General Certificate of Education Advanced Subsidiary Examination January 2013

Transcription

1 General Certificate of Education Advanced Subsidiary Examination January 013 Mathematics Unit Statistics 1B Statistics Unit Statistics 1B Friday 18 January pm to 3.00 pm F this paper you must have: the blue AQA booklet of fmulae and statistical tables. You may use a graphics calculat. Time allowed 1 hour 30 minutes MS/SS1B Instructions Use black ink black ball-point pen. Pencil should only be used f drawing. Fill in the boxes at the top of this page. Answer all questions. Write the question part reference (eg (a), (b)(i) etc) in the left-hand margin. You must answer each question in the space provided f that question. If you require extra space, use an AQA supplementary answer book; do not use the space provided f a different question. Do not write outside the box around each page. Show all necessary wking; otherwise marks f method may be lost. Do all rough wk in this book. Cross through any wk that you do not want to be marked. The final answer to questions requiring the use of tables calculats should nmally be given to three significant figures. Condensed Infmation The marks f questions are shown in brackets. The maximum mark f this paper is 75. Unit Statistics 1B has a written paper only. Advice Unless stated otherwise, you may quote fmulae, without proof, from the booklet. You do not necessarily need to use all the space provided. P56863/Jan13/MS/SS1B 6/6/6/ MS/SS1B

2 1 Bob, a church warden, decides to investigate the lifetime of a particular manufacturer s brand of beeswax candle. Each candle is 30 cm in length. From a box containing a large number of such candles, he selects one candle at random. He lights the candle and, after it has burned continuously f x hours, he recds its length, y cm, to the nearest centimetre. His results are shown in the table. x y (a) State the value that you would expect f a in the equation of the least squares regression line, y ¼ a þ bx. (1 mark) (b) (i) Calculate the equation of the least squares regression line, y ¼ a þ bx. (4 marks) (ii) Interpret the value that you obtain f b. ( marks) (iii) It is claimed by the candle manufacturer that the total length of time that such candles are likely to burn f is me than 50 hours. Comment on this claim, giving a numerical justification f your answer. ( marks) The volume of Everwhite toothpaste in a pump-action dispenser may be modelled by a nmal distribution with a mean of 106 ml and a standard deviation of.5 ml. Determine the probability that the volume of Everwhite in a randomly selected dispenser is: (a) less than 110 ml; (3 marks) (b) me than 100 ml; ( marks) (c) between 104 ml and 108 ml; (3 marks) (d) not exactly 106 ml. (1 mark) (0) P56863/Jan13/MS/SS1B

3 3 3 Stopoff owns a chain of hotels. Guests are presented with the bills f their stays when they check out. (a) Assume that the number of bills that contain errs may be modelled by a binomial distribution with parameters n and p, where p ¼ 0:30. Determine the probability that, in a random sample of 40 bills: (i) (ii) at most 10 bills contain errs; at least 15 bills contain errs; (iii) exactly 1 bills contain errs. (6 marks) (b) (c) Calculate the mean and the variance f each of the distributions B(16, 0.0) and B(16, 0.15). (3 marks) Stan, who is a travelling salesperson, always uses Stopoff hotels. He holds one of its diamond customer cards and so should qualify f special customer care. However, he regularly finds errs in his bills when he checks out. Each month, during a 1-month period, Stan stayed in Stopoff hotels on exactly 16 occasions. He recded, each month, the number of occasions on which his bill contained errs. His recded values were as follows (i) Calculate the mean and the variance of these 1 values. ( marks) (ii) Hence state with reasons which, if either, of the distributions B(16, 0.0) and B(16, 0.15) is likely to provide a satisfacty model f these 1 values. (3 marks) Turn over s (03) P56863/Jan13/MS/SS1B

4 4 4 Ashok is a wk-experience student with an ganisation that offers two separate professional examination papers, I and II. F each of a random sample of 1 students, A to L, he recds the mark, x per cent, achieved on Paper I, and the mark, y per cent, achieved on Paper II. A B C D E F G H I J K L x y (a) (i) Calculate the value of the product moment crelation coefficient, r, between x and y. (3 marks) (ii) Interpret your value of r in the context of this question. ( marks) (b) (i) Give two possible advantages of plotting data on a graph befe calculating the value of a product moment crelation coefficient. ( marks) (ii) Complete the plotting of Ashok s data on the scatter diagram on page 5. (iii) State what is now revealed by the scatter diagram. ( marks) (1 mark) (c) Ashok subsequently discovers that students A to F have a me scientific background than students G to L. With reference to your scatter diagram, estimate the value of the product moment crelation coefficient f each of the two groups of students. You are not expected to calculate the two values. ( marks) (04) P56863/Jan13/MS/SS1B

5 5 G H I J K L x y y 100 ~ Examination Marks 90 6 E 80 6 D 6 F Paper II mark (per cent) A 6 B 6 C x Paper I mark (per cent) ~ Turn over s (05) P56863/Jan13/MS/SS1B

6 6 5 Roger is an active retired lecturer. Each day after breakfast, he decides whether the weather f that day is going to be fine (F), dull (D) wet (W). He then decides on only one of four activities f the day: cycling (C), gardening (G), shopping (S) relaxing (R). His decisions from day to day may be assumed to be independent. The table shows Roger s probabilities f each combination of weather and activity. Weather Fine (F) Dull (D) Wet (W) Cycling (C) Activity Gardening (G) Shopping (S) Relaxing (R) (a) Find the probability that, on a particular day, Roger decided: (i) (ii) that it was going to be fine and that he would go cycling; on either gardening shopping; (iii) to go cycling, given that he had decided that it was going to be fine; (iv) not to relax, given that he had decided that it was going to be dull; (v) that it was going to be fine, given that he did not go cycling. (9 marks) (b) Calculate the probability that, on a particular Saturday and Sunday, Roger decided that it was going to be fine and decided on the same activity f both days. (3 marks) (06) P56863/Jan13/MS/SS1B

7 7 6 (a) The length of one-metre galvanised-steel straps used in house building may be modelled by a nmal distribution with a mean of 1005 mm and a standard deviation of 15 mm. The straps are supplied to house builders in packs of 1, and the straps in a pack may be assumed to be a random sample. Determine the probability that the mean length of straps in a pack is less than one metre. (4 marks) (b) Tania, a purchasing officer f a nationwide house builder, measures the thickness, x millimetres, of each of a random sample of 4 galvanised-steel straps supplied by a manufacturer. She then calculates crectly that the value of x is 4.65 mm. (i) (ii) Assuming that the thickness, X mm, of such a strap may be modelled by the distribution N(m, 0.15 ), construct a 99% confidence interval f m. (4 marks) Hence comment on the manufacturer s specification that the mean thickness of such straps is greater than 4.5 mm. ( marks) 7 A machine, which cuts bread dough f loaves, can be adjusted to cut dough to any specified set weight. F any set weight, m grams, the actual weights of cut dough are known to be approximately nmally distributed with a mean of m grams and a fixed standard deviation of s grams. It is also known that the machine cuts dough to within 10 grams of any set weight. (a) Estimate, with justification, a value f s. ( marks) (b) The machine is set to cut dough to a weight of 415 grams. As a training exercise, Sunita, the quality control manager, asked Dev, a recently employed trainee, to recd the weight of each of a random sample of 15 such pieces of dough selected from the machine s output. She then asked him to calculate the mean and the standard deviation of his 15 recded weights. Dev subsequently repted to Sunita that, f his sample, the mean was 391 grams and the standard deviation was 95.5 grams. Advise Sunita on whether not each of Dev s values is likely to be crect. Give numerical suppt f your answers. (3 marks) (c) Maria, an experienced quality control officer, recded the weight, y grams, of each of a random sample of 10 pieces of dough selected from the machine s output when it was set to cut dough to a weight of 80 grams. Her summarised results were as follows. X y ¼ 810:0 and X ðy yþ ¼ 110:00 Explain, with numerical justifications, why both of these values are likely to be crect. (4 marks) Copyright ª 013 AQA and its licenss. All rights reserved. (07) P56863/Jan13/MS/SS1B

8 Key to mark scheme abbreviations M mark is f method m dm mark is dependent on one me M marks and is f method A mark is dependent on M m marks and is f accuracy B mark is independent of M m marks and is f method and accuracy E mark is f explanation ft F follow through from previous increct result CAO crect answer only CSO crect solution only AWFW anything which falls within AWRT anything which rounds to ACF any crect fm AG answer given SC special case OE equivalent A,1 1 ( 0) accuracy marks x EE deduct x marks f each err NMS no method shown PI possibly implied SCA substantially crect approach c candidate sf significant figure(s) dp decimal place(s) No Method Shown Where the question specifically requires a particular method to be used, we must usually see evidence of use of this method f any marks to be awarded. Where the answer can be reasonably obtained without showing wking and it is very unlikely that the crect answer can be obtained by using an increct method, we must award full marks. However, the obvious penalty to candidates showing no wking is that increct answers, however close, earn no marks. Where a question asks the candidate to state write down a result, no method need be shown f full marks. Where the permitted calculat has functions which reasonably allow the solution of the question directly, the crect answer without wking earns full marks, unless it is given to less than the degree of accuracy accepted in the mark scheme, when it gains no marks. Otherwise we require evidence of a crect method f any marks to be awarded.

9 MS/SS1B - AQA GCE Mark Scheme (PV) 013 January Series MS/SS1B Q Solution Marks Total Comments 1 (a) a = 30 CAO 1 (b)(i) b (gradient) = 0.64 B CAO ( 0.64) b (gradient) = 0.6 to 0.7 () AWFW Treat rounding of crect answers as ISW Written fm of equation is not required a (intercept) = 31 B CAO (31) a (intercept) = 30 to 3 () AWFW Attempt at Attempt at x S xx & xy x y & S S yy xy y () & 415 (643) (all 4 attempted) 1500 & 960 (618) (both attempted) Attempt at crect fmula f b (gradient) (m1) b (gradient) = 0.64 a (intercept) = 31 (A1 A1) CAO both 4 (ii) Candle length reduces by 0.64 (cm) per hour Candle burns 0.64 (cm) each/per hour Candle reduces by 0.64 (cm) each/per hour BF1 (BF) (BF1) OE; must be in context OE; must be in context OE; must be in context OE; must be in context (double ve) F on 0.6 b 0.7 from (i) (Length, y, cm) decreases with (time, x, hours) As (time, x, hours) increases then (length, y, cm) decreases () OE; context not required B0 f reference only to crelation (iii) When x = 50, y = (31 30) = 1 When y = 0, x = = 48 to = 46.8 to 47 CAO; accept crect comparison of 3 with either AWFW AWFW Claim not justified 1 is impossible value < 50 Bdep1 OE; dependent on previous Claim cannot be answered due to uneven burning unlikely to burn completely () Extrapolation required 9

10 MS/SS1B - AQA GCE Mark Scheme (PV) 013 January Series MS/SS1B (cont) Q Solution Marks Total Comments In (a), igne the inclusion of a lower limit of 0; it has no effect on the answer Volume, V ~ N(106,.5 ) (a) P(V < 110) = PZ 5. Standardising 110 with 106 and.5; allow ( ) = P(Z < 1.6) A1 CAO; igne inequality and sign May be implied by a crect answer = A1 AWRT (0.9450) 3 (b) P(V > 100) = P(Z >.4) = P(Z < +.4) Crect area change May be implied by a crect answer by an answer > 0.5 (c) P(104 < V < 108) = P( a < Z < a) = = to 0.99 A1 AWFW ( ) P(Z < a) (1 P(Z < a)) P(Z < a) 1 = ( ) = = A1 OE; a = 0.8 is not a requirement May be implied by seen by a crect answer AWRT ( /0.1186) Condone 0.11 May be implied by a crect answer = A1 AWRT (0.5768) 3 (d) P(V 106) = 1 one unity 100% 1 CAO; accept nothing else but igne additional wds providing they are not contradicty (eg certain so = 1) Total 9

11 MS/SS1B - AQA GCE Mark Scheme (PV) 013 January Series MS/SS1B (cont) Q Solution Marks Total Comments 3 (a) E ~ B(40, 0.30) Used anywhere in (a) even only by implication from a crect value (i) P(E 10) = to A1 AWFW (0.3087) () SC F calc n of individual terms: award B f answer within above range; award f answer within range 0.3 to 0.3 (ii) P(E 15) = 1 ( ) Requires 1 Accept 3 dp rounding truncation Can be implied by 0.19 to but not by to = 0.19 to A1 AWFW (0.196) () SC F calc n of individual terms: award B f answer within above range; award f answer within range 0.18 to 0. (iii) P(E 1) = P(E 1) = Accept 3 dp rounding truncation Crect expression; may be implied by a crect answer = to A1 AWFW (0.1366) () 6 (b) Means = 3. and CAO both values; igne notation If not labelled, assume der in question Variances =.56 and 1.75 CAO each value; igne notation ISW all subsequent wking 3 (c)(i) Mean = CAO value; igne notation Variance =.54 to to.34 (SD = 1.59 to to 1.53) Any value within either range; igne notation ISW all subsequent wking (ii) B(16, 0.0) eg "One dist n " Different/larger mean Similar/same variance standard deviation Bdep1 Identification of distribution not required Both; dep on 3.,.56 /1.6 & (c)(i) B(16, 0.15) eg "Other dist n " Equal/same mean Different/smaller variance standard deviation Bdep1 Identification of distribution not required Both; dep on, 1.75/1.3 & (c)(i) SC Neither likely to provide satisfacty model Bdep1 Dep on Bdep1 and on Bdep1 3 Award Bdep1 Bdep0 Bdep0 f comparison of 3 crect means only f comparison of 3 crect variances/sds only Award up to Bdep1 Bdep1 Bdep1 f comparison of 3 crect means and f comparison of 3 crect variances/sds Total 14

12 MS/SS1B - AQA GCE Mark Scheme (PV) 013 January Series MS/SS1B (cont) Q Solution Marks Total Comments 4(a) (i) r = 0.36 to 0.35 B3 AWFW ( ) r = 0.33 to 0.3 (B) AWFW r = 0.4 to 0. () AWFW r = 0. to 0.4 () AWFW Attempt at Attempt at x x y S S xx yy & S xy Attempt at substitution into crect cresponding fmula f r y & xy () (m1) r = 0.36 to 0.35 (A1) AWFW & 4565 (all 5 attempted) & 84 (all 3 attempted) (ii) Some/little/slight/(fairly/quite) weak/ (fairly/quite) moderate negative (linear) crelation/relationship/ association/link (but not trend ) between Bdep1 Dependent on 0.4 r 0. OE; must qualify strength and state negative Igne extra wds unless contradict Bdep0 f low, small, po, unlikely, medium, average, adjective very marks/percentages in the two examination papers Context; providing 1 < r < 1 (b)(i) Identifying linear patterns/non-linear patterns/ multiple patterns/no pattern (allow trend ) Identifying outliers/anomalies B,1 OE; only one mark from each set Estimating/gives idea of value of r/sign of r B0 f reference to checking calculated value (ii) Graph (6 labelled points crect) (5 4 labelled points crect) B () Crect within a circle of radius equal to distance between grid lines Deduct 1 mark f any unlabelled increctly labelled point (iii) Two separate crelations/relationships/lines/ associations/links/sets of data (but not trends ) 1 OE; eg A to F and G to L (c) A to F: (+)0.7 to (+)0.99 AWFW; allow calculation (0.937) If not labelled, assume der A to F then G to L G to L: 0.9 to 0.5 AWFW; allow calculation ( 0.757) Total 1

13 MS/SS1B - AQA GCE Mark Scheme (PV) 013 January Series MS/SS1B (cont) Q Solution Marks Total Comments 5 Ratios (eg 3:10) are only penalised by 1 accuracy mark at first crect answer (a)(i) P(F & C) = 0.3 3/10 30% CAO (0.3) (1) (ii) P(G S) = /100 45% CAO (0.45) (1) (iii) P(C F) = 0.3 i 0.55 = 30/55 6/11 (0.54 to 0.55) (54% to 55%) A1 () CAO (6/11) AWFW ( ) (iv) P(R D) = Crect numerat Crect denominat 5/30 5/6 (0.83 to 0.834) (83% to 83.4%) A1 (3) CAO (5/6) AWFW ( ) (v) P(F C) = Crect expression 5/60 5/1 (0.416 to 0.4) (41.6% to 4%) A1 (, 3) CAO (5/1) AWRT ( ) 9 (b) P = [P(F & C)] + [P(F & G)] = A1 Attempt at sum of at least squared terms; 0 < term < 1; not a b May be implied by a crect expression a crect answer OE Igne additional terms integer multipliers May be implied by a crect answer 155/ /000 61/400 (0.15 to 0.153) (15.% to 15.3%) A1 3 CAO AWFW (0.155) Total 1

14 MS/SS1B - AQA GCE Mark Scheme (PV) 013 January Series MS/SS1B (cont) Q Solution Marks Total Comments 6 (a) L ~ N(1005, 15 ) OR V(pack) = 15 /1 5/1 75/ to 18.8 SD (pack) = 15/1 15/3 53/ 4.3 to 4.4 CAO AWFW (18.75) CAO; OE AWFW ( ) PL 1000 P 15 1 = Standardising 1000 using 1005 and 15/1 OE; allow ( ) P(Z < ) = 1 P(Z < ) = m1 Crect area change May be implied by a crect answer an answer < ( to ) = 0.13 to 0.16 A1 4 AWFW (0.1411) (1 answer) max (b)(i) 99% (0.99) z =.57 to.58 AWFW (.5758) CI f is x z n Used with z (.05 to.58), x (4.65) & (0.15) and n with n > 1 Thus A1 4 z (.05 to.06.3 to to.58), x (4.65) & (0.15) and Hence CAO/AWRT OR A1 (4.57, 4.73) 4 AWRT (b)(ii) Clear crect comparison of 4.5 with LCL CI (eg 4.5 < LCL its value 4.5 < CI its limits BF1 F on CI only providing LCL > 4.5 (ie whole of CI > 4.5) Quoting values f LCL f CI is not required BF0 f '4.5 is outside CI'; OE so Agree with manufacturer s specification Bdep1 OE; dependent on previous BF1 Total 10

15 MS/SS1B - AQA GCE Mark Scheme (PV) 013 January Series MS/SS1B (cont) Q Solution Marks Total Comments 7 (a) 10 0 a b range b 10c 0d OE; with a 4 4 b 8 with c d in equiv percentages Cannot be implied from a crect answer (justification required) SC.5 3.3(OE) 5 A1 Award f only.5 3.3(OE) 5 with no justification Award B0 f any other answer with no justification with increct justification (eg 10 = 3.16) (b) Valid statement involving: 391 and 405 OR 401 and 415 OR 4 and 10 OR 391 and 415 and 10/4 with linking statement Allow 'set weight' to imply 415 and/ mean to imply 391 B0 f 10 linked to 95.5 > (value of of.5 3.3(OE) 5) Accept rather than > Clear crect numerical comparison Neither (likely to be) crect Bdep1 Dependent on 3 (c) OR Mean y = y = 800 CAO; Variance OR = 1. = 11 AWRT CAO Award on value; igne notation SD AWRT OR 81 is similar to/within 10 of is within 100 of 800 OE; clear crect numerical comparison of 81 with 80 Allow 'set weight' to imply 80 Or OE; clear crect numerical comparison of 810 with 800 but do not accept within 10 here is similar to a value of of 3.3(OE).5 4 Clear crect numerical comparison Total 9 TOTAL 75

16 Scaled mark unit grade boundaries - January 013 exams A-level Maximum Scaled Mark Grade Boundaries and A Conversion Points Code Title Scaled Mark A A B C D E LAW03 LAW UNIT MD01 MATHEMATICS UNIT MD MD0 MATHEMATICS UNIT MD MFP1 MATHEMATICS UNIT MFP MFP MATHEMATICS UNIT MFP MFP3 MATHEMATICS UNIT MFP MFP4 MATHEMATICS UNIT MFP MB MATHEMATICS UNIT MB MMB MATHEMATICS UNIT MMB MPC1 MATHEMATICS UNIT MPC MPC MATHEMATICS UNIT MPC MPC3 MATHEMATICS UNIT MPC MPC4 MATHEMATICS UNIT MPC MS1A MATHEMATICS UNIT MS1A MS/SS1A/W MATHEMATICS UNIT S1A - WRITTEN MS/SS1A/C MATHEMATICS UNIT S1A - COURSEWORK MS1B MATHEMATICS UNIT MS1B MSB MATHEMATICS UNIT MSB MEST1 MEDIA STUDIES UNIT MEST MEDIA STUDIES UNIT MEST3 MEDIA STUDIES UNIT MEST4 MEDIA STUDIES UNIT