(i) then (a)(ii) (ii) 0.5 to Accept answers as ranges if and only if Eg: (a)(i) 0.7 to 0.9 contained entirely within given ranges (a)(ii) 0.6 to 0.4 B0 (b)(i) r = B3 3 AWRT ( ) r = 0.75 to 0.77 (B) AWFW r = 0.65 to 0.85 () AWFW Attempt at Attempt at x x y y and xy () Sxx Syy and S xy Attempt at substitution into crect cresponding fmula f r (m1) r = (A1) AWRT and (all 5 attempted) and.435 (all 3 attempted) (ii) Strong/fairly strong/moderate positive (linear) crelation/relationship/ association/link (but not trend ) between Bdep1 Dependent on 0.65 < r < 0.85 Or equivalent; must qualify strength and indicate positive Bdep0 f very strong/high/average/ medium/some etc. Circumference/size and weight of (cricket) balls Context; providing 0< r < 1 Total 7 4

107 Mark Scheme General Certificate of Education (A-level) Mathematics Statistics 1B January 011 MS/SS1B (cont) P M C = = = AWRT; accept either crect fraction (a)(i) ( ) P M = = = to AWFW; accept either crect fraction (ii) ( ) 63 1 P LD = = = to AWFW; accept either crect fraction (iii) ( ) = = Accept = A1 AWRT (iv) P( L F) (v) PML' ( ) = = Allow one arithmetic slip Allow one arithmetic slip = = = = = to A1 3 AWFW Any of these implies (b) P( L L F) = Or = AWRT (c) PL ( C ( LD+ O) ) = Crect numerat Crect denominat SC The three crect fractions identified but not multiplied M0 M0 A Note that a denominat of 3 M (second and third marks) = to A1 4 AWFW NB: 0.06 with no wking only = with no wking 4 marks Total 14 5

108 Mark Scheme General Certificate of Education (A-level) Mathematics Statistics 1B January 011 MS/SS1B (cont) 3(a)(i) AG (At least) one crect expression seen Igne contradictions = 0.99 Accept any valid equivalent (ii) = and = SC In (a)(i) and (a)(ii) allow etc Both CAO Can not be implied from (a)(i) Similar fms f lower boundary (b) Mean, x = 1.06 CAO fx = 106. Igne notation Standard deviation, s σ = B 3 AWRT fx = If B0 B0, can be awarded f attempt fx at % 0.99 z =.57 to.58 (c)(i) ( ) CI f µ is x ( z t) ± ( s σ ) n () AWFW (.5758) t = AWRT 99 ( 0.995).66 Used Must have n with n > Thus 1.06 ±.5758 A1F F on x, s / σ and z/t Hence 1.06 ± 0.01 (1.05, 1.07) A1 4 AWRT; award even if previous inaccuracies in x, s/σ z/t Dependent on A1F (ii) Volumes/ X / (parent) population may be modelled by a nmal distribution / is nmally distributed (Igne contradictions) (iii) Sample data grouped Exact sample values unknown / midpoints used x and s calculated from grouped data (d)(i) CI f µ CI in (c)(i) > 1 LCL of CI f µ LCL of CI in (c)(i) > Or equivalent; not distribution, data, values (in table), sample, n large, n simply It is stated in question σ unknown s calculated from a sample x (not µ ) and s are estimates NOT data values rounded Or equivalent; must compare CI to 1 Dependent on CI in (c)(i) > 1 (ii) all sample/ table/ data volumes/ values/ x-values/ cartons are within this range ( none/0 1 volumes outside) Total 13 6

109 Mark Scheme General Certificate of Education (A-level) Mathematics Statistics 1B January 011 MS/SS1B (cont) R B 15, (a) ( ) (i) P( R 5) = 0.6( 0) to AWFW (0.608) (ii) P( R 10) 1 P( R 10) > = Requires 1 = 1 ( ) = 0.05 to 0.06 A1 AWFW (0.055) (iii) P( R = 6) = 0.45 (a)(i) Accept 3dp rounding truncation Can be implied by 0.05 to 0.06 but not by to Can be implied by a crect answer = ( 0.45 ) ( 0.55 ) 6 = to 0.19 A1 AWFW (0.1914) (iv) P( 5 R 10) ( p ) = Accept 3dp rounding truncation 1 p p1 M0 M0 A0 1 p p M0 M0 A0 ( ) 1 p ( p ) 1 M0 A0 1 only providing result > 0 Minus ( p ) Accept 3dp rounding truncation = to A1 3 AWFW (0.8541) Or B (15, 0.45) terms stated f at least 3 values within 4 R 11 gives probability () Can be implied by a crect answer = to (A) AWFW (0.8541) (b)(i) P( S ) = 0.85 plus 1 minus CAO; requires plus minus ( ) ( ) CAO; not simply = 0.97 NB: ( ) B0 B0 ( ) + ( ) + ( ) B0 AG (ii) Accept 3dp rounding truncation P( S 48) = 0.81 to M M f the three crectly expressed terms f B (50, 0.03) B (50, 0.97) added =0.81(0) to A1 3 AWFW (0.8108) NB: Answer = only (iii) p = = 0.15 CAO; may be implied by crect answer crect expression f mean Mean, µ = = 1 CAO SC Mean = 9.6 only Total 15 7

110 Mark Scheme General Certificate of Education (A-level) Mathematics Statistics 1B January 011 MS/SS1B (cont) 5(a) Time taken is dependent upon 1 Or equivalent leaving time (b) b (gradient) = 1.8 ( 141/110) B AWRT; (CAO equivalent) (1.818) b (gradient) = 1.5 to 1.35 () AWFW Treat rounding of crect answers as ISW a (intercept) = 9.95 to 30 ( 659/) B AWFW; (CAO equivalent) ( ) a (intercept) = 9 to 31 () AWFW Thus y = x F 5 F on a and b Attempt at x x yand xy ( y and 0575 (47494) ) (All four attempted) () Attempt at Sxx and Sxy ( S 750 and 355 (510) yy) (Both attempted) Attempt at crect fmula f b gradient (m1) b (gradient) = 1.8 ( 141/110) (A1) AWRT; (CAO equivalent) a (intercept) = 9.95 to 30 ( 659/) (A1) AWFW; (CAO equivalent) Thus y = x (F) F on a and b Accept a and b interchanged only if identified crectly by a clearly shown equation If a and b are not identified anywhere in the question, then: 1.5 to to 30 (c) 7.45 am x = 15 CAO; stated, used implied y 15 = Use of 10 < x < 0 = 47 to 5 A1 AWFW (49.) Time befe 9.00 am = May be implied 9.00 ( c s y 15 ) = 3 to 8 A1 5 AWFW (5.8) SC Answer of 17 CAO NB: An answer of 8.3 to 8.37 (use of c s y 15 = 58) gains marks gains A1 M0 A0 (d)(i) y 85 = = 135 to146 1 AWFW (138.9) (ii) Extrapolation/ outside/ above range of x-values Implies leaves home at 8.55 so different traffic conditions Total 14 Or equivalent Or equivalent; 8.55 may be implied by 5 minutes 8

111 Mark Scheme General Certificate of Education (A-level) Mathematics Statistics 1B January 011 MS/SS1B (cont) 6(a) Volume, V N ( 41, 8 ) P < 400 = P Z < 8 (i) ( V ) ( Z ) ( Z ) = P < 1.5 = 1 P < 1.5 = = to Standardising 400 with 41 and 8 and/ 41 x ( ) Area change May be implied by a crect answer an answer < 0.5 A1 3 AWFW ( ) (ii) P( V 40) P( Z 1) > = > CAO but igne inequality and sign May be implied by a crect answer ( ) = 1 P Z < 1 = = 0.158to0.159 AWFW ( ) (iii) P( V = 410) = 0 zero impossible 1 Igne any wking B0 f impossible to calculate no answer (b)(i) A statement/indication that ( ) and/.363 are z-values Do not allow Φ ( 0.99) =.363, etc but allow Φ 1 ( 0.99) =.363 Do not award f z-value(s) simply embedded in standardisation statement(s) v µ A clear use of z = v = µ + σ σ with 400 and/ 40 (condone sign errs) Simple statement that z =± and/ z =±.363 sketch of nmal curve with at least one z-value marked SC Immediate algebraic use of v µ = zσ A0 The two given equations crectly derived A1 3 AG; watch f sign inconsistencies (ii) Thus 0 = ( ) σ A sensible (one that would lead to values required if completed crectly) attempt at solving the two given equations by eliminating µ σ Do NOT allow MC MR σ = 5.04 A1 AWRT (5.0366) µ = 408 A1 3 AWRT (408.84) Total 1 TOTAL 75 9

112 Version 1.0 General Certificate of Education (A-level) June 011 Mathematics MS/SS1B (Specification 6360) Statistics 1B Final Mark Scheme

113 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 events which all examiners participate in and is the scheme which was used by them in this examination. The standardisation process 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 standardisation each examiner analyses a number of candidates scripts: alternative answers not already covered by the mark scheme are discussed and legislated f. If, after the standardisation process, examiners encounter unusual answers which have not been raised 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 from: aqa.g.uk Copyright 011 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 5 6EX.

114 Key to mark scheme abbreviations M mark is f method m dm mark is dependent on one me M marks and is f method A mark is dependent on M m marks and is f accuracy B mark is independent of M m marks and is f method and accuracy E mark is f explanation ft F follow through from previous increct result CAO crect answer only CSO crect solution only AWFW anything which falls within AWRT anything which rounds to ACF any crect fm AG answer given SC special case OE equivalent A,1 1 ( 0) accuracy marks x EE deduct x marks f each err NMS no method shown PI possibly implied SCA substantially crect approach c candidate sf significant figure(s) 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. 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.

115 MS/SS1B 1 (a)(i) Mode = 53 1 CAO (ii) Median = 5 CAO Upper quartile = 53 Lower quartile = 50 CAO; either May be implied by IQR = 3 Interquartile range = 3 3 CAO; do not award if seen to be not based on 53 and 50 (b)(i) Range = 71 7 = 44 1 CAO; do not award if seen to be not based on 71 and 7 (ii) Mean, x = 51 to 51.4 B AWFW fx = 135 x = 51. Award if divis seen not to be 85 but answer in range Note: If B0 then can award f attempt at fx 85 seen Standard deviation, s = 4.1 to 4.8 B 4 Award if divis seen not to be but answer in range Igne notation and condone increct midpoints (eg upper lower limits used) AWFW fx = = 4.17 s = 4.4 (c) Interquartile range (IQR) Named Not affected by unknown/large/small/extreme/ outlying/7 & 71 values OR Bdep1 Or equivalent Dependent on previous Only negative comments on other measures Bdep0 Me than one named B0 Bdep0 Range B0 Bdep0 Standard deviation (s ) () Named Uses all data values (Bdep1) Total 11 Or equivalent Dependent on previous () Only negative comments on other measures Bdep0

116 MS/SS1B (cont) (a)(i) Diameter, D ~ N(57.15, 0.04 ) P(D < 57.) = P Z 004. Standardising 57. with and 0.04; allow ( ) = P(Z < 1.5) A1 CAO; igne inequality and sign May be implied by a crect answer = to A1 3 AWFW ( ) (ii) P(57.1 < D < 57.) = p (1 p) Allow even if increct standardising providing p (1 p) seen May be implied by a crect answer = = to 0.79(0) A1 AWFW ( ) (b)(i) P(16 balls < 57.) = p 16 with 0 < p < 1 Any probability to power 16 1 p 16 ; do not allow multiplying facts If only seen in (b)(ii), allow just = [(a)(i)] 16 = ( ) 16 = to 0.17(0) A1 AWFW ( ) (ii) Variance of D 16 = 0.04 /16 = Sd of D 16 = 0.04/ 16 = 0.01 CAO Stated used (see Notes below) CAO If only seen in (b)(i), allow just P D = P Z 001. Standardising with and 0.01 equivalent; allow ( ) = P(Z > 1) = 1 P(Z < 1) m1 = = = to A1 4 Notes: Igne partial/incomplete attempts at (ii) in (i) if followed by crect method Answer to (i) (ii) repeated Area change May be implied by a crect answer answer < 0.5 AWFW ( ) (1 answer) max Mark two complete answers in (i) as two attempts so (0 + )/ 1max Mark as per scheme; thus ( max, 0) (0, 4 max) Total 11

117 MS/SS1B (cont) 3 (a) b (gradient) = 191 B CAO b (gradient) = 190 to 19 () AWFW Treat rounding of crect answers as ISW a (intercept) = 115 B 4 CAO a (intercept) = 93 to 137 () AWFW OR Attempt at Attempt at x x y & S xx & S xy S yy xy y () & ( ) (all 4 attempted) 14 & 64 (714668) (both attempted) Attempt at crect fmula f b (gradient) (m1) b (gradient) = 191 (A1) CAO a (intercept) = 115 (A1) CAO Accept a & b interchanged only if identified and used crectly in (ii) If a and b are not identified anywhere in question, then: 190 to to 137 (ii) y 4 = = B Either; igne units ( 4699) = 4650 to 4750 () AWFW SC: ( )/ = 4673 to 4674 If B0 but clear evidence of crect use of c s equation with x = 4 () (iii) (Maximum) temperature (in February) is likely to be/will be lower/different Must imply a temperature comparison with July 1 Or equivalent; must be clear indication that (max) temperature is less than/different Extrapolation/not July/not summer/winter/etc B0 (iv) Rainfall amount/wind strength/sunshine hours/ daylight hours/opening times/day of week/ visit numbers/public holidays/school holidays/ local attractions/etc Allow if at least 1 variable crectly identified 1 Or equivalent Accept any sensible reason; do not penalise f dubious variable name so, f example, accept rainfall Minimum/average temp/etc B0 Quality price of food/ staff/etc B0 Total 8

118 MS/SS1B (cont) 3 (b) Any line (straight, freehand, curve) from (0, 1) on Figure 1 from (0, 5) on Figure Accept clear marking of (0, 1) (0, 5) with no line (i) Straight, not freehand, line from (0, 1) to (40, 5) on F1 only; allow line extensions and only very min inaccuracies in points plotted (ii) Straight, not freehand, line from (0, 5) to (10, 1) on F only; allow line extensions and only very min inaccuracies in points plotted (10, 0.5) (0, ) (30, 3.5) 3 (, 4.) (4, 3.4) (6,.6) (8, 1.8) Notes: Both lines on F1 B0 max Both lines on F B0 max >1 undeleted line on either F1 F max Total 3

119 MS/SS1B (cont) 4 (a) = 1.94 AG Fully crect expression equivalent must be seen Note: s B0 (b) (i) 96% (0.96) z =.05 to.06 AWFW (.0537) CI f is x s z n Used with 51.1 and 1.94 crectly Must have n with n > 1 Thus AF1 F on z only Hence (50.5, 51.7) (ii) Claim is > 50 Adep1 4 CAO/AWRT Dependent on AF1 but not on z so can be gained using an increct z AWRT Clear crect comparison of 50 with LCL CI so Claim is suppted/reasonable/crect/true/etc Must be consistent with c s comparison BF1 F on CI (50 < LCL CI) Bdep1 Dependent on BF1 (c) x ns = n < 50 SC: Quoted values of 49., (AWRT) so Allow any multiple of 1.94 Must clearly indicate the value of a numerical expression giving a result less than 50 Some individual packets are likely to/will contain less than 50 grams A1 Or equivalent Total 9

120 MS/SS1B (cont) 5 (a)(i) J J Total 0.35 and 0.7; CAO W W ; CAO Total and 0.; CAO Notes: Use of Venn tree diagrams without table completion B0 B0 B0 Printed table not completed but constructed and completed on Page 1/13 max Accept fractional answers Do not accept percentages (ii) P(purchases exactly one) P W J 0.15 = = = 0.5 5/100 5/0 1/4 A1 CAO Only c s equivalent to 0.10 shown and added to 0.15 Can be implied by crect answer (iii) (A) P W J = 0.8 &/ P W P J P W J = 1.35 = 0.55 (>0); accept if indicated in a Venn diagram P W P J = 1.35 >0 impossible (B) P W J = 0.79 &/ P W = 0.65 P J W = 0.85 &/ P J = 0.70 P W P J &/ P W J = 0.55 = 0.45 to 0.46 Bdep1 3 Any one of these three seen Igne contradictions, explanations & justifications Do not accept use of W and/ J AWRT Any one of these three seen Igne contradictions, explanations & justifications AWFW (b) (i) Do not allow multiplying facts in (b) P(0) = Can be implied by crect answer 1 ( ) = /1000 CAO (ii) P() = = = = = 1 ( ) M () F either method: At least two bold expressions crect Only one bold expression crect Can be implied by crect answer F second method: Must have 1 f any marks = / /500 A1 3 CAO; do not imply this from (i) Total 13

121 MS/SS1B (cont) 6 (a) X ~ B(10, 0.15) (i) P(X ) = 0.8(0) 1 AWRT (0.80) (ii) P(X ) = 1 P(X 1) = 1 ( ) Requires 1 Accept 3/ dp rounding truncation Can be implied by to but not by to 0.18(0) = to A1 AWFW (0.4557) (iii) P(1 < X < 5) = (p 1 ) Accept 3 dp rounding truncation p p 1 M0 M0 A0 (1 p ) p 1 M0 M0 A0 p 1 (1 p ) M0 A0 only providing result > 0 minus (p ) Accept 3 dp rounding truncation = to A1 3 AWFW (0.4458) OR B(10, 0.15) expressions stated f at least 3 terms within 1 X 5 gives probability () Can be implied by a crect answer = to (A) AWFW (0.4458) (b) Y ~ B(50, 0.15) Nmal approximation 0 marks (i) P(Y > 5) = 1 P(Y 5) = 1 ( ) Requires 1 Accept 3 dp rounding truncation Can be implied by 0.78(0) to but not by to 0.89 = 0.78(0) to A1 AWFW (0.7806) (ii) P(5 Y 10) = (p 1 ) Accept /3 dp rounding truncation p p 1 M0 M0 A0 (1 p ) p 1 M0 M0 A0 p 1 (1 p ) M0 A0 only providing result > 0 minus (p ) Accept 3 dp rounding truncation = A1 3 AWRT (0.7680) OR B(50, 0.15) expressions stated f at least 3 terms within 4 Y 10 gives probability () Can be implied by a crect answer = (A) AWRT (0.7680) Total 11

122 MS/SS1B (cont) 7 (a) Ryan: Value indicates that as volume increases then weight decreases Sunil: Value indicates no crelation/relationship/ association/link between volume and weight SC: If B0 B0: Would expect weight to increase with volume Would expect strong(er) positive crelation between weight and volume () Or equivalent in context Or equivalent in context Or equivalent in context (b) Ryan & Sunil: r is not affected by units/(linear) scaling Or equivalent Tim: r is not affected by sample size 0.61 > 1 impossibility Either; equivalent (c) (i) r = to B3 3 AWFW ( ) r = 0.54 to 0.55 (B) AWFW r = 0.5 to 0.6 () AWFW OR Attempt at v v w Attempt at S vv S ww & S vw Attempt at substitution into crect cresponding fmula f r w & vw & (all 5 attempted) () Accept notation of x and y & 13.5 (all 3 attempted) (m1) r = to (A1) AWFW (ii) (Quite fairly) weak/some/moderate positive (linear) crelation/relationship/ association/link (but not trend ) Bdep1 Dependent on 0.5 r 0.6 Or equivalent; must qualify strength and state positive Bdep0 f very strong/strong/high/ good/average/medium/reasonable/ po/very weak/little/etc between volumes and weights of suitcases Context; providing 0 < r < 1 Total 9 TOTAL 75

123 Version 1.0 General Certificate of Education (A-level) January 01 Mathematics MS/SS1B (Specification 6360) Statistics 1B Final Mark Scheme

124 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 events which all examiners participate in and is the scheme which was used by them in this examination. The standardisation process ensures that the mark scheme covers the students responses to questions and that every examiner understands and applies it in the same crect way. As preparation f standardisation each examiner analyses a number of students scripts: alternative answers not already covered by the mark scheme are discussed and legislated f. If, after the standardisation process, examiners encounter unusual answers which have not been raised 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 students 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 from: aqa.g.uk Copyright 01 AQA and its licenss. All rights reserved. Copyright AQA retains the copyright on all its publications. However, registered schools/colleges 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 schools/colleges 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 5 6EX.

125 Key to mark scheme abbreviations M mark is f method m dm mark is dependent on one me M marks and is f method A mark is dependent on M m marks and is f accuracy B mark is independent of M m marks and is f method and accuracy E mark is f explanation ft F follow through from previous increct result CAO crect answer only CSO crect solution only AWFW anything which falls within AWRT anything which rounds to ACF any crect fm AG answer given SC special case OE equivalent A,1 1 ( 0) accuracy marks x EE deduct x marks f each err NMS no method shown PI possibly implied SCA substantially crect approach c candidate sf significant figure(s) 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. 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.

126 MS/SS1B - AQA GCE Mark Scheme (VP) 01 January Series MS/SS1B 1 (a) Median = 10 CAO Upper quartile = 11 Lower quartile = 9 CAO; either May be implied by IQR = Interquartile range = 3 CAO; do not award if seen to be not based on 11 and 9 (b) Do not group results 1 Illustrations f : Use all values Replace 6 by use (0), 1,..., 6 Replace 1 by use 1, 13,... Recd exact values/frequencies OE statement that implies non grouping recding of all separate observed values Illustrations f B0: Recd max and/ min values Construct frequency table Use 1, 1, 13 Total 4 (a) CAO; accept minimum of Probably crect PC Pc pc pc (b) Definitely increct CAO; accept minimum of DI Di di di (c) Probably increct 3 Notes: Igne reasoning in all parts, unless it includes of the 4 statements in which case B0 If answers not labelled, then assume above der CAO; accept minimum of PI Pi pi pi Definitely wrong, etc B0 Likely crect, etc B0 Total 3

127 MS/SS1B - AQA GCE Mark Scheme (VP) 01 January Series MS/SS1B (cont) 3(a) Volume, X ~ N(3, 10 ) (i) P(X < 40) = 40 3 Standardising 40 with 3 and 10; P Z 10 allow (3 40) = P(Z < 0.8) A1 CAO; igne inequality and sign May be implied by a crect answer = A1 3 AWRT ( ) (ii) P(X > 5) = P(Z > 0.7) = P(Z < +0.7) Area change May be implied by a crect answer an answer > 0.5 = A1 AWRT ( ) (iii) P(5 < X < 40) = (i) (1 (ii) OE; allow new start igning (i) & (ii) Allow even if increct standardising providing 0 < answer < 1 May be implied by a crect answer = ( ) = Note: If (ii) is 0.4, then ( ) = M0 A0 A1 AWRT ( ) (b) P(B > 65) = P Z P Z 13.4 = P(Z > 1.65) = 1 P(Z < 1.65) m1 Attempt to change from B to X using (48 to 49), 3 and 10 Attempt to wk with distribution of B using 65, (4.8 to 4.9) and 13.4 Area change May be implied by a crect answer an answer < 0.5 = = to 0.05(0) A1 3 AWFW ( ) (c) Other fuels Other vehicles with an example (not other cars) Other types of customer Minimum purchase (policy) Purchases in integer/fixed s Customers filling fuel cans B,1 Size of car/engine/fuel tank B0 Price of fuel B0 Customer paying capacity B0 Must be two clearly different valid reasons f award of B Drivers and vehicles related eg lry drivers & lries Total 1

128 MS/SS1B - AQA GCE Mark Scheme (VP) 01 January Series MS/SS1B (cont) 4(a) U ~ B(40, 0.15) Used somewhere in (a) (i) P(U = 6) = Accept 3 dp rounding truncation = Can be implied by a crect answer = A1 3 AWRT (0.174) (ii) P(U 5) = 0.43 to AWFW (0.435) (iii) See supplementary sheet f individual probabilities P(5 < U < 10) = (p 1 ) Accept 3 dp rounding truncation but allow 0.97 p p 1 M0 M0 A0 (1 p ) p 1 M0 M0 A0 p 1 (1 p ) M0 A0 (1 p ) (1 p 1 ) (A1) only providing result > 0 MINUS (p ) Accept 3 dp rounding truncation = 0.5(00) to A1 3 AWFW (0.5003) (b) Mean = = 4.8 CAO (V =) (SD =) Either numerical expression; igne terminology May be implied by 4.08 CAO seen.0 AWRT seen (SD ) =.0 A1 3 AWRT (.0199) Do not award if labelled V (c) Mean = 7.7 CAO x 77 SD = 1.6 to 1.34 AWFW x 609 (Sample) mean is bigger / greater / different 7.7/3 = 0.4 > 0.15 and (Sample) SD is smaller / less / different Bdep1 Both; dependent on all previous 5 marks of A1 Can be sced f increct (b) re-done crectly in (c) Means & SDs different Bdep0 So model appears unsuitable Bdep1 4 OE; dependent on Bdep1 Total 14

129 MS/SS1B - AQA GCE Mark Scheme (VP) 01 January Series MS/SS1B (cont) 5 See supplementary sheet f alternative solutions and additional guidelines to parts (b), (d) and (e) (a) Calific value depends upon moisture content Moisture (content) is set/are fixed values 1 Must be in context; not it, etc Use of x and y B0 (b) b (gradient) = B AWRT; including ve sign ( ) b (gradient) = 0.07 to 0.08 () AWFW; including ve sign Treat rounding of crect answers as ISW a (intercept) = 5.35 to 5.36 B AWFW ( ) a (intercept) = 5.1 to 5.6 () AWFW Thus y = (5.35 to 5.36) 0.076x BF1 5 F on a and b even if rounded (c) a: calific value of wood with zero/no moisture dry maximum calific value OE; a 0 B0 b: each 1(%) rise in moisture content reduces calific value by MWh/tonne As x increases y decreases B () 3 In context and with values; F on b b 0 B0 Negative relationship/crelation (d) y 7 = 3.8 to 3.3 B AWFW ( ) =.5 to 3.5 () AWFW; even if by interpolation from iginal data giving likely values of (e) r(35,.5) = 0.1 to 0.19 B AWFW; including ve sign ( ) = 0.1 to 0.3 () AWFW; igne sign (f) Good/reasonable/accurate/crect/etc Accept me positive qualifying adjectives 1 OE; igne reasoning Very good () Not good (B0) (g)(i) Extrapolation/outside (observed) range (of x) 1 OE (ii) y 80 = 0.5 to 1 AWFW ( ) Negative value f calific value is impossible Me energy needed than is generated Bdep1 OE; dependent on Must be in context; negative value impossible Bdep0 Total 17

130 MS/SS1B - AQA GCE Mark Scheme (VP) 01 January Series MS/SS1B (cont) 6 See supplementary sheet f alternative solutions to parts (a)(i) and (b)(ii) (a)(i) Table Method (- way with either R C totals) A A Total E E Total Bdep ; CAO; allow fractions 0.05 and 0.3; CAO; allow fractions 0.1; AG so dependent on (ii) P( 1) = 0.9 9/10 1 CAO (iii) P(1) = = 1 ( ) = /100 7/0 1 CAO (b)(i) P(3) = OE; implied by crect answer = / /00 CAO (ii) 0.55 (1 0.3) ( ) 0.5 ( ) ( ) 0.65 At least one of these expressions values ( ) OE; implied by crect answer = 0.81 to AWFW (0.815) A1 4 CAO Total 11

131 MS/SS1B - AQA GCE Mark Scheme (VP) 01 January Series MS/SS1B (cont) 7(a) (i) 90 x 50 = CAO s 85.5 s Igne notation s = 4(.0) 4000 to 4001 A1 3 AWRT/AWFW ( ) SCs: (f no seen wking) ( = ) A1 f to 4001 A0 f to 3800 (ii) See supplementary sheet f alternative solutions x ns = n 4. 0 < 0 SC: Accept quoted values of ( 4 to 1) (n = ) ( 8.5 to 3.5) (n = 3) (both AWFW) and negative salaries are impossible A1 Allow (45 to 47) and any multiple of (3.5 to 4.5) which gives value < 0 Must clearly state the value of a numerical expression OE; must be in context Negative values impossible A0 (b)(i) Large sample n > 5 30 n = 50 OE so Must indicate CLT; dependent on CLT applies Bdep1 Indication that other than sample mean is nmally distributed Bdep0 (ii) 99% (0.99) z =.57 to.58 AWFW (.5758) CI f is x s z n Used with ( x & s) from (a)(i) and z(1.64 to.58) & n with n > 1 Thus AF1 50 F on ( x & s) with z(1.64 to to to.58) & Hence 45.8 (8.7 to 8.8) (8700 to 8800) OR (37.(0) to 37.1, 54.5 to 54.6) (37000 to 37100, to 54600) A1 4 CAO/AWFW (8.74) Igne (absence of) quoted units AWFW (c) See supplementary sheet f additional illustrations Clear crect comparison of with c s UCL CI Accept compared with c s 54.5 to 54.6 (ie different units) (6/ %) </ 0.5 5% OE; crect comparison mentioning both 1% and 5% Reject both/each of the two claims Bdep1 3 Dependent on Total 14

132 MS/SS1B - AQA GCE Mark Scheme (VP) 01 January Series MS/SS1B (cont) 4 Alternative solution (a)(iii) B(40, 0.15) expressions stated f at least 3 terms within 5 U 10 gives probability = 0.5(00) to M A1 Can be implied by a crect answer AWFW (0.5003) u (5) (10) P(U = u) (0.169) (0.0373) 3 5 Alternative solutions and additional guidelines (b) Attempt at Attempt at x x S xx & S xy S yy y & xy y & (11.33) (all 4 attempted) 4550 & 345 (6.56) (both attempted) Attempt at crect fmula f b (gradient) m1 b (gradient) = A1 AWRT a (intercept) = 5.35 to 5.36 A1 AWFW Thus y = (5.35 to 5.36) 0.076x BF1 F on a and b even if rounded Notes: 1 If a and b interchanged and equation y = ax + b used max of 5 marks If a and b interchanged and equation y = a + bx used maximum of BF1 3 Marks lost here cannot be gained from subsequent wk in parts (d) and/ (e) 5 If a and b are not identified anywhere in equation, then: 0.07 to to 5.6 (d) y 7 = (5.35 to 5.36) Clear evidence of crect use of c s equation with x = 7 = 3.8 to 3.3 A1 AWFW ( ) (e) r(35,.5) =.5 y 35 Used; allow y 35.5 =.5 {(5.35 to 5.36) } = 0.1 to 0.19 A1 AWFW ( )

133 MS/SS1B - AQA GCE Mark Scheme (VP) 01 January Series MS/SS1B (cont) 6 Alternative solutions (a)(i) Venn Diagram Method 0.55; CAO 0.10 A 0.3 E and 0.05; CAO Bdep ; AG so dependent on (a)(i) Fmula Method P( 1) = OR p = 1 OR M () Full justification f numerical expression Insufficient justification numerical expression only P(0) = 1 P( 1) OR = = p = 1 OR = 0.1 A1 3 AG; gained from M (b)(ii) 0.1 (1 0.4) 0.06 ( ) ( ) ( ) ( ) 0.04 At least one of these expressions values 1 (0.1875) OE; implied by crect answer = 0.81 to A1 4 AWFW (0.815) CAO f equivalent fraction (b)(ii) ( p) where 0 < p < 0.45 ( ) 0.5 ( ) ( ) 0.65 At least one of these expressions values OE; implied by crect answer = 0.81 to A1 4 AWFW (0.815) CAO f equivalent fraction

134 MS/SS1B - AQA GCE Mark Scheme (VP) 01 January Series MS/SS1B (cont) 7 (a)(ii) Alternative solutions P(X < 0 N(45.8, 4.0 ) = P(Z < 1.91) = 0.07 to 0.03 A1 Standardising 0 using 45.8 & 4.0 In addition to probability within range, must state that negative salaries are impossible P(X > 60 N(45.8, 4.0 )) = P(Z > 0.59) = 0.7 to 0.8 A1 Standardising 60 using 45.8 & 4.0 In addition to probability within range, must compare calculated value to 6/50 = 0.1 OE (c) Additional comment illustrations It/(claimed) mean/(claimed) value > UCL/CI B0 Must indicate % have (mean) weights between CLs so... B0 Any comparison of 60 ( ) with UCL/CI B0 Value of 60 does not refer to mean P(X > 60 N(45.8, 4.0 )) = P(Z > 0.59) = (0.7 to 0.8) > 6/50 = 0.1 B0 Assumes salaries ~ N; cf (a)(ii)

135 Version 1.0 General Certificate of Education (A-level) June 01 Mathematics MS/SS1B (Specification 6360) Statistics 1B Mark Scheme

136 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 events which all examiners participate in and is the scheme which was used by them in this examination. The standardisation process ensures that the mark scheme covers the students responses to questions and that every examiner understands and applies it in the same crect way. As preparation f standardisation each examiner analyses a number of students scripts: alternative answers not already covered by the mark scheme are discussed and legislated f. If, after the standardisation process, examiners encounter unusual answers which have not been raised 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 students 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 from: aqa.g.uk Copyright 01 AQA and its licenss. All rights reserved. Copyright AQA retains the copyright on all its publications. However, registered schools/colleges 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 schools/colleges 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 5 6EX.

137 Key to mark scheme abbreviations M mark is f method m dm mark is dependent on one me M marks and is f method A mark is dependent on M m marks and is f accuracy B mark is independent of M m marks and is f method and accuracy E mark is f explanation ft F follow through from previous increct result CAO crect answer only CSO crect solution only AWFW anything which falls within AWRT anything which rounds to ACF any crect fm AG answer given SC special case OE equivalent A,1 1 ( 0) accuracy marks x EE deduct x marks f each err NMS no method shown PI possibly implied SCA substantially crect approach c candidate sf significant figure(s) 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. 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.

138 MS/SS1B - AQA GCE Mark Scheme 01 June Series MS/SS1B 1 (a) r Sxy S S xx yy = = 0.35 A1 Crect substitution into crect fmula May be implied by a crect answer AWRT ( ) (b) SC Some / (very) weak / (very) little / (very)slight negative crelation/relationship/association/link Adep1 Dependent on OE; must qualify strength and state negative Igne extra wds unless contradict Not no, low, small, unlikely trend between width and thickness of lengths of steel Context; do not allow cms mms r = (+)0.35 A0 Adep0 max Total 4 (a)(i) Mode = 3 1 CAO (ii) Median (88 th value) = CAO Upper quartile (13 nd value) = 3 Lower quartile (44 th value) = 0 CAO; either May be implied by IQR = 3 Interquartile range = 3 3 CAO; do not award if seen to be not based on 3 and 0 (b) CAO; but only award (.3) Mean =.3 B if increct mid-points fx seen Mean = 1 to 3 () AWFW ( fx = 390.5) Standard deviation = B AWRT (s = = 6.37) Standard deviation = 5 to 7 () 4 AWFW ( fx = ) SC Only if B0 B0 B0 then award as follows but only up to a maximum total part mark of 1 At least crect mid-points 4.5, 14.5, 7, 3, 37, 44.5, 54 seen Clear use of fx/( ) (c) Mean = (c s mean from (b)) = Adding (1.6 equivalent) CAO to (c s mean from (b)) to (c s new mean) Mean = 3.9 AF1 F on (c s mean from (b)) on (c s new mean) Total 10

139 MS/SS1B - AQA GCE Mark Scheme 01 June Series MS/SS1B (cont) 3 (a) b (gradient) =.7 B AWRT (.7075) b (gradient) =. to.3 () AWFW Treat rounding of crect answers as ISW a (intercept) = 4.16 to 4. B AWFW ( ) a (intercept) = 3 to 7 () AWFW Attempt at Attempt at x x y & S xx & S xy S yy xy y Attempt at crect fmula f b (gradient) () (m1) b (gradient) =.7 (A1) AWRT a (intercept) = 4.16 to 4. (A1) 4 AWFW & (135908) (all 4 attempted) 5300 & 1035 (7608) (both attempted) Notes 1 Values of a and b interchanged and equation y = ax + b stated in (a) max of 4 marks Values of a and b interchanged and equation y = a + bx stated in (a) 0 marks 3 Values are not identified simply a = # and b = #, then. to.3 and 3 to 7 but accept, f example, as identification, [a = #, b = # with y = a + bx but no substitution f a & b] [intercept(a) = #, gradient(b) = #] 4 b = 407/1060 CAO B, otherwise if fraction equates to. to.3 (Notes 1, & 3 also apply) a = 1/53 CAO B, otherwise if fraction equates to 3 to 7 (Notes 1, & 3 also apply) 5 Some/all of marks can be sced in (b), and in c(ii) & (iii), even if some/all of marks are lost in (a) but marks lost in (a) cannot be recouped by subsequent wking in (b) (b) Crect straight line drawn on scatter diagram Crect shtened and/ freehand line drawn on scatter diagram B () Line must go from x 0 to x 70 and fall between the following lines: Lower: (10, 5) (80, 180) Upper: (10, 30) (80, 190) Notes 1 If B0 but seen crect attempt at points even if increctly evaluated If B0 but no seen evidence to suppt points (crect increct) marked on scatter diagram M0 (c)(i) Crect straight line drawn on scatter diagram Crect shtened and/ freehand line drawn on scatter diagram B () Line must go from x 0 to x 70 and fall between the following lines: Lower: (10, 60) (80, 75) Upper: (10, 65) (80, 85) Notes 1 If B0 but seen crect attempt at points even if increctly evaluated If B0 but no seen evidence to suppt points (crect increct) marked on scatter diagram M0 (ii) 7 to 9 1 AWFW (calculation 7.75) Must clearly identify x-value Thus (7 to 9, y-value) B0 (iii) At low temperatures me B (than A) dissolves At high temperatures me A (than B) dissolves Either; OE (eg a comparison using lines and/ data at a specific temperature but not at 0 C) Amount increases me rapidly f A (than B) Amount increases me slowly f B (than A) Either; OE Any comments about b a B0 Comment about rate must relate to temp Total 11

140 MS/SS1B - AQA GCE Mark Scheme 01 June Series MS/SS1B (cont) 4 Ratios (eg 194:640) are only penalised by 1 accuracy mark at first crect answer (a)(i) P(B = 3) = 194/640 97/ % 1 CAO AWRT ( ) (ii) P(T ) = = 563/640 (0.879 to 0.88) (87.9% to 88%) A1 CAO AWFW ( ) (iii) P(B = 3 & T ) = = 187/ % A1 CAO AWRT (0.9188) (iv) P(B 3 T = ) = Crect numerat (accept both 640) Crect denominat = 153/17 (0.888 to 0.89) (88.8% to 89%) A1 3 CAO AWFW ( ) Answers as fractions, percentages ratios lose accuracy (A & B) marks in (b) & (c) (b) (a)(i) (a)(ii) (a)(iii) Attempted since = 0.65 to A1 AWFW & AWRT SC Any crect fully-explained reasoning, using other than answers from part (a), which results in an inequality ( ) with both sides as numerically crect decimals (to 3 dp) (eg P(B = 3) = P(B = 3 T = ) = 7/17 = 0.419) but no/unclear/incomplete reasoning no/increct/incomplete numerical wk B0 (c) P(T 3T 4T B = 3) = Crect 3 values multiplied in numerat Crect 3 values multiplied in denominat (all AWRT) (OE products) abc multiplied by < (a, b & c) < 1 = to A1 4 AWFW ( ) Notes 1 Increct answer with no wking 0 marks The 3 crect fractions/decimals identified but not multiplied (eg added) M0 M0 A0 3 The 3 crect fractions/decimals identified together with (AWRT) M0 A0 4 A denominat of 194 C 3 = M ( nd & 3 rd marks) Total 14

141 MS/SS1B - AQA GCE Mark Scheme 01 June Series MS/SS1B (cont) 5 In (a)(i) & (c), igne the inclusion of a lower limit of 0; it has no effect on either answer (a) Weight, W ~ N(.75, 0.15 ) (i) P(W <.8) = Standardising.8 with.75 and P Z ; allow (.75.8) = P(Z < /3) A1 AWRT/CAO; igne inequality and sign May be implied by a crect answer = 0.69 to A1 AWFW ( ) (ii) P(W >.5) = P(Z > 1.67) = P(Z < +1.67) Crect area change May be implied by a crect answer an answer > 0.5 (b) Weight, X ~ N(5.5, 0.0 ) = to A1 5 AWFW (0.951) (i) P(5.1 < X < 5.3) = P(Z < 0.5) P(Z < 0.75) Must have diff of probs f each = MINUS [( ) 0.663] Accept Accept = 0.37(08) AG; do not mark simply on answer (ii) P(0 in 4) = [1 0.37] 4 Accept [1 c s (b)(i)] 4 (c) Weight, Y ~ N(10.75, 0.50 ) = = to A1 AWFW ( ) Variance of Y 6 = 0.5 /6 = to Sd of Y 6 = 0.5/ 6 = 0.04 CAO AWFW Stated used CAO AWRT 6 P Y 10.5 = P Z = Standardising 10.5 with and OE; allow ( ) P(Z < 1.) = 1 P(Z < 1.) = m1 Crect area change May be implied by a crect answer an answer < 0.5; but do not award f use of z = 0. 1 ( to ) = to 0.11 A1 4 AWFW ( ) (1 answer) max Total 13

142 MS/SS1B - AQA GCE Mark Scheme 01 June Series MS/SS1B (cont) 6 (a)(i) U ~ B(30, 0.13, ) Used crectly anywhere in (a) A1 Can be implied by a crect answer = to 0.15 A1 3 AWFW (0.1489) P(P = ) = (ii) p = 0.35 CAO P(R P > 10) = 1 ( ) Requires 1 Accept 3 dp rounding truncation Can be implied by 0.49 to but not by 0.64 to = 0.49 to A1 3 AWFW (0.49) (iii) P(5 G 10) = (p 1 ) Accept 3 dp rounding truncation MINUS (p ) Accept 3 dp rounding truncation = to 0.7 (p 3 ) A1 3 AWFW (0.719) Notes 1 p 3 0 p 3 1 M0 M0 A0 4 p 1 (1 p ) M0 A0 p p 1 M0 M0 A0 5 p 1 p M0 A0 3 (1 p ) p 1 M0 M0 A0 6 (1 p ) (1 p 1 ) (A1) (b)(i) Mean = = CAO Variance = = 17.1 to 17. AWFW (igne notation) (17.16) ISW all subsequent wking (ii).1 /= means similar/equal 0.1 /= 0. proptions similar/equal so reject claim (that p > 0.) accept that p = 0. Dependent on seen in (b)(i) (ii) Accept diff = 0.1 CAO Crect (numerical) comparison with crect conclusion (even if at end and stated as reject (both) claims ) 17.1 to 17. = 4.13 to 4.15 /= to 17. /= 17.3 to 17.4 so reject claim that not random samples accept that are random samples Bdep1 3 Comparison using two values one value + diff (0.0 to 0.04 AWFW) Comparison using two values one value + diff (0.1 to 0.3 AWFW) Dependent on previous Crect conclusion regarding randomness of sample Total 14

143 MS/SS1B - AQA GCE Mark Scheme 01 June Series MS/SS1B (cont) 7 (a) x = CAO 98% (0.98) z =.3 to.33 AWFW (.363) CI f is x z n Used with z (.05 to.58), x (5.05, ), (0.0075, 0.075, 0.75, ) and n with n > 1 Thus A1 36 z (.05 to.06.3 to to.58), x (5.05) & (0.075) x (5050) & (75) and Hence OR (5.0, 5.08) (500, 5080) Adep1 5 CAO/AWRT Dependent on previous A1 so can be sced with z.3 to.33 Igne (absence of) quoted units AWRT to 3sf accuracy Note Use of t(.43 to.7) B0 A0 A0 max (b) Clear crect comparison of with LCL CI so agree with (first) claim (about mean) Adep1 Dependent on Adep1 in (a) Must use consistent units (8/36 0. %) v (1/ %) 8 v 3.6 (3 to 4) so 8/36 OE >/ 1/10 OE so disagree with (second) claim (about individuals) Bdep1 3 Mention of a value on LHS and a value on RHS Dependent on Explicit comparison of values and crect conclusion Notes 1 It/(claimed) mean/(claimed) value < LCL/CI Adep0 Must indicate % have (mean) weights between CLs so... Adep0 3 Any reference to CI f second claim B0 Bdep0 Claim refers to individual bottles (c) Yes because volumes/bleach/litres/bottles/ (parent) population are not (stated as) nmally distributed 1 OE; but do not accept data sample it Reference to sample size only B0 (eg n > 5 n > 30) Total 9 TOTAL 75

144 Version General Certificate of Education (A-level) January 013 Mathematics MS/SS1B (Specification 6360) Statistics 1B Final Mark Scheme