General Certificate of Education June 2008 Advanced Subsidiary Examination
MATHEMATICS Unit Statistics 1B
MS/SS1B
STATISTICS Unit Statistics 1B Wednesday 21 May 2008 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 3 (enclosed). You may use a graphics calculator.
Time allowed: 1 hour 30 minutes Instructions Use black ink or black 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.
*
P5647/Jun08/MS/SS1B 6/6/6/
MS/SS1B
2
Answer all questions.
1 The table shows the times taken, y minutes, for a wood glue to dry at different air temperatures, x °C. x
10
12
15
18
20
22
25
28
30
y
42.9
40.6
38.5
35.4
33.0
30.7
28.0
25.3
22.6
(a) Calculate the equation of the least squares regression line y ¼ a þ bx .
(4 marks)
(b) Estimate the time taken for the glue to dry when the air temperature is 21 °C. (2 marks)
2 A basket in a stationery store contains a total of 400 marker and highlighter pens. Of the marker pens, some are permanent and the rest are non-permanent. The colours and types of pen are shown in the table. Colour Type
Black
Blue
Red
Green
Permanent marker
44
66
32
18
Non-permanent marker
36
53
21
10
0
41
37
42
Highlighter
A pen is selected at random from the basket. Calculate the probability that it is: (a) a blue pen;
(1 mark)
(b) a marker pen;
(2 marks)
(c) a blue pen or a marker pen;
(2 marks)
(d) a green pen, given that it is a highlighter pen;
(2 marks)
(e) a non-permanent marker pen, given that it is a red pen.
(2 marks)
P5647/Jun08/MS/SS1B
3
3 [Figure 1, printed on the insert, is provided for use in this question.] The table shows, for each of a sample of 12 handmade decorative ceramic plaques, the length, x millimetres, and the width, y millimetres. Plaque
x
y
A
232
109
B
235
112
C
236
114
D
234
118
E
230
117
F
230
113
G
246
121
H
240
125
I
244
128
J
241
122
K
246
126
L
245
123
(a) Calculate the value of the product moment correlation coefficient between x and y. (3 marks) (b) Interpret your value in the context of this question.
(2 marks)
(c) On Figure 1, complete the scatter diagram for these data.
(3 marks)
(d) In fact, the 6 plaques A, B, ..., F are from a different source to the 6 plaques G, H, ..., L. With reference to your scatter diagram, but without further calculations, estimate the value of the product moment correlation coefficient between x and y for each source of plaque. (2 marks)
P5647/Jun08/MS/SS1B
s
Turn over
4
4 The runs scored by a cricketer in 11 innings during the 2006 season were as follows. 47
63
0
28
40
51
a
77
0
13
35
The exact value of a was unknown but it was greater than 100. (a) Calculate the median and the interquartile range of these 11 values.
(4 marks)
(b) Give a reason why, for these 11 values: (i) the mode is not an appropriate measure of average; (ii) the range is not an appropriate measure of spread.
(2 marks)
5 When a particular make of tennis ball is dropped from a vertical distance of 250 cm on to concrete, the height, X centimetres, to which it first bounces may be assumed to be normally distributed with a mean of 140 and a standard deviation of 2.5 . (a) Determine: (i) PðX < 145Þ ;
(3 marks)
(ii) Pð138 < X < 142Þ .
(4 marks)
(b) Determine, to one decimal place, the maximum height exceeded by 85% of first bounces. (4 marks) (c) Determine the probability that, for a random sample of 4 first bounces, the mean height is greater than 139 cm. (4 marks)
6 For the adult population of the UK, 35 per cent of men and 29 per cent of women do not wear glasses or contact lenses. (a) Determine the probability that, in a random sample of 40 men: (i) at most 15 do not wear glasses or contact lenses;
(3 marks)
(ii) more than 10 but fewer than 20 do not wear glasses or contact lenses.
(3 marks)
(b) Calculate the probability that, in a random sample of 10 women, exactly 3 do not wear glasses or contact lenses. (3 marks) (c)
(i) Calculate the mean and the variance for the number who do wear glasses or contact lenses in a random sample of 20 women. (3 marks) (ii) The numbers wearing glasses or contact lenses in 10 groups, each of 20 women, had a mean of 16.5 and a variance of 2.50 . Comment on the claim that these 10 groups were not random samples. (3 marks)
P5647/Jun08/MS/SS1B
5
7 Vernon, a service engineer, is expected to carry out a boiler service in one hour. One hour is subtracted from each of his actual times, and the resulting differences, x minutes, for a random sample of 100 boiler services are summarised in the table. Difference
Frequency
6 4 x < 4
4
4 4 x < 2
9
2 4 x < 0
13
04x< 2
27
24x< 4
21
44x< 6
15
64x< 8
7
8 4 x 4 10
4
Total (a)
100
(i) Calculate estimates of the mean and the standard deviation of these differences. (4 marks) (ii) Hence deduce, in minutes, estimates of the mean and the standard deviation of Vernon’s actual service times for this sample. (3 marks)
(b)
(i) Construct an approximate 98% confidence interval for the mean time taken by Vernon to carry out a boiler service. (4 marks) (ii) Give a reason why this confidence interval is approximate rather than exact. (1 mark)
(c) Vernon claims that, more often than not, a boiler service takes more than an hour and that, on average, a boiler service takes much longer than an hour. Comment, with a justification, on each of these claims.
END OF QUESTIONS
P5647/Jun08/MS/SS1B
(2 marks)
Surname
Other Names
Centre Number
Candidate Number
Candidate Signature
General Certificate of Education June 2008 Advanced Subsidiary Examination
MATHEMATICS Unit Statistics 1B
MS/SS1B
STATISTICS Unit Statistics 1B
Insert Insert for use in Question 3. Fill in the boxes at the top of this page. Fasten this insert securely to your answer book.
Turn over for Figure 1
P5647/Jun08/MS/SS1B 6/6/6/
s
Turn over
2
Figure 1 (for use in Question 3)
Decorative Plaques
y
~
130 –
125 –
Width (millimetres)
120 – D
E 115 –
C
F 110 –
B A
105 –
–
–
–
–
–
225
230
235
240
245
250
Length (millimetres)
Copyright Ó 2008 AQA and its licensors. All rights reserved.
P5647/Jun08/MS/SS1B
~
–
–
0– 0
x
Version 1.0: 0608
abc General Certificate of Education
Mathematics 6360 Statistics 6380 MS/SS1B Statistics 1B
Mark Scheme 2008 examination - June 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 © 2008 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 2008 June 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.
3
MS/SS1B - AQA GCE Mark Scheme 2008 June series
MS/SS1B Q
Solution 1(a) b (gradient) = –1.01 to –1(.00) (b (gradient) = –1.05 to –0.95) a (intercept) = 53(.0) to 53.2 (a (intercept) = 52(.0) to 54(.0))
Marks B2 (B1)
Total
B2 (B1)
4
Comments AWFW
(–1.00337)
AWFW
(53.06736)
OR Attempt at
∑x, ∑x , ∑y 2
and
∑ xy
180, 3986, 297 and 5552.7 (M1)
or Attempt at S xx and S xy
386 and –387.3
Attempt at correct formula for b (gradient) b (gradient) = –1.01 to –1(.00) a (intercept) = 53(.0) to 53.2
(m1) (A1) (A1)
AWFW AWFW
Accept a and b interchanged only if then identified correctly in part (b), but B2 in (b) does not necessarily imply 4 marks in (a) (b) When x = 21,
y = 31.7 to 32.2 (y = 29.9 to 34.1)
B2 (B1)
Evidence of use of 21 in c’s equation
(M1)
2
AWFW AWFW
Special Cases (if seen): y =
33.0 + 30.7 = 31.8 to 31.9 2
(B1)
y = 31.85 without working
AWFW; or equivalent
(B1) Total
6
4
(32.0)
MS/SS1B - AQA GCE Mark Scheme 2008 June series
MS/SS1B (cont) Q
Solution 2(a)
P(Blue) =
160 2 160 = 0.4 or or 400 5 400
Marks
Total
B1
1
Comments CAO; or equivalent
In (b) to (e), method marks are for single fractions, or equivalents, only (b)
P(Marker) =
280 400
= 0.7 or
270 ≤ Numerator ≤ 290 and Numerator < Denominator ≤ 400
M1 7 280 or 10 400
A1
2
CAO; or equivalent
(c) P(B or M) = P(B ∪ M) =
160 + 280 − 119 280 + 41 321 = = 400 400 400 = 0.802 to 0.803 or (d)
= 0.35 or (e)
321 400
P(Green | Highlighter) = P(G | H) =
A1
42 120
7 42 or 20 120
P(Non-Permanent | Red) = P(P′ | R) =
= 0.233 to 0.234 or
290 ≤ Numerator ≤ 321 and Numerator < Denominator ≤ 400
M1
7 21 or 30 90 Total
2
2 9
5
(0.8025)
CAO; or equivalent Numerator = 21 and 80 ≤ Denominator ≤ 90
M1
A1
AWFW/CAO Numerator = 42 and 110 ≤ Denominator ≤ 120
M1
A1 21 90
2
AWFW/CAO
(0.2333)
MS/SS1B - AQA GCE Mark Scheme 2008 June series
MS/SS1B (cont) Q
Solution 3(a) r = 0.806 to 0.807 (r = 0.8(0) to 0.81) (r = 0.7 to 0.9)
Marks B3 (B2) (B1)
Total 3
Comments
AWFW AWFW AWFW
(0.80656)
OR
Attempt at ∑ x , ∑ x2 ,
∑y, ∑y
2
and
∑ xy
2859, 681575, 1428, 170342 and 340555
or Attempt at S xx , Syy and S xy
(M1)
Attempt at correct formula for r r = 0.806 to 0.807
(m1) (A1)
418.25, 410 and 334 AWFW Or equivalent; must qualify strength and indicate positive B0 for some/average/medium/very strong/etc
(b) Moderate/fairly strong/strong positive correlation (relationship/association)
between length and width of plaques (c) Figure 1: 6 correct labelled points (5 correct labelled points) (4 correct labelled points) (d) A to F:
r = –0.2 to +0.2
B1
B1
2
B3 (B2) (B1)
3 Deduct 1 mark if not labelled
B1
Accept ‘Zero’ but not ‘No’ correlation G to L:
r = –0.2 to +0.2
B1
Context; providing 0 < r < 1
2
AWFW (–0.0275) No penalties for calculations Statements must include a single value within range AWFW (–0.0196)
Special Cases: r = –0.2 to +0.2 with no sources r = –0.2 to +0.2 for each/both source(s) If B0 B0 but both values of r = –0.4 to +0.4
(B1)
AWFW
(B2)
AWFW; or equivalent identification
(B1)
AWFW
Total
10
6
MS/SS1B - AQA GCE Mark Scheme 2008 June series
MS/SS1B (cont) Q
Solution 4(a) Ordering: 0 0 13 28 35 40 47 51 63 77 a
Median (6th) = 40
Marks
Total
Comments
M1
May be implied by 40 and/or 63 and 13
B1
CAO
IQR = Q3(9th) – Q1(3rd) = 63 – 13 = 50
(B1) B2
(b)(i) Mode: Zero is not representative / sensible reason Wide range of (known) values Small number of values mostly different (ii) Range: Largest value, a, is unknown Cannot be calculated
4
Or equivalent
B1
B1 Total
2 6
7
Identification of 63 and 13 CAO
Or equivalent
MS/SS1B - AQA GCE Mark Scheme 2008 June series
MS/SS1B (cont) Q
Solution 5 Height X ~ N(140, 2.52)
(a)(i)
Marks
Total
Comments
M1
Standardising (144.5, 145 or 145.5) with 140 and ( 2.5 , 2.5 or 2.52) and/or (140 – x)
P(Z < 2) =
A1
2 CAO; ignore sign
0.977 to 0.98(0)
A1
145 − 140 ⎞ ⎛ P(X < 145) = P ⎜ Z < ⎟ = 2.5 ⎠ ⎝
3
AWFW
(ii) P(138 < X < 142) = P(X < 142) – P(X < 138) =
M1
Difference (142 – 138)
P(Z < 0.8) – P(Z < –0.8) =
B1
0.8 CAO
P(Z < 0.8) – {1 – P(Z < 0.8)} = (0.78814) – (1 – 0.78814) =
m1
Correct area change
0.576 to 0.58(0)
A1
(b) 0.85 (85%) ⇒ z = –1.03 to –1.04
z =
x − 140 2.5
= ±1.03 to ± 1.04 Hence (c)
x = 137.3 to 137.5
AWFW
(0.57628)
B1
AWFW; ignore sign
(–1.0364)
M1
Standardising x with 140 and 2.5; allow (140 – x)
A1
Equating z-term to the z-value
A1
2.52 = 1.56(25) 4 2.5 = = 1.25 2
4
(0.97725)
4
AWFW; CSO
(137.41)
Variance of X 4 =
B1
CAO; stated or used
⎛ 139 − 140 ⎞ ⎟ = P ( X 4 > 139 ) = P ⎜ Z > ⎜ 2.52 4 ⎟⎠ ⎝
M1
Standardising 139 with 140 and 1.25; allow (140 – 139)
P(Z > –0.8) = P(Z < 0.8) =
m1
Correct area change
SD of X 4
0.788 to 0.79(0)
A1 Total
8
4 15
AWFW
(0.78814)
MS/SS1B - AQA GCE Mark Scheme 2008 June series
MS/SS1B (cont) Q
Solution 6 Binomial distribution
Marks M1
(a)(i) M ~ B(40, 0.35)
Total
A1
P(M ≤ 15) = 0.69(0) to 0.696
A1
Comments Used somewhere in question
Used; may be implied 3
AWFW
(0.6946)
(ii) P(10 < M < 20) =
0.9637 or 0.9827
M1
Accept 3 dp accuracy
minus 0.1215 or 0.0644
M1
Accept 3 dp accuracy
= 0.84(0) to 0.843
A1
3
AWFW
(0.8422)
OR
B(40, 0.35) expressions stated for at least 3 terms within 10 ≤ M ≤ 20 Answer = 0.84(0) to 0.843 (b) W ~ B(10, 0.29)
⎛ 10 ⎞ 3 7 P(W = 3) = ⎜ ⎟ ( 0.29 ) ( 0.71) ⎝3⎠
(M1)
Or implied by a correct answer
(A2)
AWFW
B1
Used; may be implied
M1
Stated; may be implied (0.2662)
A1
p = 0.71
B1
Stated or used; may be implied by 14.2
Mean, μ = np = 14.2
B1
CAO
Variance, σ 2 = np(1 – p) = 4.11 to 4.12
B1
(c)(i) n = 20
(ii) Mean of 16.5 is greater/different or 16.5/20 = 0.825 is greater/different to 0.71
3
AWFW Note: B(10, 0.3) ⇒ 0.2668
= 0.266 to 0.2665
3
(B2,1 dep)
Variance of 2.50 is smaller/different
B1dep
Suggests claim that groups are not random samples is justified
B1dep Total
Dependent on σ 2 = 4.11 to 4.12 3 15
9
(4.118)
Dependent on μ = 14.2
B1dep
Means and variances are different
AWFW
Dependent on previous 2 marks Or equivalent
MS/SS1B - AQA GCE Mark Scheme 2008 June series
MS/SS1B (cont) Q 7(a)(i) x: f:
–5 4
Solution –3 –1 1 3 9 13 27 21
Marks
5 15
7 7
Mean ( x ) = 1.9 (0.9 to 2.9)
B2 (B1)
Standard deviation (sn-1 or σn) = 3.3(0) to 3.32 (3(.00) to 3.5(0))
B2 (B1)
If no marks scored but
∑ fx
(b)(i) 98% ⇒ z = 2.32 to 2.33 (⇒ t = 2.36 to 2.37)
or
sn −1 or σ n n or n − 1
61.9 ± 2.3263 ×
3.3 to 3.32 100 or 99
61.9 ± (0.7 to 0.8)
B1
3
(190) (1452) (3.31967) (3.30303)
AWFW AWFW
S > 1 hour or 60 minutes: Valid as 74/100 or 0.74 or 74% > 50% S >> 1 hour or 60 minutes: Not valid as UCL ≈ 1 hour (Accept Both limits ≈ 1 hour)
on (a)(i) on (a)(i); accept ‘same as’ only providing answer in (a)(i)
B1
AWFW AWFW
M1
Used; must have
A1
4
B1
1
Total TOTAL
2 14 75
10
n with n > 1
Accept 1.03 ± (0.012 to 0.013) AWFW Accept (1.01 to 1.02, 1.04 to 1.05) Actual times/values unknown Or equivalent
Must use 74 etc Or equivalent
B1
B1dep
(2.3263) (2.364)
on (a)(ii) and z/t only
A1
(61.1 to 61.2, 62.6 to 62.7)
(ii) Mean and SD based upon grouped data SD (not mean) calculated from a sample CLT used / Times (may) not (be) normal (c)
4
M1 A1
Standard deviation = 3.3(0) to 3.32
Hence
CAO AWFW
(M1)
(ii) Mean = 60 + x = 61.9
Thus
Comments
attempted
and result divided by 100
CI for μ is x ± z / t ×
Total
9 4
Dependent on UCL = 62.6 to 62.7 or UCL = 1.04 to 1.05
klm
Scaled mark component grade boundaries - June 2008 exams GCE Component Code Component Title
Maximum Scaled Mark
A
GCE MATHEMATICS UNIT PC3 GCE MATHEMATICS UNIT PC4 GCE MATHEMATICS UNIT S03 GCE MATHEMATICS UNIT S04 GCE MATHS/STATISTICS UNIT 1A - COURSEWORK GCE MATHS/STATISTICS UNIT 1A - WRITTEN GCE MATHEMATICS UNIT S1B GCE MATHEMATICS UNIT S2A - COURSEWORK GCE MATHEMATICS UNIT S2A - WRITTEN GCE MATHEMATICS UNIT S2B GCE MATHEMATICS UNIT XMCA2 GCE MATHEMATICS UNIT XMCAS
75 75 75 75 25 75 75 25 75 75 125 125
60 60 63 57 20 60 63 20 60 62 99 102
53 53 55 50 18 53 55 18 53 54 87 90
46 46 47 43 15 45 47 15 45 46 76 78
40 40 39 36 13 39 39 13 38 38 65 67
34 34 32 30 10 33 32 10 30 30 54 56
MED1 MED2 MED3 MED4 MED5 MED6
GCE MEDIA STUDIES UNIT 1 GCE MEDIA STUDIES UNIT 2 GCE MEDIA STUDIES UNIT 3 GCE MEDIA STUDIES UNIT 4 GCE MEDIA STUDIES UNIT 5 GCE MEDIA STUDIES UNIT 6
60 60 100 60 60 60
41 41 72 41 50 45
36 36 64 37 41 39
31 31 56 33 32 33
26 26 48 29 23 27
21 21 40 26 15 21
HEB1 HEB2
GCE MODERN HEBREW UNIT 1 GCE MODERN HEBREW UNIT 2
100 100
66 73
58 64
50 55
43 46
36 37
MUS1 MUS2 MUS3 MUS4 MUS5
GCE MUSIC UNIT 1 GCE MUSIC UNIT 2 GCE MUSIC UNIT 3 GCE MUSIC UNIT 4 GCE MUSIC UNIT 5
100 60 60 120 80
63 46 49 76 67
53 40 44 68 63
44 34 39 61 59
35 28 34 54 55
26 23 29 47 51
MPC3 MPC4 MS03 MS04 MS/SS1A/C MS/SS1A/W MS1B MS2A/C MS2A/W MS2B XMCA2 XMCAS
Scaled Mark Grade Boundaries B C D
E
