General Certificate of Education January 2007 Advanced Subsidiary Examination
MATHEMATICS Unit Statistics 1B
MS/SS1B
STATISTICS Unit Statistics 1B Tuesday 23 January 2007
1.30 pm to 3.00 pm
For this paper you must have: * an 8-page answer book * the blue AQA booklet of formulae and statistical tables * an insert for use in Question 7 (enclosed). You may use a graphics calculator.
Time allowed: 1 hour 30 minutes Instructions Use blue or black ink or ball-point pen. Pencil should only be used for drawing. Write the information required on the front of your answer book. The Examining Body for this paper is AQA. The Paper Reference is MS/SS1B. Answer all questions. Show all necessary working; otherwise marks for method may be lost. The final answer to questions requiring the use of tables or calculators should normally be given to three significant figures. Fill in the boxes at the top of the insert. * *
* * *
*
Information The maximum mark for this paper is 75. The marks for questions are shown in brackets. Unit Statistics 1B has a written paper only. * * *
Advice Unless stated otherwise, you may quote formulae, without proof, from the booklet.
*
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MS/SS1B
2
Answer all questions.
1 The times, in seconds, taken by 20 people to solve a simple numerical puzzle were 17 41
19 42
22 43
26 47
28 31 50 51
34 53
36 55
38 57
39 58
(a) Calculate the mean and the standard deviation of these times.
(3 marks)
(b) In fact, 23 people solved the puzzle. However, 3 of them failed to solve it within the allotted time of 60 seconds. Calculate the median and the interquartile range of the times taken by all 23 people. (4 marks) (c) For the times taken by all 23 people, explain why: (i) the mode is not an appropriate numerical measure; (ii) the range is not an appropriate numerical measure.
(2 marks)
2 A hotel has 50 single rooms, 16 of which are on the ground floor. The hotel offers guests a choice of a full English breakfast, a continental breakfast or no breakfast. The probabilities of these choices being made are 0.45, 0.25 and 0.30 respectively. It may be assumed that the choice of breakfast is independent from guest to guest. (a) On a particular morning there are 16 guests, each occupying a single room on the ground floor. Calculate the probability that exactly 5 of these guests require a full English breakfast. (3 marks) (b) On a particular morning when there are 50 guests, each occupying a single room, determine the probability that: (i) at most 12 of these guests require a continental breakfast;
(2 marks)
(ii) more than 10 but fewer than 20 of these guests require no breakfast.
(3 marks)
(c) When there are 40 guests, each occupying a single room, calculate the mean and the standard deviation for the number of these guests requiring breakfast. (4 marks)
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3 Estimate, without undertaking any calculations, the value of the product moment correlation coefficient between the variables x and y in each of the three scatter diagrams. (a)
(b) y
y
x
x
(c) y
x
(5 marks)
Turn over for the next question
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4 A very popular play has been performed at a London theatre on each of 6 evenings per week for about a year. Over the past 13 weeks (78 performances), records have been kept of the proceeds from the sales of programmes at each performance. An analysis of these records has found that the mean was £184 and the standard deviation was £32. (a) Assuming that the 78 performances may be considered to be a random sample, construct a 90% confidence interval for the mean proceeds from the sales of programmes at an evening performance of this play. (4 marks) (b) Comment on the likely validity of the assumption in part (a) when constructing a confidence interval for the mean proceeds from the sales of programmes at an evening performance of: (i) this particular play; (ii) any play.
(3 marks)
5 Dafydd, Eli and Fabio are members of an amateur cycling club that holds a time trial each Sunday during the summer. The independent probabilities that Dafydd, Eli and Fabio take part in any one of these trials are 0.6, 0.7 and 0.8 respectively. Find the probability that, on a particular Sunday during the summer: (a) none of the three cyclists takes part;
(2 marks)
(b) Fabio is the only one of the three cyclists to take part;
(2 marks)
(c) exactly one of the three cyclists takes part;
(3 marks)
(d) either one or two of the three cyclists take part.
(3 marks)
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6 When Monica walks to work from home, she uses either route A or route B. (a) Her journey time, X minutes, by route A may be assumed to be normally distributed with a mean of 37 and a standard deviation of 8. Determine: (i) PðX < 45Þ ;
(3 marks)
(ii) Pð30 < X < 45Þ .
(3 marks)
(b) Her journey time, Y minutes, by route B may be assumed to be normally distributed with a mean of 40 and a standard deviation of s . Given that PðY > 45Þ ¼ 0:12, calculate the value of s .
(4 marks)
(c) If Monica leaves home at 8.15 am to walk to work hoping to arrive by 9.00 am, state, with a reason, which route she should take. (2 marks) (d) When Monica travels to work from home by car, her journey time, W minutes, has a mean of 18 and a standard deviation of 12. Estimate the probability that, for a random sample of 36 journeys to work from home by car, Monica’s mean time is more than 20 minutes. (4 marks) (e) Indicate where, if anywhere, in this question you needed to make use of the Central Limit Theorem. (1 mark)
Turn over for the next question
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7 [Figure 1, printed on the insert, is provided for use in this question.] Stan is a retired academic who supplements his pension by mowing lawns for customers who live nearby. As part of a review of his charges for this work, he measures the areas, x m2 , of a random sample of eight of his customers’ lawns and notes the times, y minutes, that it takes him to mow these lawns. His results are shown in the table. Customer
A
B
C
D
E
F
G
H
x
360
140
860
600
1180
540
260
480
y
50
25
135
70
140
90
55
70
(a) On Figure 1, plot a scatter diagram of these data.
(2 marks)
(b) Calculate the equation of the least squares regression line of y on x . Draw your line on Figure 1. (6 marks) (c) Calculate the value of the residual for Customer H and indicate how your value is confirmed by your scatter diagram. (3 marks) (d) Given that Stan charges £12 per hour, estimate the charge for mowing a customer’s lawn that has an area of 560 m2 . (4 marks)
END OF QUESTIONS
P89732/Jan07/MS/SS1B
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Other Names
Centre Number
Candidate Number
Candidate Signature
General Certificate of Education January 2007 Advanced Subsidiary Examination
MATHEMATICS Unit Statistics 1B
MS/SS1B
STATISTICS Unit Statistics 1B
Insert Insert for use in Question 7. Fill in the boxes at the top of this page. Fasten this insert securely to your answer book.
Turn over for Figure 1
P89732/Jan07/MS/SS1B 6/6/6/
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Figure 1 (for use in Question 7) Lawn Areas and Mowing Times y
~
160 –
140 –
120 –
Time (minutes)
100 –
80 –
60 –
40 –
20 –
–
–
–
–
–
200
400
600 Area (m2 )
800
1000
1200
Copyright Ó 2007 AQA and its licensors. All rights reserved.
P89732/Jan07/MS/SS1B
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–
–
0– 0
x
Version 1.0 0107
abc General Certificate of Education
Mathematics 6360 Statistics 6380 MS/SS1B Statistics 1B
Mark Scheme 2007 examination - January series
MS/SS1B - AQA GCE Mark Scheme 2007 January series
Mark schemes are prepared by the Principal Examiner and considered, together with the relevant questions, by a panel of subject teachers. This mark scheme includes any amendments made at the standardisation meeting attended by all examiners and is the scheme which was used by them in this examination. The standardisation meeting ensures that the mark scheme covers the candidates’ responses to questions and that every examiner understands and applies it in the same correct way. As preparation for the standardisation meeting each examiner analyses a number of candidates’ scripts: alternative answers not already covered by the mark scheme are discussed at the meeting and legislated for. If, after this meeting, examiners encounter unusual answers which have not been discussed at the meeting they are required to refer these to the Principal Examiner. It must be stressed that a mark scheme is a working document, in many cases further developed and expanded on the basis of candidates’ 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 to download from the AQA Website: www.aqa.org.uk Copyright © 2007 AQA and its licensors. All rights reserved. COPYRIGHT AQA retains the copyright on all its publications. However, registered centres for AQA are permitted to copy material from this booklet for their own internal use, with the following important exception: AQA cannot give permission to centres to photocopy any material that is acknowledged to a third party even for internal use within the centre. Set and published by the Assessment and Qualifications Alliance.
The Assessment and Qualifications Alliance (AQA) is a company limited by guarantee registered in England and Wales (company number 3644723) and a registered charity (registered charity number 1073334). Registered address: AQA, Devas Street, Manchester M15 6EX Dr Michael Cresswell Director General
MS/SS1B - AQA GCE Mark Scheme 2007 January series
Key to mark scheme and abbreviations used in marking M m or dM A B E or ft or F CAO CSO AWFW AWRT ACF AG SC OE A2,1 –x EE NMS PI SCA
mark is for method mark is dependent on one or more M marks and is for method mark is dependent on M or m marks and is for accuracy mark is independent of M or m marks and is for method and accuracy mark is for explanation follow through from previous incorrect result correct answer only correct solution only anything which falls within anything which rounds to any correct form answer given special case or equivalent 2 or 1 (or 0) accuracy marks deduct x marks for each error no method shown possibly implied substantially correct approach
MC MR RA FW ISW FIW BOD WR FB NOS G C Sf Dp
mis-copy mis-read required accuracy further work ignore subsequent work from incorrect work given benefit of doubt work replaced by candidate formulae book not on scheme graph candidate significant figure(s) 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 for any marks to be awarded. However, there are situations in some units where part marks would be appropriate, particularly when similar techniques are involved. Your Principal Examiner will alert you to these and details will be provided on the mark scheme. Where the answer can be reasonably obtained without showing working and it is very unlikely that the correct answer can be obtained by using an incorrect method, we must award full marks. However, the obvious penalty to candidates showing no working is that incorrect answers, however close, earn no marks. Where a question asks the candidate to state or write down a result, no method need be shown for full marks. Where the permitted calculator has functions which reasonably allow the solution of the question directly, the correct answer without working 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 correct method for any marks to be awarded.
Jan 07
3
MS/SS1B - AQA GCE Mark Scheme 2007 January series
MS/SS1B Q Solution 1(a) Mean ( x ) = 39.3 to 39.4 Standard Deviation ( sn , sn −1 ) = 12.3 to 12.7 If neither correct but working shown, then ∑x Mean ( x ) = 20 (b) Median = 42
Median = 41.5 or 39 or 40 Interquartile Range = 55 – 31 = 24 Interquartile Range = 21 to 27 (c)(i) Mode: eg Does not exist If exists, must be > 60 or 58 All / too many different values Sparse data (ii) Range: eg Maximum value is unknown / > 60 or 58
Marks B1
Total
Comments AWFW (39.35)
B2
3
AWFW (12.358 or 12.679)
∑ x = 787 ∑ x (M1)
Used
B2
CAO
(B1)
CAO
B2
4
(B1)
4
= 34023
CAO; allow B1 for identification of 31 and 55; B0 if method shown is incorrect AWFW
B1
B1 Total
2
OE
2 9
OE; accept ‘slowest’ but not ‘smallest’
MS/SS1B - AQA GCE Mark Scheme 2007 January series
MS/SS1B (cont) Q Solution 2(a) Use of binomial in (a), (b) or (c)
Marks M1
⎛ 16 ⎞ 5 11 P(E = 5) = ⎜ ⎟ ( p ) (1 − p ) ⎝5⎠ = 0.112
Total
Can be implied Allow p = 0.45, 0.25, 0.30 or
M1 A1
(b)(i) B(50, 0.25)
3
B1
P(C ≤ 12) = 0.511
B1
Comments
AWRT (0.1123) Used; can be implied
2
AWRT (0.5110)
(ii) P(10 < B′ < 20) = 0.9152 or 0.9522
M1
Allow 3 dp accuracy
minus 0.0789 or 0.1390
M1
Allow 3 dp accuracy
= 0.836
A1
or B(50, 0.30) expressions stated for at least 3 terms within 10 ≤ B′ ≤ 20 Answer = 0.836 (c) n = 40, p = 0.7
1 3
3
AWRT (0.8363)
(M1)
Or implied by a correct answer
(A2)
AWRT
B1
Both used; can be implied
Mean µ = np = 28
B1
CAO;
Variance σ 2 = np(1 – p) = 8.4
M1
Use of np(1 – p) even if SD
Standard deviation = 8.4 or = 2.89 to 2.9
A1 Total
4 12
5
on p only
CAO; AWFW
MS/SS1B - AQA GCE Mark Scheme 2007 January series
MS/SS1B (cont) Solution Q 3(a) 0.5 ≤ Value ≤ 0.95 Positive value < 1 (and > 0)
Marks B2 (B1)
(b) –0.2 ≤ Value ≤ +0.2
B1
(c) –0.95 ≤ Value ≤ –0.5 Negative value > –1 (and < 0)
B2 (B1) Total
4(a) 90% ⇒ z = 1.64 to 1.65 or 90% ⇒ t = 1.66 to 1.67 (Knowledge of the t–distribution is not required in this unit)
CI for µ is
x ± ( z or t ) ×
( sn−1 or sn ) n
Thus 184 ± (1.6449 or 1.6649 ) ×
(
( 32 or 32.2 ) 78 or 77
)
Total
Comments Value is actually 0.8
Value is actually 0.0 5
Value is actually –0.7
5
B1
AWFW (1.6449)
(B1)
AWFW (1.6649)
M1
Used; must have
A1
n with n > 1
on z or t only
Hence 184 ± (5.94 to 6.13) or £184 ± £6 or (£178, £190) (b)(i) Likely to be valid
A1
4
B1
(ii) Different plays have different: programme prices, sales, marketing, etc B1 theatre or audience sizes, etc popularity, artists, etc so ↑Dep↑ Unlikely to be valid B1 Total
6
AWRT; ignore units Accept ‘valid’ or equivalent
3 7
Accept ‘not valid’ or equivalent
MS/SS1B - AQA GCE Mark Scheme 2007 January series
MS/SS1B (cont) Q Solution 5(a) P(D′ ∩ E′ ∩ F′ ) = 0.4 × 0.3 × 0.2
= 0.024 (b) P(D′ ∩ E′ ∩ F) = 0.4 × 0.3 × 0.8
Marks M1
Total
A1
2
M1
= 0.096
A1
Comments At least 1 probability correct
CAO; OE At least 2 probabilities correct
2
CAO; OE
(c) P(One) = (b) + P(D ∩ E′ ∩ F′ ) + P(D′ ∩ E ∩ F′ )
M1
Use of 3 possibilities; ignore multipliers
=(b) + (0.6 × 0.3 × 0.2) + (0.4 × 0.7 × 0.2)
M1
At least 1 new term correct
= 0.096 + 0.036 + 0.056 = 0.188
A1
(d) P(One or two) = (c) + (3 terms each of 3 probabilities) or = 1 – (a) – (1 term of 3 probabilities)
= 0.188 + (0.6 × 0.7 × 0.2) + (0.6 × 0.3 × 0.8) + (0.4 × 0.7 × 0.8) = 0.188 + 0.084 + 0.144 + 0.224 or = 1 – 0.024 – (0.6 × 0.7 × 0.8) = 1 – 0.024 – 0.336 = 0.64
3
CAO; OE
M1
(c) + P(Two) Used; OE; ignore multipliers 1 – (a) – P(Three)
M1
At least 1 new term correct
A1 Total
7
3 10
CAO; OE
MS/SS1B - AQA GCE Mark Scheme 2007 January series
MS/SS1B (cont) Q Solution 6(a)(i) 45 − 37 ⎞ ⎛ P(X < 45) = P ⎜ Z < ⎟ 8 ⎠ ⎝ = P(Z < 1)
Marks
Total
Standardising (44.5, 45 or 45.5) with 37 and ( 8 , 8 or 82 ) and/or (37 − x) CAO; ignore sign
M1 A1
= 0.841
A1
(ii) P(30 < X < 45) = (i) – P(X < 30)
Comments
3
AWRT (0.84134)
M1
Used; OE
= (i) – [1 – (0.80785 to 0.81057)]
m1
Area change
= 0.648 to 0.652
A1
= (i) – P(Z < –0.875)
(b) 0.12 ⇒ z = 1.17 to 1.18
3
AWFW (0.65056)
B1
AWFW; ignore sign (1.1750)
M1
Standardising 45 with 40 and σ
= 1.175
m1
Equating z-term to z-value but not using 0.12, 0.88 or 1 − z
σ = 4.23 to 4.28
A1
z=
45 − 40
σ
(c) Route A: P(X > 45) = 1 – (a)(i) Route B: P(Y > 45) = 0.12 so Monica should use Route B (smaller prob) (d) Mean of W = 18
122 =4 36 20 − 18 ⎞ ⎛ P (W > 20 ) = P ⎜ Z > ⎟ 2 ⎠ ⎝
Variance of W =
= P(Z > 1) = 0.159
B1
AWFW OE; must use 45
↑Dep↑ B1
2
on (a)(i); allow Route Y
B1
CAO; can be implied by use in standardising
B1
CAO; OE
M1
Standardising 20 with 18 and 2 and/or (18 – 20)
A1
(e) In part (d)
4
B1 Total
8
4
AWRT (0.15866);
1 17
CAO; OE
on (a)(i) if used
MS/SS1B - AQA GCE Mark Scheme 2007 January series
Question 7 (a) and (b)
Lawn Areas and Mowing Times 180
160
140
Time (y minutes)
120
100
80
60
40
20
0 0
200
400
600
800
1000
1200
Area (x m2)
(a) 8 or 7 points plotted accurately (6 or 5 points plotted accurately
B2 B1)
(b) Line plotted accurately (Evidence of correct method for ≥ 2 points
B2 M1) (Graph = 4)
9
1400
MS/SS1B - AQA GCE Mark Scheme 2007 January series
MS/SS1B (cont) Q Solution 7(a) 8 or 7 points plotted accurately (6 or 5 points plotted accurately)
Marks B2 (B1)
(b) Gradient, b = 0.114 to 0.115 (b = 0.11 to 0.12)
Intercept, a = 15.9 to 16.1 (a = 13 to 19) Attempt at
∑x, ∑x , ∑y 2
and
Total 2
Comments
B2 (B1)
AWFW (0.11469)
B2 (B1)
AWFW (16.00824)
∑ xy
4420, 3230800, 635 and 441300
or Attempt at S xx and S xy
(M1)
Attempt at correct formula for b b = 0.114 to 0.115 a = 15.9 to 16.1
(m1) (A1) (A1)
788750 and 90462.5 AWFW AWFW
Accept a and b interchanged only if then identified correctly later in question
(c)
Line plotted accurately (Evidence of correct method for ≥ 2 points)
B2 (M1)
ResH = yH – YH = 70 – (a + b × 480)
M1
Used; or implied by correct answer; allow for YH – yH shown
A1
AWFW (–1.06)
= –1.5 to –0.5 Point H is (almost) on / just below the line (d) Y = a + b × 560 or reading from scatter diagram
= 79 to 81 Cost = Y ×
12 Y or 60 5
= £15.8 to £16.2
B1
6
3
At least from x = 200 to 1000
Accept near / close / just above or equivalent
M1
Used
A1
AWFW (80.2)
M1
Used
A1 Total TOTAL
10
4 15 75
AWFW; ignore units
(£16.05)
AQA January Examinations 2007 Scaled Mark Component Grade Boundaries (GCE Specifications) Generally, scaled mark boundaries are the same as raw mark boundaries as there is no scaling of marks. However, they may be different if a unit of assessment consists of more than one component.
Component Code MM2B MPC1 MPC2 MPC3 MPC4 MS/SS1A/W MS/SS1A/C MS1B MS2A/W MS2A/C MS2B
Component Title MATHEMATICS UNIT MM2B MATHEMATICS UNIT MPC1 MATHEMATICS UNIT MPC2 MATHEMATICS UNIT MPC3 MATHEMATICS UNIT MPC4 STATISTICS 1A - WRITTEN STATISTICS 1A - COURSEWORK MATHEMATICS UNIT MS1B STATISTICS 2A - WRITTEN STATISTICS 2A - COURSEWORK MATHEMATICS UNIT MS2B
Maximum Scaled Mark 75 75 75 75 75 75 25 75 75 25 75
A 62 59 59 60 62 61 20 61 60 20 61
Scaled Mark Grade Boundaries B C D 54 46 39 51 43 35 52 45 38 53 46 39 54 47 40 54 46 39 18 15 13 53 46 39 53 45 39 18 15 13 54 47 40
E 32 28 31 32 33 31 10 32 33 10 33
MED1 MED2 MED3 MED4 MED5 MED6
MEDIA STUDIES UNIT 1 MEDIA STUDIES UNIT 2 MEDIA STUDIES UNIT 3 MEDIA STUDIES UNIT 4 MEDIA STUDIES UNIT 5 MEDIA STUDIES UNIT 6
60 60 100 60 60 60
41 41 72 45 50 40
35 35 64 40 41 36
29 30 56 36 32 32
23 25 48 32 23 28
18 20 40 28 15 24
MUS1
MUSIC UNIT 1
100
71
61
51
41
31
PHYSICS A UNIT 1 PHYSICS A UNIT 2 PHYSICS A UNIT 3 - WRITTEN
50 50 50
40 34 37
35 30 33
30 26 30
26 23 27
22 20 24
PA01 PA02 PHA3/W
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