Version.0 klm General Certificate of Education June 00 Mathematics MS04 Statistics 4 Mark Scheme
Mark schemes are reared by the Princial Examiner and considered, together with the relevant questions, by a anel 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 resonses to questions and that every examiner understands and alies it in the same correct way. As rearation for the standardisation meeting each examiner analyses a number of candidates scrits: alternative answers not already covered by the mark scheme are discussed at the meeting and legislated for. If, after this meeting, examiners encounter unusual answers which have not been discussed at the meeting they are required to refer these to the Princial Examiner. It must be stressed that a mark scheme is a working document, in many cases further develoed and exanded on the basis of candidates reactions to a articular aer. Assumtions about future mark schemes on the basis of one year s document should be avoided; whilst the guiding rinciles of assessment remain constant, details will change, deending on the content of a articular examination aer. Further coies of this Mark Scheme are available to download from the AQA Website: www.aqa.org.uk Coyright 00 AQA and its licensors. All rights reserved. COPYRIGHT AQA retains the coyright on all its ublications. However, registered centres for AQA are ermitted to coy material from this booklet for their own internal use, with the following imortant excetion: AQA cannot give ermission to centres to hotocoy any material that is acknowledged to a third arty even for internal use within the centre. Set and ublished by the Assessment and Qualifications Alliance. The Assessment and Qualifications Alliance (AQA) is a comany limited by guarantee registered in England and Wales (comany number 6447) and a registered charity (registered charity number 074). Registered address: AQA, Devas Street, Manchester 5 6EX
Key to mark scheme and abbreviations used in marking M m or dm A B E mark is for method mark is deendent on one or more M marks and is for method mark is deendent on M or m marks and is for accuracy mark is indeendent of M or m marks and is for method and accuracy mark is for exlanation or ft or F follow through from revious incorrect result MC mis-coy CAO correct answer only MR mis-read CSO correct solution only RA required accuracy AWFW anything which falls within FW further work AWRT anything which rounds to ISW ignore subsequent work ACF any correct form FIW from incorrect work AG answer given BOD given benefit of doubt SC secial case WR work relaced by candidate OE or equivalent FB formulae book A, or (or 0) accuracy marks NOS not on scheme x EE deduct x marks for each error G grah NMS no method shown c candidate PI ossibly imlied sf significant figure(s) SCA substantially correct aroach d decimal lace(s) No Method Shown Where the question secifically requires a articular method to be used, we must usually see evidence of use of this method for any marks to be awarded. However, there are situations in some units where art marks would be aroriate, articularly when similar techniques are involved. Your Princial Examiner will alert you to these and details will be rovided on the mark scheme. Where the answer can be reasonably obtained without showing working and it is very unlikely that the correct answer can be obtained by using an incorrect method, we must award full marks. However, the obvious enalty to candidates showing no working is that incorrect answers, however close, earn no marks. Where a question asks the candidate to state or write down a result, no method need be shown for full marks. Where the ermitted calculator has functions which reasonably allow the solution of the question directly, the correct answer without working earns full marks, unless it is given to less than the degree of accuracy acceted in the mark scheme, when it gains no marks. Otherwise we require evidence of a correct method for any marks to be awarded.
MS04 Differences are: 0.5, 0.5, 0.7, 0., 0., 0., 0., 0.5 Mean 0.875 s 0.9594 A H 0 : μ d 0. Accet μs μ A H : μ d > 0. t 0.875 0. calc.7 0.9594 A 8 0.05 ν 7 t.998 crit Insufficient evidence to A 0 accet coach s belief Total 0 ( x x) 56.54 s 6.804 ν 9 χ 9 (0.05).700 ft on v 0 χ 9 (0.975) 9.0 95% CL for σ are (a) s.506 ( ) 9.506, 9.506 9.0.700 A 95% CI for σ is (.7, 4.58) (Accet 4.57) A 6 (b)(i) H 0 : Var(X) Var(Y) or σ X σ orσ Y X σy H : Var(X) > Var(Y) or σ X > σ or σ Y X > σy (ii) s 6.65.59 9 A s.847 F calc.506²/.59².40 0.04 ν ν 9 on ν 0,0 F crit (0.95).79 Reject H 0 ; sufficient evidence to suggest that Nadia s times are less variable. A 7 Total 4
MS04 (cont) (a) x y 0.86 s 0.958 + 0.87 0.0957 7+ 6 ν t.06 A AWFW (0.09, 0.0) s 0.049 ft ν 0.86 ±.06 0.0957 + 7 6 ( 0.045,0.68) A 7 AWFW ( 0.04,0.68) (b) Random samles / Indeendent E Common variance E Normal distributions E (c) Insufficient evidence to suort belief E since 0 CI E Total 5
MS04 (cont) 4(a)(i) x( x ) q + 6q + q +... q (ii) + + + q( q 6 q...) q( q) q q E[X(X ) q E(X²) E(X) ( ) (AG) A q E(X²) q+ + q + + q Var(X) q ( ) (AG) A (b)(i) EX ( ) EX ( ) Var( X) 4 Var( X ) 9 A (AG) 4 A 5 (ii) N < 0 5 N 5 > A Working with and obtaining correct answer gets /4. 0 N 7 ma 4 Accet trial and imrovement. No working award B. Total 4 6
MS04 (cont) 5(a) x λx λx x λe dx e 0 0 e A (b) 0.05, 0.08 (Accet 0.05) (c) O i E i 4.48 0 9.0 9.58 Probabilities 80 6 7.0 4.6 Combining classes 9 6.57 H 0 : Exonential Distribution with arameter 0.5 is an aroriate model ( O E) χcalc E Use of correct formula 0.970 (5 th and 6 th ) A Correct value (Or.67 (4 th and 5 th )) ν 5 4 χ crit 7.779 ft on v 5 0.970 (or.67) < 7.779 Accet H 0 So the exonential distribution with arameter 0.5 may be an aroriate A 8 model Total 7
MS04 (cont) 6(a) Var (X) π² 8 π² π² 8 A (b)(i) E( X ) π π 8 Var( X ) n (ii) E( X ) π unbiased E Var( X ) 0 as n consistent E (c)(i) π 8 RE(M wrt X ) 5 π 07 5 Any sensible value for π 0.565 or 0.566 A Prefer X since RE(M wrt X ) < E or Var( X ) < Var (M) (ii)(a) π > 6. π >. A is A0 (B) x.0 m. both (C) X is the more efficient estimator, imlying that for the majority of samles it will be closer than M to π. E However, for this articular samle m is closer to π than x. E 5 Total 4 TOTAL 75 8