Mark Scheme (Results) January Pearson Edexcel International A Level Mathematics. Statistics 1 (WST01)

Transcription

1 Mark Scheme (Results) January 017 Pearson Edexcel International A Level Mathematics Statistics 1 (WST01)

2 Edexcel and BTEC Qualifications Edexcel and BTEC qualifications come from Pearson, the world s leading learning company. We provide a wide range of qualifications including academic, vocational, occupational and specific programmes for employers. For further information, please visit our website at Our website subject pages hold useful resources, support material and live feeds from our subject advisors giving you access to a portal of information. If you have any subject specific questions about this specification that require the help of a subject specialist, you may find our Ask The Expert service helpful. Pearson: helping people progress, everywhere Our aim is to help everyone progress in their lives through education. We believe in every kind of learning, for all kinds of people, wherever they are in the world. We ve been involved in education for over 150 years, and by working across 70 countries, in 100 languages, we have built an international reputation for our commitment to high standards and raising achievement through innovation in education. Find out more about how we can help you and your students at: January 017 Publications Code WST01_01_1701_MS All the material in this publication is copyright Pearson Education Ltd 017

3 General Marking Guidance All candidates must receive the same treatment. Examiners must mark the first candidate in exactly the same way as they mark the last. Mark schemes should be applied positively. Candidates must be rewarded for what they have shown they can do rather than penalised for omissions. Examiners should mark according to the mark scheme not according to their perception of where the grade boundaries may lie. There is no ceiling on achievement. All marks on the mark scheme should be used appropriately. All the marks on the mark scheme are designed to be awarded. Examiners should always award full marks if deserved, i.e. if the answer matches the mark scheme. Examiners should also be prepared to award zero marks if the candidate s response is not worthy of credit according to the mark scheme. Where some judgement is required, mark schemes will provide the principles by which marks will be awarded and exemplification may be limited. Crossed out work should be marked UNLESS the candidate has replaced it with an alternative response.

4 EDEXCEL IAL MATHEMATICS General Instructions for Marking 1. The total number of marks for the paper is 75.. The Edexcel Mathematics mark schemes use the following types of marks: M marks: method marks are awarded for knowing a method and attempting to apply it, unless otherwise indicated. A marks: Accuracy marks can only be awarded if the relevant method (M) marks have been earned. B marks are unconditional accuracy marks (independent of M marks) Marks should not be subdivided. 3. Abbreviations These are some of the traditional marking abbreviations that will appear in the mark schemes. bod benefit of doubt ft follow through the symbol will be used for correct ft cao correct answer only cso - correct solution only. There must be no errors in this part of the question to obtain this mark isw ignore subsequent working awrt answers which round to SC: special case oe or equivalent (and appropriate) dep dependent indep independent dp decimal places sf significant figures The answer is printed on the paper The second mark is dependent on gaining the first mark 4. All A marks are correct answer only (cao.), unless shown, for example, as A1 ft to indicate that previous wrong working is to be followed through. After a misread however, the subsequent A marks affected are treated as A ft, but manifestly absurd answers should never be awarded A marks. 5. For misreading which does not alter the character of a question or materially simplify it, deduct two from any A or B marks gained, in that part of the question affected.

5 6. If a candidate makes more than one attempt at any question: If all but one attempt is crossed out, mark the attempt which is NOT crossed out. If either all attempts are crossed out or none are crossed out, mark all the attempts and score the highest single attempt. 7. Ignore wrong working or incorrect statements following a correct answer.

6 January 017 WST01 Mark Scheme Question Scheme Number 1. (a) 5 small sq = 5 tomatoes or 1 large square = 5 tomatoes or fd=5 for ~3 5 or 0 or or = 4 (b) '4' or , so probability = (condone 91%) 100 Marks A1, A1 (c) (7 6.5) or (a) 516 (6.5 5) , 100, A1 (d) Since (or equivalent reason) and state median 6.5 (e) Median mean, so negative skew (f) (a) 3 Freq. for (5.5 weight < 7) = (7 5.5) '16' or '3', probability P (both weigh between 5.5 and 7) = (o.e.) or awrt , A1 A1 (4) [1 marks] Notes A correct answer with no working scores A1 in parts (a)~(c) for a correct: statement linking area with frequency or calculation or at least values on the fd scale on axis or at least frequencies on/in histogram bars. A1 for an answer of 4 (if not in script, can be awarded if 4 seen correctly on histogram). If answers on both diagram and script contradict, the script has preference. (b) (c) (d) for 100 (5 + (a) ) ft 0 their (a) 10 or for a correct method for finding the sum of the areas of all the bars above 3 (condone one slip if 5 terms seen) fully correct expression (possibly ft their (a)) and need division by 100 (o.e.) A1 for 11 0 or 0.55 (o.e.) [Allow 55% or ratio 55:100] for Q 6.5 with reason based on (c) where 0.5 < their (c) < 1 [comparison of 55 & 50] (e) for stating median > mean and negative skew (independent of (d)) (f) 1 st for method to find the frequency between 5.5 and 7 (Implied by the 4 used) e.g. ( ) ( ) based on 1 st A1 for 4 (o.e.) 100 nd '4' '4' 1 for ft their 4 but must have numerator < denominator of nd A1 for (o.e.) or awrt NB scores A1M0A0 [ alone 0/4]

7 Question Number. (a) Scheme (The event that) the integer selected is prime and ends in a 3 (and is between 1 and 50 inclusive) Marks (b) (or equivalent e.g. 0.30) [condone 30%] (c) 1 50 (or equivalent e.g. 0.4) [condone 4%] 7 P( AC) 50 7 (d) P( AC ), 30 P( C) 30 50, A1 (e) 15 7, so not independent , A1 P( B AC) 50 (f) P( B ( AC)), 7 P( AC) 7 50, A1 [9 marks] (d) for a correct ratio expression (may be in words) with at least one correct probability substituted or correct ratio expression and 7 m or where 7 < n or m < 30 n 30 or fully correct ratio of probabilities. A1 for 7 or any exact equivalent e.g. 0.3 but 0.33 is A0 (Correct ans only = A1) 30 (e) for correctly comparing their (b) with their (d), can be in words or symbols e.g. P(A) P(A C) in symbols. A1 dependent on a correct (b) and (d) (or awrt 0.33 in (d)) and for concluding not independent SC For a correct test using correctly labelled P( A), P( C) and P( AC) with all correct probabilities and (o.e.) seen leading to not independent score M0A1 (f) for a correct ratio expression (may be in words) with at least one correct probability r substituted or correct ratio expression and or where r < 7 or < t 7 t or fully correct ratio of probabilities A1 for or an exact equivalent. Allow awrt 0.86 here as well.(correct ans. only = A1) 7

8 Question Number 3. (a) 7 [ y ].5, Var(Y) 1 Scheme 6.98 (.5) 1 Marks, = (allow ) awrt A1 (3) (b)(i) (504)( 7) Sxy or ' 56.7' r, awrt 0.98 (1674)(.3), A1 (ii) Negative correlation, so Priya s belief is incorrect. (4) (c) ' 56.7' b [= ] a ' b' or a.5 ' ' 4 1 1, a =awrt 0.87,A1 y x A1(dep on M) (4) (d) [ y (3) ] 1.9 C awrt 1.9 (no fractions) (e) ( w 3) x (o.e.) 1.8 w x A1 (f)(i) Var(W) = 1.8 Var(Y), = awrt 0.60, A1 (ii) ryx rwx 0.98 ft (3) [17 marks] Notes (a) either fraction or exact decimal equivalent [must see mean separately to earn this mark] 6.98 for expr for variance y [ft y ] or Syy, (allow s i.e. Syy = awrt 0.03) [No ] For in (b)(i) and 1 st in (c) do not allow ft for S xy = (b)(i) Correct expression for S xy or 56.7 (May be implied by a correct value for r) for correct express for r with 1674,.3 and their S xy [Correct ans. only 3/3, r = 0.93 is /3] (ii) for Priya s belief not supported and reason e.g. negative correlation or r is negative or r is close to 1 or as salinity (or x) increases, temperature (or y) decreases (c) 1 st (e) (f)(i) (ii) for correct expression for b f.t. their S xy (May be implied by correct answer) nd for correct use of a y bx to find a (f.t. their value of b)( Implied by 0.87 ) 1 st A1 for a = awrt 0.87 (no fraction) nd A1 for an equ n in the form y = a + bx with their a and b = awrt (no fraction) ( w 3) for substituting 1.8 for y (o.e.) in their regression equation A1 for a correct equation for w in terms of x with c = awrt 31 and d = awrt for 1.8 Var(Y) f.t. their (a) (if > 0) ][Allow use of s awrt 0.66 to score A1] ft their answer to (b)(i) to at least sf (Must see a value written down here)

9 Question Scheme Marks Number 4. (a) [E( X ) ] , = 6.9, A1 (b) [E( X ) ] = 1.09, A1 Var( X ) 48.7 '6.9', (3) (c) Var(3 X) ( ) Var( X), = 4.36, A1 (d) (e) P( X Y) , [ E(Y) ] = 6.5 or 13 (o.e.) 1 (oe) 4, A1 (f) P( X Y) P( X 6Y 6) P( X 7 Y 7) P( X 8Y 8) = A1 (3) Notes (a) for a correct expression for E(X ) (Correct answer only is A1) [13 marks] (b) 1 st for attempting a correct expression for E( X ), sum of at least 3 correct products seen The first can be implied by 48.7 Stating Var(X) = the expression for E( X ) can score M0A0 and may get in (c) nd for correct use of Var ( X ) E( X ) [E( X )] f.t. their E( X ) A1 for 1.09 (Correct answer only is A1) (c) for ( ) Var( X ) or ( ) '(b)' [if (b) > 0] (condone no brackets if final answer is > 0) or a fully correct expr for Var(3 X) based on 3 x Prob A1 for 4.36 (Correct answer only with no working scores A1) (e) for an expression for P( X Y) (at least 3 of the 4 products correct). May be implied by a correct answer. (f) 1 st for a correct probability formula (as in scheme) or complete list of X > Y [e.g. X = 6 and Y=5; X =7 and Y =6; X =7 and Y =5; X = 8 and Y =5; X = 8 and Y = 6; X = 8 and Y =7] nd for a correct probability expression(i.e. correct values in formula) 1 NB alternative expressions e.g from listing Y < X rather than X > Y 4 The 1 st may be implied by scoring the nd 19 A1 for or 40 SC/ (Y >X) 3 1 Only apply if they reach [P(Y > X) = = ]

10 Question Scheme Number 5. (a) Age Computer use 0.80 Use computer every day Marks p (1 p) < (0.0) 0.55 Does not use computer every day Use computer every day (0.45) Does not use computer every day (b) p (1 p) 0.55 = 0.70 p = 0.6 A1 (c) (a) P( 50 use computer daily) '0.6' 0.80 P( 50 use computer daily) P(use computer daily) A1oe [6 marks] Notes Allow undefined letters for labels e.g. U(use) and U or N and NE Allow labels on branches and probabilities at the ends Condone 80% and 55% etc on tree diagram and in (b) 1 st for correct shape ( branches then 4 branches) and correct labels on first set of branches (p, < 50 and 50 but condone > 50 ) nd for correct labels on second set of branches (0.80, 0.55, daily and not daily) Allow 0.8p and 0.55(1 p ) on or at the end of the appropriate branches. NB they do not require the probabilities in brackets for either of these two marks. (b) for a correct equation to find p using their tree diagram. A1 for 0.6 [ condone 60%] (Correct answer only will score A1) (c) for a correct expression with 0.70 substituted correctly and numerator < denominator 7 or correct ratio of probabilities f.t. their p provided 0 p A1 for 48 4 or an exact equivalent e.g. (Correct answer only is A1) Allow awrt following a correct expression. [68.6% is A0] 8

11 Question Number Scheme Marks 6. (a) 98% (Condone 0.98) (b) z = (or better: calculator gives ) awrt.58 (c) [P(X < 46 X > 54) =] (b) z =.33 z = PZ " " or P Z ".579" ".579" Z Z P 1.55 or 1P( ) P(both bags outside range) = (0.11) =, awrt /7 Notes A1 A1 d, A1 (3) (4) [8 marks] for or better seen and used, can be with (may be implied by = awrt.579) for standardising with 56 or 44, 50 and and equating to a z-value z > A1 for awrt.58 from correct working. Use of z =.33 leads to can score B0A1 Special case: use of z =.3 from tables gives.586 = awrt.59 can score B0A1 Ans only A1 can be awarded for sight of at least = awrt.5791 or awrt.579 (c) 1 st for attempt to find sum of the area above 54 and below 46 or area above 54 or area below 46 ( needed) Allow ft of their provided> 1 st A1 for awrt 0.1 (NB = is A0 here and 1 st M0 too) nd d for p dependent on previous nd A1 for awrt (use of calculator value) or SC for those who use 1 tail only and get 0.06 but then do (0.06 ) Score as M0A0A0 Do not award for ( ) or 3 ( )

12 Question Number 7. (a) (b) Sum of probabilities = 1 gives Scheme a b a b 3a b 4a b e.g. 10 a 4 b 1 leading to 5ab 30* P(X = 1) + P(X = ) + P(X = 3) = 0 6a 3b 13 or 60 0 e.g. (6a 3b 39) (5a b 30) 3 leading to 3a 1 7 or P(X = 4) = 0 4a b (4a b 1) (5a b 30) (o.e.) leading to 3a 1 Marks A1cso A1 d a = 4 and b = 5 (c) [y] [<1] 1 [ y < 4] 4 [ y < 9] 9 [ y <16] [ ] 16 [F(y)] [ 0 ] Notes A1 (4) cao A1 (4) [10 marks] (a) 1 st for use of sum of probabilities = 1 to form a linear equation in a and b (4 terms seen) (b) A1 cso for fully correct solution with no errors or omissions seen and at least one intermediate line of working seen 1 st for use of P X i or P(X = 4) = to form a nd equation in a and b 0 0 i1 1 st A1 for a correct 3 term nd equation in a and b with a and b terms collected. nd d dependent on 1 st for solving relevant linear equations i.e. eliminating a or b leading to a linear equation in 1 variable. Allow 1 numerical or sign slip. nd A1 for both a = 4 and b = 5 (Correct answer only can score all 4 marks) (c) 1 st for all y-values, can allow label of x (accept 1, 4, 9 and 16 or 1,, 3, 4 ) nd 9 cao for FY = oe but must be clearly labelled as cdf linked to Y = 1 but not for 60 P(Y = y) or P(Y = 1) for a correct method to find FY(4) or FY(9) ft their a and b [dep on correct y-values seen] A1 for fully correct cumulative distribution function allow F = , F(4) = 60, F(9) =, F(16)=1 60 y 7 y NB: Probability distribution of X NB F( y) for y =1,4,9,16 (o.e.) 60 x Is OK for all marks only with y values given P(X = x)

13

14 Pearson Education Limited. Registered company number 8788 with its registered office at Edinburgh Gate, Harlow, Essex CM0 JE