GCE Mathematics. Mark Scheme for June Unit 4732: Probability and Statistics 1. Advanced Subsidiary GCE. Oxford Cambridge and RSA Examinations

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1 GCE Mathematics Unit 47: Probability and Statistics Advanced Subsidiary GCE Mark Scheme f June 05 Oxfd Cambridge and RSA Examinations

2 OCR (Oxfd Cambridge and RSA is a leading UK awarding body, providing a wide range of qualifications to meet the needs of candidates of all ages and abilities. OCR qualifications include AS/A Levels, Diplomas, GCSEs, Cambridge Nationals, Cambridge Technicals, Functional Skills, Key Skills, Entry Level qualifications, NVQs and vocational qualifications in areas such as IT, business, languages, teaching/training, administration and secretarial skills. It is also responsible f developing new specifications to meet national requirements and the needs of students and teachers. OCR is a not-f-profit ganisation; any surplus made is invested back into the establishment to help towards the development of qualifications and suppt, which keep pace with the changing needs of today s society. This mark scheme is published as an aid to teachers and students, to indicate the requirements of the examination. It shows the basis on which marks were awarded by examiners. It does not indicate the details of the discussions which took place at an examiners meeting befe marking commenced. All examiners are instructed that alternative crect answers and unexpected approaches in candidates scripts must be given marks that fairly reflect the relevant knowledge and skills demonstrated. Mark schemes should be read in conjunction with the published question papers and the rept on the examination. OCR will not enter into any discussion crespondence in connection with this mark scheme. OCR 05

3 47 Mark Scheme June 05 S June 05 Final mark scheme Note: ( sfs means answer which rounds to... to sfs. If crect ans seen to > sfs, ISW f later rounding Penalise over-rounding only once in paper. "oe" means " equivalent" (i S xx = 00.9 S yy = 4 S xy =.9 r = M ".067" ".0" "97." M 9 66 Crect sub in a crect S fmula 5 crect value of one S seen Crect sub in crect S fmulae and a crect r fmula = 0.9 ( sf A No wking: 0.9 MMA [] Igne comment about 0 < r < 0. (ii (F these 6 clubs Allow without F these 6 clubs & top Allow "Salary has little effect on points" No/little/po/weak oe relationship/cr n/link oe "no strong cr'n between etc" Igne all else including "positive" between (top salaries and no. of points B In context. NOT if use "goals" instead of "points" [] (iii Extrapolation oe Outside range of values. Salary is less than the others. NOT "Cr'n does not imply causation" Cr n po/weak no rel'nship/link oe r small r close to 0 r not close to NOT "Could be other facts" Points not close to line Results do not crelate well NOT if use "goals" instead of "points" Small sample only (top 6 clubs oe B B Any two; allow without context [] (i 5 B Allow 0 to 40 inclusive [] 50 (ii 00 oe M NOT NOT = % to % A NOT (unless sub from [] (iii eg 7.5, 7.5 5, , 5-90 B any values in intervals 5-0 & 5-90 NOT "Because it's cumulative frequency" "Classes" "intervals" "groups" "mid- B No raw data given. Not have each data value NOT "Because it's a line of best fit" points" "bounds" seen Exact values not given can't be read off oe NOT "Because graph is difficult to read" Data lost oe NOT "because graph is a curve" [] Igne all else f nd B, not st B NOT "Cont data has no exact data pts"

4 47 Mark Scheme June 05 (iv Median = 9 ± drawn B stated Mark diagram even if contradicts Quartiles = 5 ±, 55 ± drawn B stated statements of values in (iv (iii Ends in ranges 5-0 & 5-90 drawn Bf ft (iii If no diagram, award max BBB f Crect B&W plot ± drawn Bf ft (iii mark intention (allow unruled lines statements of med, quartiles & ends [4] (v Stretched out at top end oe B Positive skew, NOT any of below: Not symmetrical Skewed to right ( to higher values me large extremes than small extremes Me concentrated towards lower end Larger skewness at top IQR is towards the lower end Me values ( data in lower half of range Larger plums me spread than smaller ones skewed to the left ( to lower values Median closer to lowest value majity below 9 Average towards lower end Igne all else distribution towards lower end Me plums have lower masses No need f context Majity of distribution towards lower end Me below 50 ( 45 Upper whisker longer than lower whisker [] (i One set reversed, max 4 mks, eg Year Y Y Age Q A Quality A Q similar Attempt ranks M Attempt ranks M Crect ranks A Allow both sets of ranks reversed Increct ranks A0 Attempt Σd (= M Attempt Σd (= M 6 "" 9( M 4 6 "" 9( M = ( sf A NB 0.9 is crect ( sf A [5] (ii Older is better oe newer is wse oe B No ft from (i As age increases, quality increases -0.9 in (i leads to same conclusion as +0.9 in (ii NOT as year increases quality increases Must imply older is better oe, Nothing contradicty seen, ie NOT igne all else NOT High/strong/good cr'n/agreement/ ie "good ( positive cr'n between age and In context; no need to include "rank" rel'nship between age and quality oe quality" is not enough []

5 47 Mark Scheme June 05 4 (i S xx = 4. S xy = 4.65 b = y Alternative method: = a b M = 60.5a + 4.b M M Crect sub in any crect S xx S xy fmula crect value of either S hence a = 7. b = -0.0 oe y = 0.0x + 7. A = (x M 60.5 M a = Crect sub in both Ss and in a crect b fmula y = -0.0x + 7. ( ( sf A y = 95 x + 7./9 79 Must include "y =" no wking, crect ans MMMA Allow y = -0.x + 7. (awrt sf If find x on y line, can sce first M only ans x = -4.y seen first M only [4] 4 (ii M = (km/l ( sf Aft ft their equn from (i but no ft from x on y line [] 4 (iii (7.56, 5.6 ( sf ( 6, , [] 4 (iv Use reg line of x on y (either equn line M Must specify imply x on y, otherwise M0A0 If calc x on y reg line (allow errsm NOT "Use either x on y y on x" Sub y = 5. fuel = 5. km/l = 5. A NOT "and read off y cod" Subst 5. into their x on y line A [] Igne all else 5 (i (a ( M alone = 0 09 ( sf A [] 60.5 A 5

6 47 Mark Scheme June 05 5 (i (b ( 0.7 M alone NOT ( P(X =,,,4,5,6,7, all terms crect NOT - (-0.7 (= = ( sf A [] 5 (ii Bin stated B implied by C C 6 ( 0.7 a 0.7 b (a+b= by ans Allow "Bio" C ( B NOTE. Must see sub in fmula f this B Allow crect ( sf B Crect ans, no wking: BB0B [] 5 (iii Their (ii 0.7 seen together M ( C ( seen together Their (ii 0.7 ( M C ( ( C ( ie wholly crect method ft(ii ie wholly crect method SC: ( oe alone = 0.00 ( sf Aft Allow 0.0; ft their (ii only M0MA0 [] 6 (i 7! P 7 seen M 5!( 6 C + 6 NOT 5! 6 C 7! 7! M 5! (6C alone M > 5 crect fracs mult: 6 crect fracts mult... = ( sf A 5040 oe [] 6 (ii (a 5 B Igne any wking seen [] 6 (ii (b 5 C alone ( C M alone, eg NOT 5 C... 5 C +... But allow 5 C as denom of prob MA0 6 C (! 7 7 C 5 P = 0 A [] 6 (ii (c C M 6 C + "5" 7 C - "0" 7 C - 5 C Allow as denom of a prob MA0 = 5 Af ft (a &/ (b only if wking seen [] M 6

7 47 Mark Scheme June 05 7 (i (a Binomial seen implied B by tables 0 C 0 C 7 by 0.5 a 0.75 b (a + b = C ( M = 0.50 ( sf A Allow 0.5 [] 7 (i (b M P(X =,4,5,6,7,,9,0 all crect terms NOT (P(X > from table (( ( C ( Allow 0 C instead of 0 C = ( sf A [] 7 (ii seen M Their (i(b seen, result of -(i(b seen eg B(6, P(X > = ( oe M P(X =,,,4,5,6 all crect terms seen = ( sf Af ft from (i(b [] If 0. and and 0.7 similar used, can sce (i BB0(ii MA0 (iii MMMA0 (iv B (i Crect structure with no extra branches B Allow extra branches with crect 0 & igne probs and R & B Probs and R and B all crect Bdep Igne other probs [] (ii M + NOT = ( sf A No ft from tree f A [] ft their tree, eg "without replacement" gives 5 4 (= 5 MA0 7

8 47 Mark Scheme June 05 (iii There are basically 6 cases, as in LH column. Some group these into cases, as in middle column. Others use 9 cases - treat these as grouped into the 6 cases below and mark accdingly. Must decide which case a candidates is using, and use the cresponding scheme. NB. Listing Adnan and Beryl separately gains no marks. They must be combined into cases. F example ( = is crect f P(Adnan 4 throws and Beryl throw, but it could also come from P(Adnan gets RRRB which sces no marks by itself. If crect 6 ( cases, equiv, are given, but extra cases also given, award st M, and possibly nd M, but no me. No ft from tree, but if clearly state cases may sce st M only. All six cases seen implied: All three cases soi: All four cases soi: &; &, &; 4&, 4&, 4& R & B; RR & RB; RRR & RRB M B&B; RB & RB; RRB & RRB; RRRX & RRRX &; & (< ; 4& (< 4 M R & B; RR & RB; RRRR & RRB; RRRB & RRB ie & & & 4&4 M ie cases: (> & (> & (4 & P(& = 7 P(R & B = 9 M ( +( +( +( P(& = ( 4 NB Must be clearly part of -case method P(& = ( 4 P(RR & RB = ( P(4& = ( ( P(RRR & RRB = ( ( 79 ( 4 ( ( ( Both these crect expressions results and add all 79 cases oe ie completely crect method M P(4& = ( 6 4 ( 4 ( ( sf A 79 P(4& = ( May see other groupings of 6 cases into cases eg ( 4 ( ( Crect expressions ( results f of these 6 probs 9 M 7 7 Crect expressions ( results f the other of these 6 probs & no extra cases, and add all 6 cases ie completely crect method 4&( &( & M oe oe M ie completely crect method M ( sf A ( sf A See next page f me [4] M 97 all crect 79 M ( M ( sf A 79 NB = 9 often seen, usually sces 0. Must be clearly part of -case method to sce.

9 47 Mark Scheme June 05 (iii cont 4 COMMON INCORRECT METHODS: ANOTHER EXAMPLE ANOTHER INCORRECT METHOD RB All six cases seen implied: M B RRB BRB RRRB BRBRRB oe MM0 ie & &( 4&( RRRR BRBRRB = A0 This sces the st M f all cases soi + 4 ( ( ( ( + ( + ( + ( ( ( (The last two "fractions" together make the rd case ie attempt 4&, &, &; &, 4&; 4& ( ( M0M0M0M0 Some of these overlap, but st, 5th, 6th crect. P(& + P(& + P(4& Overall MMM0A0 = + ( + ( 4 ( ( ( M0MM0A0 (iv Unlimited number of throws oe B Not fixed number of throws Allow Throw die until blue obtained Not stop at 4 throws oe Turn continues until blue obtained NOT Continue until st success NOT "Not stop at 4 throws when blue obtained" [] Igne all else 9 (i a + b, a + b, a + b B All three seen a + b + a + b + a + b = oe Bdep Must see this line oe befe final answer Must include "= " "Probabilities add up to " oe stated (a + 6b = AG [] 9 (ii a + b + (a + b + (a + b = 5 M ft their probs 6a + 4b = 5 a + 4b = 5 Af any crect three term equn, ft their probs eg 6 6b 6a b = 5 b = a 6 = 5 a = A any crect equn in a b only. cao a =, b = 6 A cao [4]

10 OCR (Oxfd Cambridge and RSA Examinations Hills Road Cambridge CB EU OCR Customer Contact Centre Education and Learning Telephone: Facsimile: general.qualifications@ocr.g.uk F staff training purposes and as part of our quality assurance programme your call may be recded monited Oxfd Cambridge and RSA Examinations is a Company Limited by Guarantee Registered in England Registered Office; Hills Road, Cambridge, CB EU Registered Company Number: OCR is an exempt Charity OCR (Oxfd Cambridge and RSA Examinations Head office Telephone: Facsimile: OCR 05

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