2014 Applied Mathematics - Statistics. Advanced Higher. Finalised Marking Instructions

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1 04 pplied Mathematics - Statistics dvanced Higher Finalised ing Instructions Scottish Qualifications uthority 04 The information in this publication may be reproduced to support SQ qualifications only on a noncommercial basis. If it is to be used for any other purposes written permission must be obtained from SQ s NQ ssessment team. Where the publication includes materials from sources other than SQ (secondary copyright), this material should only be reproduced for the purposes of examination or assessment. If it needs to be reproduced for any other purpose it is the centre s responsibility to obtain the necessary copyright clearance. SQ s NQ ssessment team may be able to direct you to the secondary sources. These ing Instructions have been prepared by Examination Teams for use by SQ ppointed ers when marking External Course ssessments. This publication must not be reproduced for commercial or trade purposes.

2 Part One: General ing Principles for pplied Mathematics Statistics dvanced Higher This information is provided to help you understand the general principles you must apply when marking candidate responses to questions in this Paper. These principles must be read in conjunction with the specific ing Instructions for each question. (a) (b) s for each candidate response must always be assigned in line with these general marking principles and the specific ing Instructions for the relevant question. ing should always be positive ie, marks should be awarded for what is correct and not deducted for errors or omissions. GENERL MRKING DVICE: pplied Mathematics Statistics dvanced Higher The marking schemes are written to assist in determining the minimal acceptable answer rather than listing every possible correct and incorrect answer. The following notes are offered to support ers in making judgements on candidates evidence, and apply to marking both end of unit assessments and course assessments. These principles describe the approach taken when marking dvanced Higher pplied Mathematics papers. For more detailed guidance please refer to the detailed ing Instructions. The main principle is to give credit for the skills demonstrated and the criteria met. Failure to have a correct method may not preclude a candidate gaining credit for their solution. The answer to one part of a question, even if incorrect, can be accepted as a basis for subsequent dependent parts of the question. The following are not penalised: working subsequent to a correct answer (unless it provides firm evidence that the requirements of the question have not been met) legitimate variation in numerical values/algebraic expressions. 4 Full credit will only be given where the solution contains appropriate working. Where the correct answer might be obtained by inspection or mentally, credit may be given. 5 Sometimes the method to be used in a particular question is explicitly stated; no credit will be given where a candidate obtains the correct answer by an alternative method. 6 Where the method to be used in a particular question is not explicitly stated in the question paper, full credit is available for an alternative valid method. (Some likely alternatives are included but these should not be assumed to be the only acceptable ones.) Page

3 Part Two: ing Instructions for each Question Section Question Expected nswer(s) Max dditional Guidance (a) P(S) P(Pos).P(S Pos) P(Neu).P(S Neu) P(Neg).P(S Neg) (b) P(S Pos) P(Pos S) P(S) P(Pos S) P(S) P(Pos).P(S Pos) P(S) (0 46) 59 (other methods are acceptable) Page

4 Question Expected nswer(s) Max dditional Guidance (a) In order to take a simple random sample, a list of all sixth year pupils in Scottish schools would be required and such a list is not readily available. (b) simple random sample of the 76 schools would be taken. You could then visit each of the chosen schools to obtain the required information from sixth year pupils. (c) random sample of schools would be taken in each region. You could then visit each of the selected schools to obtain the required information from sixth year pupils. (d) n advantage of cluster sampling is that it can reduce the cost of obtaining data through reducing travel costs for interviewers. n advantage of stratified sampling is that it yields more precise estimates. Page 4

5 Question Expected nswer(s) Max dditional Guidance Ho: there is no difference in population blood group distribution H: there is a difference 6 Group O B B O i E i ( O E) x E (90 0) (49 868) x (8 5 4) (0 8 7) with d.f. The critical value of chi-squared at the 0 % level is Since 48 5 exceeds this value Ho is rejected and the data provide very strong evidence that the blood group proportions amongst people of Hawaiian ethnic origin differ from those amongst the Caucasian segment of the US population eg the Hawaiians have significantly more people in group and less in the other three groups. Page 5

6 Question Expected nswer(s) Max dditional Guidance 4 Rank sum for yields obtained using compost = = 44 6 W n ( n ) P( W n( n ) 8) p-value = = < 0 05 so we would reject Ho at the 5% level and conclude that there is evidence that the median yields differ. 5 (a) The assumption required is that the sample is random. ˆ( ˆ) ˆ 96 p p p n (00,0 6) (b) f ( x) x( x) f ( x) x 0 when x 0 5 M max n 9605 n n 5 (c) M n n Page 6

7 Question Expected nswer(s) Max dditional Guidance 6 (a) x 555 s x t 46 s 5778 n 0 The critical region is t 6 Since the observed value of t lies in the critical region, the null hypothesis that = 540 would be rejected. The assumption required is that tensile strength is normally distributed. 6 (b) Since 540 does not lie in the interval the null hypothesis H 0 : = 540 would be rejected at the 5% level of significance. 7 (a) T 0 X Y 0 Y 4 40 T X Y 45 T X Y r 9 7 (b) r kt k 68 9 P(T 60) (c) If one takes longer to shop one might buy more goods and take longer to checkout. Plot a scattergraph of Y on X and check for a correlation. Page 7

8 Question Expected nswer(s) Max dditional Guidance 8 (a) P(Coin deemed biased coin is fair) P( X 0) where X ~ B(00,0 5) Y ~ N(00, 50) P( X 0) P( Y 095) P( Z 4) (b) P(Coin deemed fair coin is biased) P( X 0 X ~ B (00, 0 55) P( Y 09 5 Y ~ N (0, 495) P( Z 007) (c) The risks could be reduced by increasing the number of tosses from (a) For each additional crow fly mile one can expect to travel, on average, an additional 54 miles by road on the optimum route. 9 (b) n urban area has almost certainly got more roads. Page 8

9 Question Expected nswer(s) Max dditional Guidance 9 (c) Yˆ a bx i i 7 The required prediction interval is given by ˆ ( xi x) Yi t /, ns n S xx (40 055) (56 0,66 0) The required confidence level is given by ˆ ( xi x) Yi t /, ns n S xx (40 055) (588,6 ) For a school with crow fly distance 40 miles, the PI indicates that one can be 95% confident that the optimum road distance will lie between 56 0 and 66 0 miles. The CI indicates that one can be 95% confident that mean optimum road distance for schools with crow fly distance 40 miles will lie between 58 8 and 6 miles. 9 (d) Since 0 97 exceeds 0 05 the implication is that the null hypothesis that = 0 cannot be rejected at the 5% level of significance. The linear model could therefore be simplified to Y x E( Y ) x i i i i i Thus a crow fly distance of 0 corresponds to an expected optimum road distance of 0 as would be anticipated. [END OF SECTION ] Page 9

10 Section B (Mathematics for pplied Mathematics) Question Expected nswer(s) Max dditional Guidance B y x x dy x. d x x d x dx dx dx x. x x Gradient given by dy when x = 0, dx Gradient = product rule first correct term second correct term evaluation (accept decimal equivalent to minimum of sf) B (a) B k 9 5 evaluation B (b) 0 k k 0 det = (0 + ) (0 + 5) + 4(k 0) = k form of determinant evaluation B (c) 0 6 BC evaluation B (d) BC = I. B = C or C = B identity matrix connection or mention of inverse relationship correct Page 0

11 Question Expected nswer(s) Max dditional Guidance B I xsin x dx 5 u = x du = dv = sinx v sinx cos x evidence of integration by parts correct choice of u, dv I x. cos x. cos x dx correct substitution x cosx cosx dx x cosx sinx 9 final integration correct x I0 cosx sin x 9 cos6 sin 6 0 sin evaluation Page

12 Question Expected nswer(s) Max dditional Guidance B r r r r using n n n n r * r 6 80 r r = correct substitution into * evaluation (using incorrect formula this mark available if of equivalent difficulty eg n r r n n B 5 (a) (e x + ) 4 x 4 x x e e e x x e. e x x x x e e e e ccept Binomial expansion or Pascal s Triangle correct coefficients correct powers of e x and simplification B 5 (b) x e 4 dx 4x x x x e 8e 4e e 6 dx 4x x x e 8e 4e x e 6x c 4 correct integration of composite fractions (at least one correct term involving exponential) completion of integral (+ c not essential) Page

13 Question Expected nswer(s) Max dditional Guidance B 6 (a) people. B 6 (b) 0000 B N 0000 N N 0000 N 5 appropriate form of partial fractions = (0 000 N) + BN, B correct values of and B Using N N dn dt gives dn dt N 0000 N separate variables Integrating, dn N 0000 N dt nn n 0000 N t c N n t c 0000 N starts integration eg dn correct N completes integration (moduli signs not required) Page

14 Question Expected nswer(s) Max dditional Guidance B 6 (c) N Using n t c 0000 N 4 N gives 0000 N e tc N 0000 N t c Ke where K e accurately converts to exponential form (stating explicitly K= e c not required) When t = 0, N = K 99 K Hence N 0000 N 99N = (0 000 N) e t N(99 + e t ) = 0 000e t t e 99 interprets initial condition K valve correctly gathers N terms 0000e N 99 e t t [END OF SECTION B] [END OF QUESTION PPER] Page 4

2014 Applied Mathematics - Mechanics. Advanced Higher. Finalised Marking Instructions

2014 Applied Mathematics - Mechanics. Advanced Higher. Finalised Marking Instructions 04 Applied Mathematics - Mechanics Advanced Higher Finalised ing Instructions Scottish Qualifications Authority 04 The information in this publication may be reproduced to support SQA qualifications only