AS Statistics. SS1B Mark scheme June Version 1.0 Final Mark Scheme

AS Statistics SS1B Mark scheme 6380 June 016 Version 1.0 Final Mark Scheme

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.

MARK SCHEME AS STATISTCS SS1B 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 13

MARK SCHEME AS STATISTCS SS1B JUNE 016 General f SS1B 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 0.0031 as 3.1 10-3 ) 4 of 13

MARK SCHEME AS STATISTCS SS1B JUNE 016 1 (a) r = 0.608 B3 AWRT (0.60810) = 0.6 to 0.6 (B) AWFW = 0.5 to 0.7 (B1) AWFW Attempt at x x y Attempt at S xx S yy & S xy y & xy (M1) 0.5 41.0455 11.30 1.786 &.8983 (all 5 attempted) 0.0395 0.017 & 0.0158 (all 3 attempted) Attempt at substitution into crect cresponding fmula (m1) f r r = 0.608 (A1) AWRT 3 Some/moderate positive (linear) crelation/relationship/association Bdep1 1 Only accept phrase stated; igne additional comments unless contradicty Any mention of strong weak Bdep0 3 Use of: quite/fairly/relatively/reasonably moderate Bdep0 4 Use of: high big good low small po medium average Bdep0 Dependent on 0.5 r 0.7 Must qualify strength and state positive between trunk and tail lengths of male African B1 Context; providing 1 < r < 1 elephants 1 As trunk lengths of elephants increase so do tail lengths (OE) Bdep0 B1 As trunks/x increase so do lengths/y (OE) Bdep0 B0 Total 5 5 of 13

MARK SCHEME AS STATISTCS SS1B JUNE 016 (a) Mean = 48.1 B1 CAO ( x = 481) Var(n) = 3.31 Var(n 1) = 3.68 B AWRT (3.31) ( x = 3169.) CAO (3.680) Var(n) Var(n 1) = 3.1 to 3.9 (B1) 3 1 Value of variance stated as 1.81 to 1.93 and not evaluated B1 Value of variance standard deviation stated as 1.81 to 1.93 B0 480 to 48 10 3 If, and only if, B0 B0, then award M1 f seen attempt at ( ) AWFW Mean = ( their mean ) 0.354 M1 Can be implied by a crect answer = 16.9 to 17.1 A1 AWFW (17.074) Var(n) Var(n 1) = ( 3.1 to 3.9) 0.354 M1 Can be implied by a crect answer = 0.415 to 0.416 0.461 to 0.46 A1 AWFW (0.41505 0.46116) 4 1 New Sd = (1.8199 to 1.9183) 0.354 = (0.644 to 0.680) M0 New Var = [(1.8199 to 1.9183) 0.354] = (0.644 to 0.680) M1 3 If no marks sced, then seen multiplication of data values by 0.354 M1 only Total 7 6 of 13

MARK SCHEME AS STATISTCS SS1B JUNE 016 3 Accept the equivalent percentage answers with %-sign (see GN5) (a)(i) 960 955 Standardising 960 with 955 and 5; P(X < 960) = P Z < M1 5 allow (955 960) (ii) (iii) (iv) = P(Z < 1) = 0.841 A1 AWRT (0.84134) () P(X > 946) = P(Z > 1.8) = P(Z < 1.8) B1 CAO; igne sign P(X = 950) = 0 zero nought nothing nil B1 P(946 < X < 960) = P( 1.8 < Z < 1) = = 0.964 B1 AWRT (0.96407) () (1) CAO; accept nothing else but igne zeros after decimal point (eg 0.00) Igne additional wds providing they are not contradicty (eg impossible so = 0) (i) (1 (ii)) (i) + (ii) 1 0.841 (1 0.964) 0.841 + 0.964 1 M1 OE; providing 0 < answer < 1 Can be implied by a crect answer AWRT = 0.8 to 0.81 A1 AWFW (0.80541) () Part (a) Total 7 7 of 13

MARK SCHEME AS STATISTCS SS1B JUNE 016 3 Continued Part (a) Total 7 (i) P(10 bottles > 946) = [(a)(ii)] 10 (0.964) 10 M1 Providing 0 < (a)(ii) < 1 (ii) ( ) 5 5 = 0.693 to 0.694 A1 AWFW (0.69356) () V X =.5 CAO 10 10 OR B1 Can be implied by what follows 5 10 5 Sd( B ) = 1.58 CAO/AWRT (1.58114) 10 95.5 955 P( X < 95.5) = P Z <.5 M1 Standardising 95.5 with 955 and.5 ( OE ) ; allow (955 95.5) = P(Z <.5) = 1 P(Z < 1.58) m1 Can be implied only providing M1 sced and answer < 0.5 = 0.056 to 0.058 A1 AWFW (0.05693) (4) 6 1 In (ii), award of B0 0/4 marks Use of distribution of total in (ii): B1 f Sd = 5 10 (OE); M1 f P(Z < (955 9550)/(5 10)) P(Z <.5) ; m1 f 1 P(Z < 1.58) A1 f 0.056 to 0.058 (AWFW) Total 13 8 of 13

MARK SCHEME AS STATISTCS SS1B JUNE 016 4(a) Accept the equivalent percentage answers with %-sign (see GN5) (i) P(CW) = 110/400 = 55/00 = 11/40 = 0.75 B1 CAO; either of four listed answers (1) (ii) P(SW H) = 56/400 = 8/00 = 14/100 = 7/50 = 0.14 B1 CAO; any one of five listed answers (1) (iii) 30 + 4 + 4 + 6 104 M1 Numerat CAO P(B (H C)) = = = 400 400 104/400 = 5/00 = 6/100 = 13/50 = 0.6 A1 CAO; any one of five listed answers () (iv) P(SW C) = 45 400 45 = M1 Fraction CAO 10 400 10 45/10 = 15/40 = 9/4 = 3/8 = 0.375 A1 CAO; any one of four listed answers () (v) P((E C) W) = P(W C) = ( + + + ) ( + ) 3 17 1 14 400 84 = 150 110 400 60 M1 M1 Numerat CAO Denominat CAO 4 1 = 130 65 (M) 4/130 = 1/65 = 0.33 A1 CAO/AWRT (0.3308) (3) 9 45 + 5 70 400 400 ( ) p B1 1 7 CAO; OE, 0.175 40 Seen anywhere, even in an increct expression P(B H) = 30 + 4 54 400 400 ( ) p B1 7 CAO; OE, 0.135 00 Seen anywhere, even in an increct expression SCs 1 Prob = ( p ) ( p ) M1 Providing 0 < p1, p < 1 p p p p M0 ( ) 1 3 4 4 6 m1 = 0.00334 to 0.00335 A1 AWFW (0.0033488) 5 1 Answer of 0.00056 (AWRT) without wking B1 B1 M1 m0 A0 Answer of 0.036 to 0.0363 (AWFW) without wking B1 B1 M0 m0 A0 3 In each of the following (increct) expressions, ( + ) and igne the value of n: 70 69 54 53 70 69 54 53 n B1 B1 and n B1 400 400 400 400 400 399 398 397 Total 14 9 of 13

MARK SCHEME AS STATISTCS SS1B JUNE 016 5(a) (i) b (gradient/slope) = 0.37 to 0.373 B AWFW (0.3735) b (gradient/slope) = 0.3 to 0.4 (B1) AWFW a (intercept) = 6.94 to 6.95 a (intercept) = 6 to 9 B (B1) AWFW (6.94648) AWFW (ii) SC (iii) Attempt at x Attempt at S xx & S xy x y & xy Attempt at substitution into crect cresponding fmula f b (M1) (m1) 34 89.70 04 & 5573.05 (all 4 attempted) ( y = 3493.64) 174.70 & 65.05 (both attempted) (S yy = 5.64) b = 0.37 to 0.373 a = 6.94 to 6.95 (A1 A1) AWFW ( x = 7 & y = 17) 4 1 Written fm of equation is not required Award 4 marks f y = (6.94 to 6.95) + (0.37 to 0.373)x f (6.94 to 6.95) + (0.37 to 0.373)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 are not identified, then B0 B0 6 Some/all of marks can be sced in (a)(ii), (a)(iii), & (c)(i), even if some/all of marks are lost in (a)(i), but marks lost in (a)(i) cannot be recouped by subsequent wking in (a)(ii), (a)(iii), (c)(i) but see Note 3 Each/every/one degree ( C) rise in ground temperature results in B1 increase per degree ( C) is (on average) b vibrations per second BF1 F on b providing 0.3 b 0.4 1 To sce any marks, an explanation must indicate change in x affecting change in y, not change in y affecting change in x Accept, f example, 10 C and 10b vibrations 3 Reference only to crelation B0 BF0 1 As x/temperature increases (by c) then y/vibrations increases by b (OE; value of b (0.3 b 0.4) must be stated but context and/ units are not required) B1 Given: When temperature/x < 15 C = 0 C value of y = 0 Equation: When temperature/x = 0 C vibrations/value of y = 6 to 9 B1 BF1 Must be stated clearly AWFW F on a providing 6 a 9 1 B1 is f a clear statement of infmation given in the question in terms of temperature/x and y BF1 is f a clear statement of the value of vibrations/y shown by the equation when temperature/x = 0 Part(a) Total 8 10 of 13

MARK SCHEME AS STATISTCS SS1B JUNE 016 5 Continued Part (a) Total 8 y(3) = 15.4 to 15.6 B1 AWFW (15.51059) 1 Note 1 Igne any method shown (c) (i) res(8.6) = 17.0 a b 8.6 = 0.55 to 0.65 = 0.5 to 0.7 B (B1) AWFW; do not igne sign ( 0.59576) AWFW; igne sign Note 1 If, and only if, B0, then attempted use of ±(17.0 a b 8.6) M1 providing 0.3 b 0.4 and 6 a 9 (ii) Value will be/is always: 0 zero nought nothing nil B1 1 Total 1 CAO; accept nothing else, but igne zeros after decimal point (eg 0.00) Igne any explanation 11 of 13

MARK SCHEME AS STATISTCS SS1B JUNE 016 6 Accept 3 dp rounding of probabilities from tables Accept the equivalent percentage answers with %-sign (see GN5) (a) Use of B(30, 0.8) B(30, 0.45) M1 Indicated by an expression by any one crect probability in (a) 30 0.8 1 0.8 3 3 30 3 P(Vans = 3) = ( ) ( ) = 4060 0.0195 0.000140597 M1 Crect expression Can be implied by a crect answer Igne additional expressions Note (c) = 0.01 to 0.013 A1 AWFW (0.0153) 3 P(Cars <15) = 0.644 to 0.645 0.769 M1 AWFW AWRT = 0.644 to 0.645 A1 AWFW (0.6448) 1 F calculation of individual terms no method: award B f 0.644 to 0.645 (AWFW); B1 f 0.769 to 0.77 (AWFW) P(Van HGV) = 0.4 B1 CAO; stated identified from below P(Vans HGVs 10) = 1 0.1763 M1 = 0.83 to 0.84 A1 AWFW (0.837) Note (d) Note = 1 0.915 0.708 to 0.709 (M1) 3 1 F calculation of individual terms no method: award B3 f 0.83 to 0.84 (AWFW); B f 0.708 to 0.709 (AWFW) P(0 < M 5) = P(M 5) P((M 0); but p = 0.85 is not tabled so must use calculat P(M > 0) = P(M < 10) and P(M 5) = P(M 5) so P(0 < M 5) = P(5 M < 10) = P(M 9) P(M 4); and p = 0.15 is tabled may use calculat Using p = 0.15 gives Using p = 0.85 gives B1 Either CAO Stated identified from below 0.9903 0.9971 (p 1 ) 0.4755 0.894 M1 MINUS 0.545 0.7106 (p ) 0.0097 0.009 M1 = 0.464 to 0.466 A1 AWFW (0.4658) 4 1 F calculation of individual terms no method: award B4 f 0.464 to 0.466 (AWFW); B3 f 0.47 to 0.473 (AWFW); B3 f 0.79 to 0.81 (AWFW); B3 f 0.86 to 0.87 (AWFW) (1 p ) (1 p 1 ) (B1) M1 M1 A1 (B1) M1 M1 (B1) M1 3 Answer of 1 0.4658 = 0.534 to 0.536 B1 M1 M1 A0 B3 Total 1 1 of 13

MARK SCHEME AS STATISTCS SS1B JUNE 016 7(a) (i) x = 140 30 = 408 B1 CAO s = 397 9 = 137 s = 11.7 σ = 397 30 = 13 σ = 11.5 B1 AWRT (136.9655 & 11.7033) Igne any notation AWRT (13.4 & 11.5065) 98% (0.98) z =.3 to.33 B1 AWFW (.363) CI f µ is.3 to.33 Igne any notation.05to.06 ( 137 11.7 13 11.5) 408 ± M,1.45to.47 M0 if CI is not of the fm: 30 9 ( 1 ee).14 to.16 C ± ( zt) ( D 30 9 ) ; allow any combination in last term Hence (z) 408 ± 5 CAO/AWRT (4.95 to 5.8) Adep1 Dependent on award of M Note (ii) (403, 413) AWRT 6 1 If award of M0 is followed by a numerically crect CI possibly solutions 0.5% above 400 of 400 40 B1 CAO Clear crect comparison of 40 with CI {eg 40 < CI 40 < LCL} BF1 Statement must include reference to 40 F on CI providing it is above 40 Must have found an interval in (a)(i) but quoting values f CI CLs is not required (c) Sample meets requirement Yes Bdep1 Dependent on BF1 3 1 Statement must clearly indicate that 40 is below the CI OE Statements of the fm 40 is within 98% of the data/values/loaves/weights/grams B1 BF0 Bdep0 3 Comparison of 40 with 408 comparison of 40 with CI which includes 40 B1 BF0 Bdep0 4 Use of 40 (5%) 600 (50%) B0 BF0 Bdep0 Number < 388 = 4 which is greater than 3 Percent < 388 = 13 which is greater than 10 B1 Requires 4 & 3 Requires 13(AWRT) & 10 Sample does not meet requirement BF1 Dependent on B1 CLT used in part (a)(i) first part construction of CI B1 1 First question B0 Igne additional wds providing they are not contradicty 1 13 of 13