# Final. Mark Scheme. Mathematics/Statistics MS/SS1B. (Specification 6360/6380) Statistics 1B. General Certificate of Education (A-level) June 2013

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1 Version 1.0 General Certificate of Education (A-level) June 2013 Mathematics/Statistics MS/SS1B (Specification 6360/6380) Statistics 1B Final Mark Scheme

3 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 A2,1 2 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.

4 1(a)(i) Mean = 62.2 to 62.3 B1 AWFW (62.25) SD = 17.4 to to 16.9 B1 2 AWFW ( ) (ii) Mean = to BF1 AWFW (16.806) F on (a)(i) only providing 45 < mean < 65 SD = 9.66 to to 9.39 BF2 3 AWFW ( ) F on (a)(i) only providing 10 < SD < 20 (b) r xy = B1 CAO Award on value only; igne any explanation wking r xy = r uv with no value stated B0 r not affected by change(s) in/different units Accept Fmula It f r and reference to linear is not necessary r not affected by linear scaling B1 2 Accept Fmula It f r but reference to linear is necessary Scaling/coding/transfmation/change/ conversion to u and v is linear Total 7 OE; but reference to linear is necessary 4

5 Accept percentage equivalents in (a) Weight, X ~ N(421, ) 2(a)(i) P(X = 421) = 0 zero nought 0% B1 CAO; accept nothing else but igne additional wds providing that they are not contradicty (eg impossible so = 0) (ii) P(X < 425) = P Z < 2.5 Standardising 425 with 421 and 2.5 but allow ( ) = P(Z < 1.6) = to A1 AWRT ( ) (iii) P(418 < X < 424) = P( a < Z < a) = P(Z < a) (1 P(Z < a)) 2 P(Z < a) 1 = ( ) = = A1 OE; a = 1.2 crect standardising are not required May be implied by (AWRT) seen anywhere by a crect answer AWRT ( / ) Implied by a crect answer = to 0.77 A1 6 AWFW ( ) (b) 0.98 z = 2.05 to 2.06 B1 AWFW (2.0537) x = 2(.0) to 2.4 x = 426 to A1 3 Standardising x with 421 and 2.5 but allow (421 x); and equating to a z-value (igne sign) AWFW (426.13) Must be consistent signs throughout (c) 0.01 z = 2.33 to 2.32 B1 AWFW; (igne sign) ( ) z = 410 µ Standardising 410 with µ and ( ) but allow (µ 410) 410 µ Equating to a z-value (igne sign) 3.0 = 2.6 to 2.3 A1 µ = 417 Adep1 4 Total 13 AWRT (416.98) Dependent on previous A1 Must be consistent signs throughout 5

6 Accept percentage equivalents except f 27 3(a)(i) O ~ B(40, p) P(NS 10) = 0.97 B1 1 AWRT (0.9701) (ii) P(LPE 25) = 1 ( ) Requires 1 Accept 3 dp rounding Can be implied by ( to 0.077) but not by (0.04 to ) = A1 2 AWRT (0.0769) (iii) P(UPE = 2) = ( ) ( ) Crect expression; may be implied by a crect answer Igne extra terms = A1 2 AWRT (0.0160) (iv) p = = 0.35 B1 CAO; award on value only May be implied by any of four probabilities below by a crect answer P(10 < X < 15) = (p 1 ) MINUS (p 2 ) Accept 3 dp rounding Accept 3 dp rounding = 0.45 to A1 4 AWFW (0.4506) (b) p = = p = B1 CAO; may be implied by 27 Each can be found in several ways CAO; may be implied by Number = = 27 B1 2 CAO; can be found in several ways Total 11 6

7 4(a)(i) in r gy = = 0.91 to (a)(i) (a)(ii) (c)(i) A1 AWFW ( ) (ii) r ly = = 0.64 to A1 3 AWFW ( ) (b) (Very) Strong positive crelation Bdep1 Dependent on 0.9 r gy < 1 (Some) Moderate positive crelation between girth and weight and/ length and weight Bdep1 Dependent on 0.6 r ly 0.7 Bdep0 f any mention of strong B1 3 At least one interpretation in context (c)(i) r xy = = 0.98 to B1 AWFW ( ) Most strongly crelated with y is x Bdep1 2 CAO; dependent on 0.97 r xy < 1 (ii) x = = 124 to 125 A1 2 AWFW (124.52) (iii) b = = 1 to A1 116/115.4 (= 1.005) M0 A0 AWFW ( ) a = b = 0.3 to 0.6 B1 3 AWFW ( ) (iv) r xy /nearly/almost/close to (+)1 very strong/almost exact (positive) crelation (Stating r xy =0.98 to Bdep0) b =/ /nearly/almost/close to (+)1 a /nearly/almost/close to 0 (Stating a = 0.4 to 0.6 Bdep0) Bdep1 Bdep1 Bdep1 OE Dependent on 0.97 r xy < 1 OE; strong is not sufficient OE; must reference value of 1 unity Dependent on A1 in (c)(iii) OE; must reference value of 0 igin Dependent on B1 in (c)(iii) Estimate (not 'it' 'this' 'value', etc) is (very/highly/likely to be) accurate/precise/ reliable (almost) exact/crect Bdep1 4 Total 17 OE; dependent on scing at least 2 of the previous 3 marks in (c)(iv) Fairly accurate, good approximation, (quite) likely, (very) close, reasonable, etc Bdep0 7

8 5(a)(i) P(A = 2) = = 0.85 to 0.86 B1 AWFW ( /200 OE) (ii) P(A = 1) = ( ) + ( ) = 1 [ ( )] Do not igne extra terms = 0.14 A1 3 CAO (7/50 OE) (b)(i) P(A W D W ) = = 0.72 A1 2 CAO (18/25 OE) (ii) P(A B D B ) = (b)(i) 0.95 ( 1) = ( 1) = (a)(i) to A1 2 AWFW ( /250 OE) (iii) P(A T D T ) = = 0 B1 1 CAO; award on value only (iv) P(neither) = P([A W D W ] [A T D T ]) (1 0.90) (1 0.15) (1 0.95) (1 0) P(neither) = P(A W A T ) P(D W A W ) P(D T A T ) (1 0.90) (1 0.95) (1 0.15) (1 0) m1 () (m1) Accept /200 OE Award and m1 on value(s) only Accept /20 OE Accept /200 OE Award and m1 on value(s) only Accept /20 OE = OE = to A1 3 AWFW ( /4000 OE) Total 11 8

9 6(a)(i) x = = 19.9 B1 CAO 25 98% (0.98) z = 2.32 to 2.33 B1 AWFW (2.3263) σ CI f µ is x ± z n Used with z (2.05 to 2.58), x ( to 21) and σ (0.4) and n with n > 1 Thus ± A1 25 z (2.05 to to to 2.58), x (19.9) and σ (0.4) and Hence 19.9 ± 0.2 (19.7, 20.1) A1 5 CAO/AWRT ( ) AWRT (ii) Clear crect comparison of 20 with CI eg 20 is within CI LCL < 20 < UCL so Agree with claim no reason to doubt claim F on CI providing it contains 20 BF1 Quoting values f CI is not required Bdep1 2 OE; dependent on previous BF1 (iii) Weight of sand in a bag X/x iginal distribution parent population is nmal B1 1 It/mean/data/sample/infmation/sand is nmal B0 Reference only to sample size standard deviation B0 9

10 Accept percentage equivalent probabilities 6(b)(i) Y ~ N(25.25, ) V(mean) = / to SD (mean) = 0.35/ to P( Y 25) P < = Z < B1 CAO/AWFW ( ) CAO/AWFW ( ) Standardising 25 using and 0.35/ 10 OE but allow ( ) = P(Z < ) = 1 P(Z < ) m1 Crect area change an answer < 0.5 = 1 ( to ) = to A1 4 AWFW ( ) (0.987 to 0.989) B1 m0 A0 (ii) P( Y > 25) = P Z > 0.35 Standardising 25 using and 0.35 but allow ( ) = P(Z > ) = P(Z < ) = to A1 AWFW ( ) (0.236 to 0.239) A0 P(Y > 25 in each of 10) = p 10 Any p 10 providing 0 < p < 1 = to A1 4 AWFW ( ) Total 8 TOTAL 75 10

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