# hij Mark Scheme Mathematics 6360 Statistics 6380 General Certificate of Education MS/SS1B Statistics 1B 2009 examination - January series

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1 Version 1.0: 0109 hij General Certificate of Education Mathematics 6360 Statistics 6380 MS/SS1B Statistics 1B Mark Scheme 2009 examination - January series

4 MS/SS1B 1 (a) 247 B1 fx Mean = x 52 = ¾ B2 CAO (4.75 = 5 ISW) B2 52 If B0 but evidence of Median (26, 26½) = 5 fx 52 If B0 but evidence of cumulative frequencies F: (0) If data assumed continuous so use of x 4 + where 0 < x < 2 7 () B2 (B1) () CAO Stated identification of 26 26½ Need to see attempt at 4 F values (4 < median < 4.29) Mode(s) = 4 and 6 B1 5 CAO both (so mode = 5 B0) (b) Mode(s) B1 CAO Me than one mode/value Two modes/values No unique mode/value B1dep Or equivalent; eg not unique Dep only on previous B1 sced Notes: If data treated as two separate sets, then Modes = 1 and 13 B0 in (a) but only marks available are B1 B1dep in (b) B1 B1dep 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 2 Total 7 4

5 2 (a)(i) r = to B3 AWFW ( ) r = 0.02 to 0.03 (B2) AWFW (ii) OR Attempt at Attempt at r = 0.1 to 0.1 x xy 2 x y S xx S yy & S xy 2 y & (B1) () AWFW Attempt at crect fmula f r (m1) r = to (A1) 3 AWFW (Almost/virtually) no/zero (linear) crelation (relationship/association/link) between length and (maximum) diameter of carrots B & (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 B1 2 Context; providing 1 < r < 1 (b) Unlikely/wrong/increct/invalid B1 Or equivalent Would expect a positive value Would expect weight to increase with B1 length Would imply shter carrots are heavier 2 Total 7 Or equivalent reason 5

6 3 X ~ N(5.08, ) (a)(i) (ii) P( X < 5) = = 1 P( Z 16. ) P Z < 005. = P Z < 1. 6 ( 5< < 510) = P( X 510. ) P X. = P( Z 0. 4) ( ) Standardising (4.5, 4.95, 5, ) with 5.08 and ( 005., ) and/ (5.08 x) < = 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 2 AWFW ( ) (b)(i) Variance of X 4 = /4 = SD of X 4 = 0.05/2 = P( X 4 > 5. 05) = P Z > B1 CAO; stated used Standardising 5.05 with 5.08 and 0.025; allow ( ) = P( Z > 1. 2) = P( Z 12. ) < m1 Area change; may be implied AWFW ( ) = to A1 4 (1 answer) B1 max (ii) Zero B1 1 CAO; equivalent (igne any wking) (c) 1% (0.01) z = 2.33 to 2.32 B1 AWFW; igne sign ( ) 5 μ z = 005. = A1 Standardising 5 with μ and ; allow (μ 5) Only allow: ±2.05 to ±2.06 ±2.32 to ±2.33 ±2.57 to ±2.58 μ = 5.11 to 5.12 Note: 5 μ = B1 A1 A0 A1 AWFW (5.1163) 4 Total 14 Or equivalent inconsistent signs 6

7 4 P(C) = 0.6 P(C B) = 0.25 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) = = 0.4 B1 1 CAO; equivalent (ii) P(C B ) = Can be implied by crect answer = 1 ( ) = 0.35 A1 2 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) (M2) 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) 0.25] 0.3 (0.12) 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 2 AWRT ( ) Total 12 7

8 5 (a) Mean = = 54 B1 1 CAO; cannot be gained in (b) (b) 98% (0.98) z = 2.32 to 2.33 B1 AWFW (2.3263) CI f μ is x ± z σ Used n Must have n with n > 1 Thus 54 ± F on x (but not 1620) and z only 30 A1F Allow x = 54 even if B0 in (a) Hence 54 ± (3.38 to 3.42) CAO & AWFW (54 & 3.4) A1 4 (50.58 to 50.62, to 57.42) 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) B1, (c) A1 max Use of n = 1 with (b) (c) not identified (b) B1, (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, to 72.64) Note: Accept sensible non-symmetric intervals such as: (0, ) = (0, 70.4 to 70.5) A1F 2 CAO & AWFW (54 & 18.6) If z-value increct, then must use 54 ± 8 z from( b) AWFW (35.4, 72.6) (d) Nowhere No B1 1 Total 8 CAO; equivalent (igne any reasoning) 8

9 6(a) Figure 1: 3 crect labelled points B2 2 crect labelled points (B1) 2 Deduct 1 mark if not labelled (b) OR Attempt at Attempt at b (gradient) = B2 AWRT ( ) b (gradient) = 0.68 to 0.69 (B1) AWFW a (intercept) = B2 AWRT ( ) a (intercept) = 0.34 to 0.35 (B1) AWFW x S xx & xy 2 x y & xy & (all 4 attempted) () S 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 42) (70, 47¼ to 49) (80, 54 to 56) If B0 but evidence of use of line f 2 points within range 50 x 80 (d)(i) Residual = y (a + bx) [ (a + bx) y] H I J 2.5 to 4(.0) 2.5 to 4(.0) 2(.0) to 4(.0) B2dep Dep on B1 B1 A1 A0 in (b) At least from x 55 to 70 Any two () 2 Calc n points shown on graph A2,1 ( 1 EE) Used implied; equivalent (using graph); 1 residual crect (2.98) AWFW; igne signs only (3.19) providing all the same (2.70) (ii) y 65 = a + b 65 y 65 = 44 to 45.5 Mean = 2.3 to 4(.0) A1dep 4 + [(d)(i)] [2.95 to 2.97] 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 B1 AWFW; do not igne sign (2.96) Dep on previous A2 sced Use shown AWFW (44.9) Use shown AWFW; igne sign of mean residual = 46 to 50 A1 AWFW (47.8) (B2) 3 Total 15 y M = x 478. OR no evidence of method {from (d)(i) and/ (d)(ii)} Evidence of increct method B0 9

10 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 = to A1 3 AWFW (0.0215) Accept 3 dp accuracy from tables P(S < 10) = B1 calculation = to B1 2 AWFW (0.2424) P(S = 3) = (. ) (. ) (iii) P(15 S 20) Accept 3 dp accuracy = (p 1 ) p 2 p 1 M0 M0 A0 p 1 (1 p 2 ) M0 A0 minus (p 2 ) 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 20 gives probability = to () (A2) 3 Or implied by a crect answer AWFW (iv) Mean, μ = np = = ½ Variance, σ 2 = np(1 p) = = 12.3 to 12.4 B1 2 B1 CAO (22.5 = ISW) Accept AWFW (12.375) (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 B1 1 B1 1 Total 12 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

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