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8/14/2019 Stpm Trial 2009 Maths2 Q&A (n9)
1/14
CONFIDENTIAL* 2
1. A discrete random variableXhas a probability distribution as shown in the tablebelow.
X=x 0 m 2 3
P(X=x) 0.1 2n 0.3 0.2
(a) Find the value ofn.
[1 mark]
(b) Given that ( ) 1 6 E X .= , find the value ofm.
[1 mark]
(c) Given that1 2
Y X X+ whereX1 andX2( )E Y
are independent random variables ofX,
find .
[2 marks]
2. The mean height of 100 students selected randomly in a college was found to be
150 cm with a standard deviation of 5 cm.(a) Estimate the mean and standard deviation of the mean height of all the students in
the college.
[3 marks](b) Estimate the standard error of the mean height of the students.
[2 marks]
3. The demand for tiger prawns in Malaysia depends on the price.
Price per kg (RM) 27 26 25 24 23 22 21 20
Sales ( 000kg ) 10 12 15 19 27 37 44 59
Calculate the Pearsons correlation coefficient. [5 marks]
4. The number of laptop computers that are sold in a week by 16 representatives in a
town is as follows:
6 10 9 59 22 14 25 26 11 27 50 27 37 38 19 38
(a) Draw a stemplot to represent the above data.
[3 marks]
(b) Hence, find the median and the semi-inter quartile range of this distribution.
[4 marks]
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CONFIDENTIAL* 3
5. The following table shows the number of digital camera sold in a departmentalstore for the year 2007 and 2008.
Type of digital camera2007 2008
Price(RM) Quantity Price(RM) Quantity
A 350 53 450 81B 500 36 550 75
C 800 35 900 42
D 1200 60 1500 50
By using 2007 as the base year,
(a) calculate the average of relative quantity index of digital camerasA, B, Cand
D for the year 2008.[3 marks]
(b) calculate the Paasche price index for the year 2008 and explain your answer.
[3 marks]
6. EventsA andB are such that
( )1
3P A = , ( )
3|
4P B A = , ( )
1'
4P A B =
Find
(a) ( )P A B ,
(b) ( )P B ,
Determine whether eventsA andB are independent. Give reasons for your answer.
[7 marks]
7. The relationship between two variablesx andy are found to be as the following:
x 15 16 17 18 19 20 21 22
y 2.4 2.5 2.6 2.6 3.0 3.5 3.6 3.4
(a) Plot the scatter diagram for the above data and state the relationship between
x andy.
[3 marks]
(b) Find the equation of the regression line ofy onx in the form ofy = a + bx,where a and b are expressed correct to two decimal places. Draw the graph of the
regression line on your scatter diagram.
[7 marks]
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CONFIDENTIAL* 4
8. The following table shows the activities for a project and their preceding activities andduration.
Activity Preceding Activities Duration (days)
A - 3
B - 3C - 7
D A 1
E D,J 2
F B 2
G C 1
H E,F,G 1
J B 1
(a) Draw an activity network for the project showing the earliest start time and the
latest start time for each activity.
[6 marks](b) State the critical path and the minimum completion time.
[2 mark]
9. The table below shows the duration in minutes of 160 telephone calls made in onemonth by a trading company.
(a) Calculate the mean call duration by the trading company.
[2 marks](b) Plot the cumulative frequency curve for the grouped data above.
Hence, estimate the median for the durations of the telephone calls.
[4 marks]
(c) Describe the skewness of the distribution.[2 marks]
Duration (minutes) Frequency
0.0 2.9 12
3.0 5.9 33
6.0 8.9 459.0-11.9 38
12.0-14.9 19
15.0-17.9 7
18.0-20.9 6
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CONFIDENTIAL* 5
10. The following table shows the quarterly profits (RM000) of a company.
YearQuarter
1 2 3 4
2003
20042005
26
3234
44
5658
100
120122
46
5052
(a) Calculate the centred four-quarter moving averages for the above data.[4 marks]
(b) Calculate the quarterly seasonal variation index using the multiplicative model.
[4 marks]
(c) Predict the amount of profit for the first quarter of the year 2006.[4 marks]
11. A bakery shop bakes two types of breads,A andB, which are made of three types of
ingredients: P, Q andR. One loaf of typeA bread needs 3 units of ingredient P,1 unit of ingredient Q and 3 units of ingredientR. One loaf of typeB bread needs
4 units of ingredient P, 2 units of ingredient Q and 8 units of ingredientR. The bakeryhas 480 units of ingredient P, 180 units of ingredient Q and 640 units of ingredientR.The profit for TypeA bread is RM2.00 per loaf, whereas the profit of type B bread is
RM3.00 per loaf.
(a) Formulate the linear programming problem to obtain maximum profit.
[4 marks]
(b) Using the graphical method, determine the number of loaves of each type of breadthat must be baked to obtain the maximum profit and find this maximum
profit. [9 marks]
12. The mass of a type of pill produced by a pharmacy store has a normal distribution
with mean g and standard deviation 0.2 g
(a) Given 7= g, find the probability that the mean mass of a random sample of 16
pills exceed 7.05g.
[4 marks]
(b) If the mean mass of the random sample of the 16 pills is 7.2 g, find a 96%
confidence interval for the population mean, . State with reason whether the
managers claim that 8 = g is true or false.[6 marks]
(c) Determine the minimum sample size needed so that the difference between sample
mean and true mean is less than 0.1 g at a 90% confidence interval.
[5 marks]
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CONFIDENTIAL* 1
1. (a) 0.1 + 2n +0.3+0.2 = 1
Mark Scheme
n = 0.2 B1
(b) 0(0.1) + m(0.4) + 2(0.3) + 3(0.2) = 1.6
m = 1 B1
(c) E(X1) =E(X2
) = 1.6
( ) ( ) ( )1 21.6 1.6
3.2
E Y E X E X = +
= +
=
2. (a) Estimate of population mean = sample mean
= 150 cm
Estimate of population standard deviation
( )
2
2
1
100 5
99
5 025 5 03
ns
n
or. .
=
=
=
(b) Estimate of the standard error of the mean
( )
2
25
100
0 5
s
n
.
=
==
M1
A1
B1
A1
M1
M1
A1
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CONFIDENTIAL* 2
3.xy
x yS xy
n=
( ) ( )188 2234952
8
288 5.
=
=
( )22
xx
xS x
n=
( ) 21884460
8
42
= =
( )22
yy
yS y
n=
( ) 222383458
2128 875.
= =
Coefficient of correlation, rxy
xx yy
S
S S=
( ) ( )288 5
42 2128 875
0 9648
.
.
.
==
or
2 2188 223 4952 4460 8345x , y , xy , x , y= = = = =
r=2 2 2 2[ ( ) ][ ( ) ]
n xy x y
n x x n y y
= 8(4952) 188(223)
336 17031
(his value)
= 0.9648
M1
A1
A1
M1 method to calculate any of the Sxy, Sxx,
B1 - Any three of the summation correct
A1 All the values correct
M1
M1 calculating any of the Sxy, Sxx, SyyA1 all values correct
B1-Any three correct
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CONFIDENTIAL* 3
4(a)
0 6 9
1 0 1 4 9
2 2 5 6 7 7
3 7 8 84
5 0 9
Key: 2|5 = 25
(b) Median = 25.5 B1
Q1 = 12.5 , Q3
Semi-inter quartile range =
= 37.5 (both) B1
1
2(37.5 - 12.5) M1
= 12.5 A1
5(a)
Type of digital
camera
Sales QuantityI n
o
q
qq = 1002007 (qo 2008 (q) n)
A 53 81 152.83
B 36 75 208.33
C 35 42 120.00
D 60 50 83.33
Average Relative Quantity Index =qI
n
=152.83 208.33 120.00 83.33
4
+ + +M1 (his value)
= 141.13 A1
(b) Paasche price index 2006 20062005 2006
100p q
p q=
( ) ( ) ( ) ( )( ) ( ) ( ) ( )
450 81 550 75 900 42 1500 50100
350 81 500 75 800 42 1200 50
119.473
+ + +=
+ + +
=
The price of the items have increased by 19.47%
M1 uniform scale, Key
A1 all correct entries
B1 Stem for 4
M1 (strict)
A1
B1
B1 correct values for I
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CONFIDENTIAL* 4
6 (a) ( )3
|4
P B A =
( )( )
( ) ( )
( )
3
4
34
3 1
4 3
1
4
P B A
P A
P B A P A
P A B
=
=
=
=
(b) ( )1
'4
P A B =
( ) ( )
( )
( )
1
41 1
4 4
1
2
P B P A B
P B
P B
=
=
=
Since ( ) ( ) ( )P A B P A P B or ( ) ( )|P B A P B thus A and B are not
independent.
7(a) Uniform scale and at least 5 correct points D1
All correct D1
Positive linear relationship B1
(b) y a bx= + 2
148 23 6 444 4 2780 x , y . , xy . , x= = = = B1
( )
( )
2 2
2
1
1
1 148 23.6444.4
8 8 8
1 1482780
8 80.186
xy x ynb
x xn
=
=
=
23.6 1480.186
8 8
0.486
a y bx=
=
=
The equation of the regression line: 0.486 0.186y x= +
( ) ( )18 5 2 95 x , y . , .= D1
Straight line passing through his ( ) x , y D1
M1
M1
M1
A1
B1 deduction
B1 - reason
A1
M1 usage for the formula (his value)
B1 (his value of
A1
A1
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CONFIDENTIAL* 5
8(a)
D1 correct sequence and flow of the network diagram
D1 correct activities and duration
B1 correct values of EST
B1 correct values of EFT
B1 correct values of LST
B1 correct values of LFT
(b) B1 - Critical Path : C-G-H (strict)
B1 - Minimum completion Time = 9 days
9)(a) Mean1384
160= M1
= 8.65 A1
(b)
Less than Cumulative Frequency
2.95 12
5.95 45
8.95 90
11.95 128
14.95 147
17.95 15420.95 160
B1- correct boundaries and cumulative frequencies
Graph
D1 uniform scale and correct points (his values)
D1 all correct
B1 - median = 8.4 0.2
(c) X= 8.65,median = 8.0
The distribution is skewed to the right or positively skewed B1
because median < mean B1
1
0 0
4
7 7
6
8 8
3
3 5
5
4 6
2
3 5
A
B
C
D
E
F
G
3
1
3
7
2
2 1
7
9 9
J
1
1
H
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CONFIDENTIAL* 6
3
6
9
12
15
18
21Ma
Cumulative Frequency
80
60
40
20
0
120
140
160
100
X
X
X
X
X
X
X
X
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CONFIDENTIAL* 7
10 (a)
Year QuarterOriginal Data,
Y
Moving
Average
Trend,
T
Y
T
2003
1 26
2 44
54.0
3 100 54.75 1.83
55.5
4 46 57.00 0.81
58.5
2004
1 32 61.00 0.52
63.5
2 56 64.00 0.88
64.5
3 120 64.75 1.8565.0
4 50 65.25 0.77
65.5
2005
1 34 65.75 0.52
66.0
2 58 66.25 0.88
66.5
3 122
4 52
(b)
YearQuarter
1 2 3 4
2003 1.83 0.81
2004 0.52 0.88 1.85 0.77
2005 0.52 0.88
Total 1.04 1.76 3.68 1.58
Average 0.52 0.88 1.84 0.79
Correction factor 0.9926 0.9926 0.9926 0.9926
Seasonal Variation 0.52 0.87 1.83 0.78
A1M1M1
M1 adjusted seasonal variation
A1 all correct
B1: Correction factorM1: Average
A1
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CONFIDENTIAL* 8
(c) Range of trend values = 66.25 54.75 = 11.5
Average change in each period =11.5
7
Thus, the projected trend value = 66.25 + 3(11.5
7) = 71.18
Forecast value for the first quarter of 2006
71.18 0.52
37.01
=
=
Forecast profit for the first quarter of 2006 is RM 37 010
11(a) x = number of loaf of typeA bread.
y = number of loaf of typeB bread
3x + 4y 480 B1x + 2y 180 B13x + 8y 640 B1
0 0x , y
Maximizing, P = 2x + 3y B1
(b) 5 lines (including axis and search line) 1 mark for every correct line D5
Correct region B1
Number of loaf of Type A bread = 120
Number of loaf of Type B bread = 30
Maximum profit( ) ( )2 120 3 30
330 00RM .
= +
=
OR
All 4 extreme points (0,80) , (80,50) , (120,30) (160,0) M1
Substitute in obj. function M1
x = 120, y =30 (both) A1
RM 310.00 A1
A1
M1
M1 , A1
M1 (substitution)
A1
B1 dependent on searc
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CONFIDENTIAL* 9
100
120
2x + 3y = 90
x + 2y = 180
X
3x + 8y = 640
X
y
80
60
40
20
0
X
X
3x + 4y = 480
x
20
40
60
80
100
120
140
160
180
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CONFIDENTIAL* 10
12. X~N(, 0.22
(a)
)
7= , 16n = 20 2
716
.X ~ N ,
( )2
7 05 77 05
0 2
16
.P X . P Z
.
> = >
( )1
0 1587
P Z
.
= >
=
(b) 0 02 2 054.Z .=
96% confidence interval for the population mean , is
( )
( )
20 27 2 2 05416
7 2 0 1027
7 097 7 303
.. .
. .
. , .
=
=
=
At the level of confidence of 96%, the managers claim
that 8= is false because the interval (7.097,7.303)
does not contain 8 g
(c)2
2
0 1Z .n
20 21 645 0 1
.. .
n
( )2
21 6450 2
0 1
10 82
.n .
.
n .
The minimum sample size is 11
B1
M1
A1
B1
M1 for 7 2. K
A1
B1 False, B1 Reason
M1 - substitution
M1 for 0.1
M1
A1
B1 z-value 1.645
B1 for 7 05X .>
M1 for the correct term K
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