AS Mathematics. MS1B Statistics 1B Mark scheme June Version 1.0: Final Mark Scheme

Transcription

1 AS Mathematics MS1B Statistics 1B Mark scheme 6360 June 016 Version 1.0: Final Mark Scheme

2 Mark schemes are prepared by the Lead Assessment Writer and considered, together with the relevant questions, by a panel of subject teachers. This mark scheme includes any amendments made at the standardisation events which all associates participate in and is the scheme which was used by them in this examination. The standardisation process ensures that the mark scheme covers the students responses to questions and that every associate understands and applies it in the same crect way. As preparation f standardisation each associate analyses a number of students scripts. Alternative answers not already covered by the mark scheme are discussed and legislated f. If, after the standardisation process, associates encounter unusual answers which have not been raised they are required to refer these to the Lead Assessment Writer. It must be stressed that a mark scheme is a wking document, in many cases further developed and expanded on the basis of students reactions to a particular paper. Assumptions about future mark schemes on the basis of one year s document should be avoided; whilst the guiding principles of assessment remain constant, details will change, depending on the content of a particular examination paper. Further copies of this mark scheme are available from aqa.g.uk. Copyright 016 AQA and its licenss. All rights reserved. AQA retains the copyright on all its publications. However, registered schools/colleges f AQA are permitted to copy material from this booklet f their own internal use, with the following imptant exception: AQA cannot give permission to schools/colleges to photocopy any material that is acknowledged to a third party even f internal use within the centre.

3 MARK SCHEME AS MATHEMATICS MS1B JUNE 016 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. 3 of 1

4 MARK SCHEME AS MATHEMATICS MS1B JUNE 016 General f MS1B GN1 GN GN3 GN4 GN5 GN6 There is no allowance f misreads (MR) miscopies (MC) unless specifically stated in a question. In general, a crect answer (to accuracy required) without wking sces full marks but an increct answer ( an answer not to required accuracy) sces no marks. In general, a crect answer (to accuracy required) without units sces full marks. When applying AWFW, a slightly inaccurate numerical answer that is subsequently rounded to fall within the accepted range cannot be awarded full marks. Where percentage equivalent answers are permitted in a question, then penalise by one accuracy mark at the first crect answer but only if no indication of percentage (eg %) is shown. In questions involving probabilities, do not award accuracy marks f answers given in the fm of a ratio odds such as 13/47 given as 13:47 13:34. GN7 Accept decimal answers, providing that they have at least two leading zeros, in the fm c 10 -n (eg as ). GN8 Where a candidate's response to a part of a question is simply to label the part (eg (d)(i)) with nothing else (ie no attempt at a solution), then this is still treated as a response and marked as 0 rather than NR. Also, deleted wk, if not replaced, should be marked and not treated as NR. Specific f MS1B 1. Question Unless clearly identified assume that the der is 'measure of average' then 'measure of spread'. Marks of B0 B0 in (b) have maj implications as to the marks available in (c) using the SC.. Question 3 Despite the answers to (a)(v) and (b) not being exact, the final accuracy marks are CAO (3 dp) and CAO (3 sf) respectively. 3. Question 4 Take care not to miss the awarding of f the SC in (b). 4. Question 5 In (a)(iii), the mark is either B B0. Take careful note of 1, and 3 after (c)(ii); the switching of answers is not unusual n are confused answers. Thus, f example, 0.99 to is seen in (i), and 0.84 (0.99 to 0.995) followed by (0.99 to 0.995) 6 is seen in (ii); the latter step is not ISW. 5. Question 6 In (a)(iii) and (a)(iv) the f the value of p can be inferred from a probability. Thus, f example, in (a)(iii), the probability of sces and to 0.06 sces. 6. Question 7 If * is not sced in (b)(i), then it can be sced in (b)(ii) f 166 to 167, by inference, from (OE) 7 8. In (b)(i), statements of the fm 400 is It lies within the CI in (a). BF0 Bdep0. In (b)(i), having previously calculated ( )CI as (309, 453), then statements of the fm The claim that the mean is ( )400 is valid as this lies within the CI. BF0 Bdep0. 4 of 1

5 MARK SCHEME AS MATHEMATICS MS1B JUNE (a) r = B3 AWRT ( ) = 0.95 to 0.97 (B) AWFW = 0.9 to 0.99 () AWFW Attempt at x x y Attempt at S xx S yy & S xy y & xy () & (all 5 attempted) & 986 (all 3 attempted) (b) Attempt at substitution into crect cresponding fmula (m1) f r r = (A1) AWRT 3 (Very/extremely) strong positive Bdep1 Dependent on 0.9 r 0.99 (linear) crelation 1 Statements must include the wds strong and positive together with crelation association relationship ; igne additional comments unless clearly contradicty The only acceptable qualifiers f strong are very extremely 3 The use of: highly/moderately/quite/fairly/relatively/reasonably/respectively strong Bdep0 4 The use of: high big good some moderate medium average Bdep0 between Height(s) and arm span(s) of men aged 1 to 40 1 As heights of men (aged 1 to 40) increase so do arm spans (OE) Bdep0 As heights/x increase so do arm spans/y (OE) Bdep0 B0 Total 5 Context; providing 1 < r < 1 Must contain at least the 4 emboldened wds 5 of 1

6 MARK SCHEME AS MATHEMATICS MS1B JUNE 016 (a)(i) Mode = 6 CAO 1 1 Mode is 6 (visits) because largest frequency/number of days is 13 (OE) Modes are 13 and 6 (OE) B0 (ii) x F: Median = 5 CAO IQR = 6 4 = CAO 1 Median is at CF = 7 to 8, UQ is at CF = 41 to 4 and LQ is at CF = 13 to 14 An answer of 5 /and with clearly shown increct method(s) B0 B0 B0 B0 (b) Mean CAO; accept nothing else (c) SC CAO; accept naming of only one of Range Standard deviation Variance these three measures; nothing else 1 Unless clearly identified as to which is the measure of average and which is the measure of spread, assume der is as requested in question (ie average then spread) so (eg simply range and mean B0 B0 and so Bdep0 Bdep0 in (c) but the possibility, using SC below, of in (c)) Accept arithmetic mean 3 Do not accept abbreviations such as Sd / Var symbols such as x / µ / w/ s/ s / s / s Mean = 5.6 Bdep CAO fx = 1408 Dependent on 1 st in (b) Mean = 5 to 6 (dep) AWFW Range = 45 CAO Bdep1 Dependent on nd in (b) Sd(n) = 5.6 Sd(n 1) = 5.31 AWRT ( ) fx = Var(n) = 7.7 Var(n 1) = 8. AWRT ( ) 3 1 Unless identified, by name, abbreviation symbol, as to which is the measure of average and which is the measure of spread, assume der is as requested in question (ie average then spread) so (eg simply 5.6 and 5.6 Bdep0 Bdep0) F the measure of average, the only valid answer is 5.6 (CAO) 5 to 6 (AWFW) 3 F the measure of spread, award Bdep1 f any seen (CAO/AWRT) value that cresponds to the one measure of spread named crectly in (b) so (eg Variance named in (b) and 5.31 and 8. seen in (c) Bdep1 but Variance named in (b) and 5.31 only seen in (c) Bdep0) 1 If, and only if, Bdep0 Bdep0 sced, then, igning labels: award f 5 to 6 (AWFW) and f one of 45, 5.6, 5.31, (AWRT) Total 8 6 of 1

7 MARK SCHEME AS MATHEMATICS MS1B JUNE 016 3(a) Accept the equivalent percentage answers with %-sign in parts (a)(i) to (a)(iv) but not in parts (a)(v) and (b) (see GN5) (i) P(A ) = 176/500 = 88/50 = 44/15 = 0.35 CAO; any one of four listed answers (1) (ii) P(A 66 B ) = 68/500 = 34/50 = 17/15 = CAO; any one of four listed answers (1) (iii) Numerat CAO P(A B 1 ) = = = A1 CAO () (iv) P(A 41 B ) = ( ) 500 ( ) Fraction CAO = 0.79 A1 AWRT (0.7931) () (v) P(B A 65 ) = 5 + ( 0) Numerat CAO ( ) Denominat CAO ( ) (Accept numerat and denominat each 500) (M) = 0.67 A1 CAO (3 dp only) (0.6667) (3) 9 (b) P(A B >0 ) = ( p 1 ) CAO; OE,, Seen anywhere, even in an increct expression P(A 66 B 1 ) = ( ) p CAO; OE ( 0.4 ) Seen anywhere, even in an increct expression SCs Providing 0 < p Prob = ( p1) ( p) ( p ) 1, p < 1 1 p Must be equivalent to product of two squared probabilities with no extra terms 4 6 m1 = A1 CAO (3 sf only) ( ) 5 1 Answer of (AWRT) even without wking mo A0 Answer of (AWRT) even without wking M0 mo A0 3 In each of the following (increct) expressions, ( + ) Igne der of terms and/ value of n providing n is an integer p3 p4 n and p3 p4 p5 n p3 p4 p5 n Use of diviss of 504, 503, 50, and 501 max of m1 but much me likely to be M0 mo 5 A final answer of does sce A1 Total 14 7 of 1

8 MARK SCHEME AS MATHEMATICS MS1B JUNE (a) b (gradient/slope) = 3(.00) to 3.01 B AWFW (3.0040) b (gradient/slope) =.95 to 3.05 () AWFW a (intercept) = 181 to 18 a (intercept) = 179 to 183 B () AWFW ( ) AWFW (b) SC (c) (d) Attempt at x Attempt at S xx & S xy x y & xy Attempt at substitution into crect cresponding fmula f b () (m1) & (all 4 attempted) ( y = 194) 3575 & (both attempted) (S yy = 351) b = 3(.00) to 3.01 a = 181 to 18 (A1 A1) AWFW ( x = 47.5 & y = 34) 4 1 Written fm of equation is not required Award 4 marks f y = (181 to 18) + (3 to 3.01)x f (181 to 18) + (3 to 3.01)x 3 Values of a and b interchanged and equation y = ax + b stated used in (c) max of 4 marks 4 Values of a and b interchanged and equation y = a + bx stated used in (c) 0 marks 5 Values of a and b are not identified (eg y = (181 to 18) + (3 to 3.01) (181 to 18) + (3 to 3.01)) 0 marks Answers as fractions ( ) 3 and 181 can sce at most m Some/all of marks can be sced in (b), (c) & (d), even if some/all of marks are lost in (a), but marks lost in (a) cannot be recouped by subsequent wking in (b) (c) but see Note 3 Each/every/one degree rise in water temperature results in increase per degree is (on average) b grams BF1 F on b providing.95 b 3.05 Accept, f example, 5 C and 5b grams f BF1 1 To sce any marks, an explanation must indicate change in x affecting change in y, not change in y affecting change in x Reference only to crelation B0 BF0 1 As x/temperature increases (by k) then y/mass increases by b (OE; value of b (.95 b 3.05) must be stated but context and/ units are not required) y(68) = 385 to 386 AWFW ( ) 1 1 Igne method used If linear interpolation from data is used, then y(68) = 384 B0 3 If a and b are interchanged, then y(68) = 1150 to 1450 B0 Residuals are relatively small/less than 10% Percentage residuals are small Residuals are small relative/compared to y-values/estimate in (c) Accept any value within 3% to 10% Estimate is (likely to be) (relatively) accurate Bdep1 Dependent on 1 Accept the use of: moderately/quite/fairly/reasonably accurate Residuals are small Residuals are small relative to x-values B0 Bdep0 3 Residuals show points are "close to the line" "not far from the line" (OE) B0 Bdep0 4 Conflicting reasons justifying both "accurate" and "not accurate" B0 Bdep0 Total 9 8 of 1

9 MARK SCHEME AS MATHEMATICS MS1B JUNE 016 5(a) Accept the equivalent percentage answers with %-sign (see GN5) (i) P(X < 1540) = P Z < 9.6 Standardising 1540 with 155 and 9.6 but allow ( ) (ii) (iii) (iv) (b) P(X > 1535) = = P(Z < 1.56(5)) = 0.94 to 0.94 A1 AWFW ( ) () P(Z > 1.04(17) = 1 P(Z < 1.04) Area change; can be implied by any final answer < 0.5 = = to 0.15 A1 AWFW ( ) () P(1515 < X < 1540) = P( 1.04 < Z < 1.56) = ( ) P(X 1500) = 1 one unity 100% = 0.79 to B AWFW (0.7913) () 10% (0.1) z = 1.8 (1) 7 CAO; accept nothing else but igne zeros after decimal point (eg 1.00) Igne additional wds providing that they are not contradicty (eg certain so = 1) AWRT (1.816) Seen; igne sign ( 1535 ( µ x x) ) ± 9.6 ( 1.8 to 1.9) = ± Standardising 1535 with µ / x and 9.6; allow (( x ) 1535) equating to ±(1.8 to 1.9) µ and µ = 15 to 153 A1 AWFW (15.697) Reduction = =.3 Adep1 CAO (1 dp only); dependent on A1 4 Note 1 Award max of A0 A0 if the signs are not consistent throughout (answer f µ is likely to be ) Parts (a) & (b) Total 11 9 of 1

10 MARK SCHEME AS MATHEMATICS MS1B JUNE Continued Parts (a) & (b) Total 11 (c) Note (i) Each pack contains a random sample of bottles Packs contain random samples of bottles Must contain at least the 3 emboldened wds and clearly infer that bottles in a pack are a random sample (1) This mark can be sced anywhere in (c) 1 Samples (of bottles) are random (OE) Each bottle is randomly selected (OE) B0 Packs are selected at random (OE) Each pack is selected at random (OE) B0 (Stated in question!) 3 Packs/bottles are selected independently (OE) Packs/bottles are nmally distributed (OE) B0 p = P(bottle > 505) = P Z > 3.5 = P(Z > 1) = P(Z < 1) = 0.84 AWRT ( ) Can be implied by a crect answer P(6 bottles > 505) = 6 p Providing 0 < p < 1 (ii) = 0.35 to A1 AWFW ( ) (3) 1 Calculation of ( ) = B0 Calculation of ( ) 6 B0 A0 ( ) V B =.04 CAO/AWRT (.04167) 6 6 Can be implied by what follows 3.5 Sd ( B ) = 1.43 CAO/AWRT (1.4887) P( B 505) P > = Z > Standardising 505 with and ( OE ) ; allow ( ) SC = P(Z > 6) = P(Z >.45) = 0.99 to A1 AWFW (0.9985) (3) 7 1 Do not give BOD f unclear/dubious/questionable identifications of (i) & (ii) If answers to (i) & (ii) are not identified, then mark as (i) followed by (ii) 3 If answers to (i) & (ii) are switched, then the only mark available is the f stating the necessary assumption 4 In (ii), award of B0 0/3 marks 1 Use of distribution of total in (ii): f Sd = (OE); f P(Z > ( )/(3.5 6)) P(Z > 6) P(Z >.45) A1 f 0.99 to (AWFW); award of B0 0/3 marks Total of 1

11 MARK SCHEME AS MATHEMATICS MS1B JUNE 016 6(a) Accept 3 dp rounding of probabilities from tables Accept the equivalent percentage answers with %-sign (see GN5) (i) 50 Crect expression 4 46 P(Red = 4) = ( 0.18 ) ( 0.8 ) Can be implied by a crect answer 4 Igne additional expressions = (ii) (iii) = 0.06 to 0.07 A1 AWFW (0.063) P(Yellow 10) = 0.88 AWRT (0.8801) 1 P(Blue Green) = 0.5 CAO; indicated as a value of p implied by any one of the probabilities opposite P(Blue Green 30) = = to 0.10 A1 AWFW (0.1013) Note (iv) = to 0.06 () 3 1 F calculation of individual terms no method: award B3 f to 0.10 (AWFW); B f to 0.06 (AWFW) Using p = 0. gives Using p = 0.8 gives Either CAO; indicated as a value of p implied by any one of the probabilities opposite (p 1 ) One of either pair MINUS MINUS (p ) One of matching pair from above (b) = 0.89 to 0.89 A1 AWFW (0.8913) 4 1 F calculation of individual terms no method: award B4 f 0.89 to 0.89 (AWFW); B3 f 0.9 to 0.9 (AWFW); B3 f 0.95 to 0.95 (AWFW) Answers involving (1 p ) (1 p 1 ) () A1 () () 3 Answers involving 1 (p 1 p ) even after (p 1 p ) (eg 1 ( ) = ) max of Mean = = 54 CAO Variance = = 44. to 44.3 AWFW (44.8) 1 Igne any subsequent wk following crect statement of variance (eg So Sd = 6.6 to 6.7 ) The statement 44. to 44.3 followed by Variance = 6.6 to 6.7 Variance = 1960 to 1961 B0 Total 1 11 of 1

12 MARK SCHEME AS MATHEMATICS MS1B JUNE (a) 99% (0.99) z =.57 to.58 AWFW (.5758) CI f µ is.57 to.58.3 to ±.70 to to.43 ( to 148.) Hence ± (58.50 to 60.50) (57.00 to 59.00, to ) M,1 ( 1 ee) Adep1 Igne any notation M0 if CI is not of the fm: ( z t) (( s s ) ) ± Allow any combination in last term NB: = CAO/AWFW; 1 dp only Dependent on award of M AWFW; 1 dp only 4 1 If award of M0 is followed by a numerically crect CI solutions Use same rules as above f t =.7 to.71 (AWFW) ± (6.00 to 63.00) (54.50 to 55.50, to ) (b) (i) 400 equates to ( ) 333 to 334 AWFW ( ) * * This mark may be sced in (b)(ii) * CI ( ): 381 ± (70 to 73) (308 to 311, 451 to 454) CAO/AWFW ( dp not required) AWFW ( dp not required) Clear crect comparison of (333 to 334) with CI in (a) {eg is within CI in (a)} (Must be clear that comparing like with like) Clear crect comparison of (400) with CI in (b)(i) {eg 400 is within CI in (b)(i)} BF1 Statement must include reference to 333 to 334 F on CI providing it is includes 333 to 334 Must have found an interval in (a) but quoting values f CI CLs is not required Statement must include reference to 400 F on CI providing it is includes 400 Must have found an interval in (b)(i) but quoting values f CI CLs is not required (ii) Agree with accept claim Claim is (likely to be) true/crect/right/valid/accurate Bdep1 OE; dependent on BF1 (3) 1 Statement must clearly indicate that (333 to ) is within the cresponding CI OE Statements of the fm It/this/mean/value/etc is within the (cresponding) CI BF0 3 Statements of the fm (333 to ) is within 99% of the data/values/pounds/euros BF0 4 Statements such as Claim is likely to be reasonable/suppted/crect/true/possible/valid Bdep1 providing BF1 00 equates to ( ) 166 to 167 AWFW ( ) * Award if 1 st not sced in (b)(i) * Can be implied by (OE) 7 8 Per cent < 00/ = seen with seen with 10 ( )00 seen with ( )5 ( )166 to 167 seen with ( )187.5(0) Requires both crect numbers (OE) from any of these pairs eg 17.5 & 5, 0. & 0.5, 8 & 10 7 & 5 only B0 Agree with accept claim Claim is (likely to be) true/crect/right/valid/accurate Bdep1 () 5 OE; dependent on Total 9 1 of 1