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

Transcription

1 GCE Mathematics Unit 4732: Probability and Statistics Advanced Subsidiary GCE Mark Scheme f June 204 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 204

3 Annotations and abbreviations Annotation in scis Meaning Blank Page this annotation must be used on all blank pages within an answer booklet (structured unstructured) and on each page of an additional object where there is no candidate response. and BOD Benefit of doubt FT Follow through ISW Igne subsequent wking M0, M Method mark awarded 0, A0, A Accuracy mark awarded 0, B0, B Independent mark awarded 0, SC Special case ^ Omission sign MR Misread Highlighting Other abbreviations in mark Meaning scheme E Mark f explaining U Mark f crect units G Mark f a crect feature on a graph M dep* Method mark dependent on a previous mark, indicated by * cao Crect answer only oe Or equivalent rot Rounded truncated soi Seen implied www Without wrong wking

4 Subject-specific Marking Instructions f GCE Mathematics (OCR) Statistics strand a Annotations should be used whenever appropriate during your marking. The A, M and B annotations must be used on your standardisation scripts f responses that are not awarded either 0 full marks. It is vital that you annotate standardisation scripts fully to show how the marks have been awarded. F subsequent marking you must make it clear how you have arrived at the mark you have awarded. An element of professional judgement is required in the marking of any written paper. Remember that the mark scheme is designed to assist in marking increct solutions. Crect solutions leading to crect answers are awarded full marks but wk must not be judged on the answer alone, and answers that are given in the question, especially, must be validly obtained; key steps in the wking must always be looked at and anything unfamiliar must be investigated thoughly. Crect but unfamiliar unexpected methods are often signalled by a crect result following an apparently increct method. Such wk must be carefully assessed. When a candidate adopts a method which does not crespond to the mark scheme, award marks accding to the spirit of the basic scheme; if you are in any doubt whatsoever (especially if several marks candidates are involved) you should contact your Team Leader. c The following types of marks are available. M A suitable method has been selected and applied in a manner which shows that the method is essentially understood. Method marks are not usually lost f numerical errs, algebraic slips errs in units. However, it is not usually sufficient f a candidate just to indicate an intention of using some method just to quote a fmula; the fmula idea must be applied to the specific problem in hand, eg by substituting the relevant quantities into the fmula. In some cases the nature of the errs allowed f the award of an M mark may be specified. A Accuracy mark, awarded f a crect answer intermediate step crectly obtained. Accuracy marks cannot be given unless the associated Method mark is earned ( implied). Therefe M0 A cannot ever be awarded. B Mark f a crect result statement independent of Method marks. E 2

5 A given result is to be established a result has to be explained. This usually requires me wking explanation than the establishment of an unknown result. Unless otherwise indicated, marks once gained cannot subsequently be lost, e.g. wrong wking following a crect fm of answer is igned. Sometimes this is reinfced in the mark scheme by the abbreviation isw. However, this would not apply to a case where a candidate passes through the crect answer as part of a wrong argument. d e When a part of a question has two me method steps, the M marks are in principle independent unless the scheme specifically says otherwise; and similarly where there are several B marks allocated. (The notation dep * is used to indicate that a particular mark is dependent on an earlier, asterisked, mark in the scheme.) Of course, in practice it may happen that when a candidate has once gone wrong in a part of a question, the wk from there on is wthless so that no me marks can sensibly be given. On the other hand, when two me steps are successfully run together by the candidate, the earlier marks are implied and full credit must be given. The abbreviation ft implies that the A B mark indicated is allowed f wk crectly following on from previously increct results. Otherwise A and B marks are given f crect wk only differences in notation are of course permitted. A (accuracy) marks are not given f answers obtained from increct wking. When A B marks are awarded f wk at an intermediate stage of a solution, there may be various alternatives that are equally acceptable. In such cases, exactly what is acceptable will be detailed in the mark scheme rationale. If this is not the case please consult your Team Leader. Sometimes the answer to one part of a question is used in a later part of the same question. In this case, A marks will often be follow through. In such cases you must ensure that you refer back to the answer of the previous part question even if this is not shown within the image zone. You may find it easier to mark follow through questions candidate-by-candidate rather than questionby-question. f Wrong missing units in an answer should not lead to the loss of a mark unless the scheme specifically indicates otherwise. Candidates are expected to give numerical answers to an appropriate degree of accuracy. 3 significant figures may often be the nm f this, but this always needs to be considered in the context of the problem in hand. F example, in quoting probabilities from Nmal tables, we generally expect some evidence of interpolation and so quotation to 4 decimal places will often be appropriate. But even this does not always apply quotations of the standard critical points f significance tests such as.9,.4, 2.7 (maybe even 2. but not 2.7) will commonly suffice, especially if the calculated value of a test statistic is nowhere near any of these values. Sensible discretion must be exercised in such cases. Discretion must also be exercised in the case of small variations in the degree of accuracy to which an answer is given. F example, if 3 significant figures are expected (either because of an explicit instruction because the general context of a problem demands it) but only 2 are given, loss of an accuracy ("A") mark is likely to be appropriate; but if 4 significant figures are given, this 3

6 should not nmally be penalised. Likewise, answers which are slightly deviant from what is expected in a very min manner (f example a Nmal probability given, after an attempt at interpolation, as 0.4 whereas 0.47 was expected) should not be penalised. However, answers which are grossly over- under-specified should nmally result in the loss of a mark. This includes cases such as, f example, insistence that the value of a test statistic is (say) 2.47 merely because that is the value that happened to come off the candidate's calculat. Note that this applies to answers that are given as final stages of calculations; intermediate wking should usually be carried out, and quoted, to a greater degree of accuracy to avoid the danger of premature approximation. The situation regarding any particular cases where the accuracy of the answer may be a marking issue should be detailed in the mark scheme rationale. If in doubt, contact your Team Leader. g Rules f replaced wk If a candidate attempts a question me than once, and indicates which attempt he/she wishes to be marked, then examiners should do as the candidate requests. If there are two me attempts at a question which have not been crossed out, examiners should mark what appears to be the last (complete) attempt and igne the others. NB Follow these maths-specific instructions rather than those in the assess handbook. h Genuine misreading (of numbers symbols, occasionally even of text) occurs. If this results in the object and/ difficulty of the question being considerably changed, it is likely that all the marks f that question, section of the question, will be lost. However, misreads are often such that the object and/ difficulty remain substantially unaltered; these cases are considered below. The simple rule is that all method ("M") marks [and of course all independent ("B") marks] remain accessible but at least some accuracy ("A") marks do not. It is difficult to legislate in an overall sense beyond this global statement because misreads, even when the object and/ difficulty remains unchanged, can vary greatly in their effects. F example, a misread of.02 as 0.2 (perhaps as a quoted value of a sample mean) may well be catastrophic; whereas a misread of.74 as.74 may have so slight an effect as to be almost unnoticeable in the candidate's wk. A misread should nmally attract some penalty, though this would often be only mark and should rarely if ever be me than 2. Commonly in sections of questions where there is a numerical answer either at the end of the section to be obtained and commented on (eg the value of a test statistic), this answer will have an "A" mark that may actually be designated as "cao" [crect answer only]. This should be interpreted strictly if the misread has led to failure to obtain this value, then this "A" mark must be withheld even if all method marks have been earned. It will also often be the case that such a mark is implicitly "cao" even if not explicitly designated as such. 4

7 On the other hand, we commonly allow "fresh starts" within a question part of question. F example, a follow-through of the candidate's value of a test statistic is generally allowed (and often explicitly stated as such within the marking scheme), so that the candidate may exhibit knowledge of how to compare it with a critical value and draw conclusions. Such "fresh starts" are not affected by any earlier misreads. A misread may be of a symbol rather than a number f example, an algebraic symbol in a mathematical expression. Such misreads are me likely to bring about a considerable change in the object and/ difficulty of the question; but, if they do not, they should be treated as far as possible in the same way as numerical misreads, mutatis mutandis. This also applied to misreads of text, which are fairly rare but can cause maj problems in fair marking. The situation regarding any particular cases that arise while you are marking f which you feel you need detailed guidance should be discussed with your Team Leader. Note that a miscopy of the candidate s own wking is not a misread but an accuracy err.

8 S June 204 Mark Scheme Final (without introduction) Note: (3 sfs) means answer which rounds to... to 3 sfs. If crect ans seen to > 3sfs, ISW f later rounding Penalise over-rounding only once in paper. Question Answer Marks Guidance (i) Median = 7.4 (m) B cao IQR = M allow even though this is an increct method 7 =.0 (m) allow.7. NOT.3 (ii) B* A These pairs of values only, and subtract, f M eg allow but only if med = , =.0 BMA 7.4, =.7 BMA 7.4, 7.. =.3 BMA0 7.4, 7.7. =.2 BM0A0 crect digits in crect leaves, igne der, allow one omitted extra misplaced increct digit 7.4, = 0. BMA0 74., = 0. B0MA 74., = 0. B0MA 74., 77.7 =.7 B0MA 7.4, 7 = 3 BMA0 74., 7 = 3 B0MA0 74., 77 = 2 B0M0A0 Allow a separate diag with leaves to left of stem. If only a separate diag is drawn, with leaves to right of stem: all crect including der, alignment and key: B Complete crect diag including der and Bdep key: eg 4 means. (B) and.4 (A) If all digits are in crect rows and ders,

9 Question Answer Marks Guidance key and alignment allow just means. NOT means. Allow means, if consistent with (i) & crect key, award this mark unless EITHER:. eg a 2 nd digit in one row is clearly aligned with a 3 rd digit in another OR 2. st, 3rd, 4th & th rows are very [2] different lengths, eg because of crossing out and replacement (iii) One crect comment on size: B. One crect comment on spread shape: B. The following are examples only. Igne any wking; mark the statements only. Allow "First set" "Right" f A, "Second set" "Left" f B. A higher overall A has me taller trees fewer shter A has higher median (mean, ave, medium) B me evenly spread distributed B me spread out B has larger range IQR sd Ranges of both are similar A is nearer to nmal A is negatively skewed A has a (unique) mode, modal class peak; (B doesn't) B B B shter overall B has fewer taller trees me shter B has lower median (mean, ave, medium) A less evenly spread distributed A less spread out A has smaller range IQR sd Allow A's heights are me consistent Not other comment about skew Igne any other reference to mode most common NOT A higher than B NOT B has shter trees than A Allow just quoting the two medians, even if wrong, so long as med of A is gter than med of B. Similarly if quote IQRs NOT any reference to outliers NOT any reference to sample size NOT any reference to indiv trees NOT two comments on size NOT two comments on spread [2] Igne all else even if increct eg highest on both is. B0 2 (a) ( ) M last two terms crect. NOT eg (0) M2 st 3rd term crect M M allow (any number) 2, dep +ve result = 2.4 A 3 M0M0A0 2 (b) (i) 2k + 3k + 4k + k = oe B 4k = oe "= " is essential NOT just = 4 so k = 4 7

10 Question Answer Marks Guidance (k = 4 AG) Allow verification, eg stating that [] = (b) (ii) 2, 3 4 4, 4 4, 4 2 4,, 2, 20 B > 3 crect 2k, k,2k, 20k B Σxp M > 3 crect terms added 2k +k +2k +20k 40k M 4 M0A0 = (3 sf) oe, eg 7 4 A SC (=2.43) B0MA0 3 (i) Use of instead of. f last value of x: all M-marks can be sced, but no A-marks. (ans: gives 2.32 and.23; gives 2.39 and.40) Use of and instead of. (probably with freqs 9400/2) could lead to crect mean MA, but possibly MMA0 f sd. Σfx Σf attempted (= ) M = 2.3 (3 sf) A Σfx 2 attempted (= Σf = ) M 3 terms of Σfx crect.. and Σf Allow increct Σf NOT Σx 3 terms of Σfx 2 crect and Σf Allow increct Σf NOT Σx M0A0 Σf ( x x) Σf 2 3 terms of num crect and Σf M2 ( ), ( ) 2 terms of num crect and Σf M Allow increct Σf but NOT if Σf = Σx "2.3" 2 (=.70 to.72, 3 NB not requ'd f MM sf) M dep +ve result M0M0A0 s.d. =.3.30 (3 sf) A allow.3 Crect answer(s) without wking sce full marks []

11 Question Answer Marks Guidance 3 (ii) 2 B Igne wking f both, even if 3 B allow IQR = 3 = 2, ie UQ = 3 implied Increct [2] NB 3, 2 B0B0 unless labelled crectly 4 If 2 is interpreted consistently as , max marks: (i)(a) MMA0 3 (i)(b) B0 (i)(c) Bft Bft (ii) BMMA0 4 (i) (a) Binomial seen implied M by use of table 9 C ( 3 2 ) p ( 3 ) q (p + q = 9) Eg 0.22 seen M 9 C ( 3 ) 3 ( 3 2 ) = (3 sf) A (i) (b) (3 sf) B NB (from ) rounds to 0.30 so B [] 4 (i) (c) Bft 2 Bft NB 2, B0B0 unless labelled crectly [2] 4 (ii) 27 seen B not necessarily in a statement B(27, 2 3 ) seen implied M (i) 27 C ( 3 ) 9 ( 2 3 ) M attempt eg P(X = ) P(X 2 = ) P(X 3 = 9), P(X = 2) P(X 2 = 7) P(X 3 = 9), P(X = 3) P(X 2 = ) P(X 3 = 9), etc >3 sets with X +X 2 +X 3 = (not nec'y added) M = 0. (3 sf) A [4] S xx = (= 4200) S yy =.. (=.) 2 NB P(X = ) P(X 2 = ) P(X 3 = ) = = M0M0A0 (= 0.074) M0M0A

12 Question Answer Marks Guidance S xy = 30 (= ) M Crect sub in a crect S fmula. r = " " M Crect sub in 3 crect S fmulae and a "4200" "." crect r fmula = 0.9 (3 sf) A Crect ans with no wking M2A Igne comment about < r < 0.9 (ii) eg As you move further away, prices drop B High prices go with sht distances oe Both variables must be in context ; miles & enough Allow "Strong ( high good equiv) neg cr'n between price and distance" Igne all else, even if increct NOT just neg cr'n between price & dist [] (iii) None B Igne all else, even if increct [] (iv) b = " "4200" " (= ) M ft their S xy & S xx from (i) f M-marks only fresh start crect method Y. = (x y = 0.0x +. (3 sf) oe A oe y = 20 x 40 (v) Values of x are chosen befehand x is independent controlled 30 ) oe M a = B [] (i) B [] oe allow y = 0.0x +. ( figs which round to these) (NOT y = 0.0x +. NOT y = 0.02x +. ) Crect ans with no wking M2A x is fixed given set predetermined oe (ii) Σd 2 = 0 f first teams M May be implied by use of Σd 2 = 2 Σd 2 = 2 B 2 d ( 2 ) M ft their Σd 2 ( 0) Must have "y = " Allow figures in equn which round to the crect figures to either 3 sf 2 sf, even if they result from arith errs. Not "x is constant." Not just "y depends on x" Igne all other, even if increct using ranks from (i) can sce 2nd M only 0

13 Question Answer Marks Guidance 7 (i) (ii) = (3 sf) A [4] n = n+4 n : 4 = : 3 3 : 4 = : n n = 7 4 "7" "7" M A [2] + alone oe M - 3F =4 & n= F; 4 4; "7" oe Completely crect method " 0" 7 oe "0" crect first step involving n complete crect method f finding n 4+ "7" + 2+ "7" "0" "0" "7" "0" = (3 sf) oe 0 Aft ft their integer answer to (i) eg if their (i) is 2, ans 0. MAft (iii) (a) 0 oe ( 0 + seen ( 2 ) ) [2] M = A 7 (iii) (b) FA + MC FC + MA Either 4 2 (iiia) 0 2 NB ft their C C2 C 0 2 oe ie allow M if 2 is omitted OR if instead of, but not both errs oe oe allow M f crect num denom [2] NB long methods may be crect, eg ( 4 0 ) ( ) same as M Allow NB ft their (iii)(a) 4 ie allow instead of AND allow one case with 2 both cases without 2 0 ie allow and one of these two errs

14 Question Answer Marks Guidance cf scheme f (iii)(a) (i) ( = + ) = (3 sf) A cao C 2 oe seen anywhere num= 0 alone 0 C C2 C + 4 C C2 C oe oe 300 allow M if one of these fracts crect NB C 2 in denom NOT M, cf (iii)(a) [2] NB see note on long methods in 7(iiia) M 4 20 oe seen alone... eg 2 4 M 7 C2 oe 2 C4 P4 C 4! M oe all crect C ! 2 7 oe 4 2 oe all crect NB oe all crect M2 increct C 4 does not sce = (3 sf) A Crect ans sces MMA regardless of method. (ii)! 2 alone! 2 alone oe M2 M f!! P 720 seen NB! sces M0 unless!! 2 = 440 A (iii)! 4 alone! 2 2 alone M2 M f! P 720 seen! seen but NOT from! 3! = 20 A 9 If 0.3 and 0.7 are interchanged consistently through all four parts, all M-marks can be sced, but no A-marks. M f 7! 2 alone NB 7! sces M0 unless 7! 2 alone!: M0 unless!! 2! If 0.3 is calculated increctly (eg ) consistently, lose the A-mark in (i) but all other marks are available on ft, so long as 0 < ans <. 2

15 Question Answer Marks Guidance 9 (i) alone M = (3 sf) 7203 oe A allow [2] 9 (ii) ( ) 0.3 M2 M f term omitted, wrong extra. must add terms, not mult. = (3sf) 499 oe 0000 A Allow (iii) 0.7 M2 M f 0.7 alone 0.7 (= 0.32) (= 0.9) ( ) M2 ( ) M M Just : M0 ( ) M2 term omitted, wrong extra M 0.3( ) M2 (ii) + 0.3( ) M2 (i) + (ii) + 0.3( ) M2 one term omitted extra: must add terms, not mult. M NB ans 0.32 might be MM0A0 from omitting last term. Could be, eg, their (ii) + 0.3( ) crect wking, but subtr from : M = 0.2 (3 sf) A 9 (iv) ( 0.2 ) oe M (0.7 ) 2 ( 0.7 ) ( 0.7) Not (0.7 ) 2 their "0.2" 0.3( ( )) Completely crect method = (3 sf) Aft allow ft their "0.2" except if [2] 3

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