trial math m 950-2 2015 jong srawak
DESCRIPTION
matematik m stpmTRANSCRIPT
950
CONFIDENTIAL*
PAGE
8
CONFIDENTIAL*/SULIT
Section A
[45 marks]
Answer all questions in this section.
1The following are the numbers of customers visited a departmental store on eleven working days.
2032363738394041424562(a) Find the median, the lower and upper quartiles. [3 marks]
(b) Construct a boxplot for the data.[5 marks]
(c) The number of customers visited a departmental store is more than 40 times
for 75% of working days. Comment on the statement. [2 marks]
2A random variable X has the following probability distribution.
x
-1
0
1
2
P(X = x)
a
8
2
8
3
b
Given that
4
3
)
X
(
E
=
, find the values of a and b. [4 marks]
3The variables x and y are believed to be linearly correlated. Ten measurements of x and y are summarized as follows:
=
1370
x
=
1670
y
=
189006
2
x
=
289736
2
y
=
232545
xy
Calculate the coefficient of determination and comment on the result.
[6 marks]
4The table below shows the quantity and prices of three electrical items produced by a company from 2012 to 2014
Electrical item
Quantity (in thousands)
Price (RM per unit)
2012
2013
2014
Air conditioner
40
50
60
1300
Fan
30
40
50
90
Television
40
50
50
1100
(a) By using price as the weights and the year 2012 as the base year, calculate Laspeyres quantity index for the years 2013 and 2014.[4 marks]
(b) Briefly comment on the different rates of output growth of the company between the year 2013 and 2014.[2 marks]
5The table below shows the quarterly sales (RM x 1000) of a motorbike between 2011 to 2014 and an additive model seasonal variation for each quarter.
Quarter
Year
1
2
3
4
2011
60
84
42
72
2012
64
86
43
74
2013
67
90
44
76
2014
69
98
45
78
Seasonal variation
-2.58
21.71
-24.58
5.46
(a) Fit a trend line to the deseasonalised time series.[6 marks]
(b) Forecast the sales of a motorbike for the quarter 3 in year 2015.[3 marks]
6 (a) There are 5 boys and 4 girls on a sport committee. A subcommittee of 3 students
is selected from the committee at random. Find the probability that all are boys given that all are the same sex.[4 marks]
(b) A and B are two events such that P(A) = 0.5, P(B) = 0.7 and
()
PAB
= 0.3. Calculate
(i)
()
PAB
(ii)
()
PAB
Show that event A and B are not independent. [6 marks]
Section B
[15 marks]
Answer any one question in this section.
7.In a production process of flower vases, is known that on average 45 flower vases have cracks in every 1000 flower vases produced.
(i) Find the probability that there are at most two flower vases having cracks
in a random sample of size 5.[5 marks]
(ii) If the probability that no flower vases, in n flower vases selected at
random, having cracks is 0.1, determine the value of n.[4 marks]
(i) By using suitable approximation, find the probability that more than three flower vases have cracks in a random sample of size 100.[6 marks]
8The relationship between two variables x and y are found to be as follows:
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.[3 marks]
(b) Calculate the coefficient of correlation between x and y, and comment on your result[4 marks]
(c) Find the equation of the regression line of y on x in the form y = a + bx, where a and b are expressed correct to two decimal places. Draw the graph of the regression line on your scatter diagram.[6 marks]
(d) Estimate the value of y when x = 40. Comment on your result.[2 marks]
END OF QUESTION PAPER
ANSWER
1
(a) Median =
observtion
N
2
=
observtion
2
11
= 5.5 observation
= 6th observation
= 39 times..[1 mark]
Lower quartile,
1
Q
=
observtion
N
4
=
observtion
4
11
= 2.75 observation
= 4th observation
= 36 times..[1 mark]
Upper quartile,
3
Q
=
observtion
N
4
3
=
observtion
4
)
11
(
3
=8.25 observation
=9th observation = 42 times..[1 mark]
(b) Lower boundary =
)
(
2
3
1
3
1
Q
Q
Q
-
-
=
)
36
42
(
2
3
36
-
-
= 27 51
Upper boundary =
)
(
2
3
1
3
1
Q
Q
Q
-
+
=
)
36
42
(
2
3
42
-
+
= 51
Outlier = 20 and 62[2 marks]
Boxplot[3 marks]
(c ) Not ture. Since the lower quartile is 36 times and therefore 75% above is not 40 times.
2.
4
3
)
X
(
E
=
a(-1) +
8
2
(0)+
8
3
(1)+ b(2) =
4
3
-8a + 16b = 3 (1)..[1 mark]
=
=
1
)
(
x
X
P
a+
8
2
+
8
3
+ b = 1
8a + 8b = 3(2)..[1 mark]
Solve (1) and (2) a =
8
1
, b =
4
1
..[2 marks]
3
(
)
(
)
-
-
-
=
n
y
y
n
x
x
n
y
x
xy
r
2
2
2
2
n
n
r
2
2
)
1670
(
289736
)
1370
(
189006
10
1670
1370
232545
-
-
-
=
= 0.994..[2 marks]
Coefficient o determination ,
988
.
0
)
994
.
0
(
2
2
=
=
r
..[2 marks]
Hence, 98.80% of the variation of y is explained for due the variation of x.
..[2 marks]
4 Laspeyres quantity index for the years 2013 =
100
12
12
12
13
p
q
p
q
100
)
1100
(
40
)
90
(
30
)
1300
(
40
)
1100
(
50
)
90
(
40
)
1300
(
50
+
+
+
+
=
100
98700
23600
=125.2 ..[2 marks]
Laspeyres quantity index for the years 2014 =
100
12
12
12
14
p
q
p
q
100
)
1100
(
40
)
90
(
30
)
1300
(
40
)
1100
(
50
)
90
(
50
)
1300
(
60
+
+
+
+
=
100
98700
137500
=139.3 ..[2 marks]
The companys output has increased 25.2% from the year 2012 to 2013. From the year 2012 to 2014, the companys output increased by 39.3%. ..[2 marks]
5
Quarter
Year
1
2
3
4
2011
60
84
42
72
2012
64
86
43
74
2013
67
90
44
76
2014
69
98
45
78
Seasonal variation
-2.58
21.71
-24.58
5.46
Year
Coded quarter
(t)
Sales (RM X 100) (y)
Seasonal variation (s)
Deseasonalised sales (d)
1
60
-2.58
62.58
2
84
21.71
62.29
3
42
-24.58
66.58
4
72
5.46
66.54
5
64
-2.58
66.58
6
86
21.71
64.29
7
43
-24.58
67.58
8
74
5.46
68.54
9
67
-2.58
69.58
10
90
21.71
68.29
11
44
-24.58
68.58
12
76
5.46
70.54
13
69
-2.58
71.58
14
98
21.71
76.29
15
45
-24.58
69.58
16
78
5.46
72.54
..[2 marks]
(a)
58
.
197
=
t
96
.
1091
=
d
9964
.
13605
=
td
8964
.
5903
2
=
t
954
.
74718
2
=
d
(
)
-
-
=
n
t
t
n
d
t
td
b
2
2
16
)
58
.
197
(
8964
.
5903
16
)
96
.
1091
)(
58
.
197
(
9964
.
13605
2
-
-
=
= 0.03512..[2 marks]
t
b
d
a
-
=
-
=
16
58
.
197
3512
.
0
16
96
.
1091
a
= 67.81..[1 mark]
T = a + bt
The trend line is T = 67.8 +0.0351t..[1 mark]
(b) For 3rd quarter in year 2015; t = 19
T = 67.8 +0.0351(19) =68.4669 and
3
S
=-24.58
Forecast value =68.4669 + (-24.58) = 43.8869..[2 marks]
Hence, the forecast value for the sales of a motorbike for the quarter 3 in year 2015 is RM43886.90...[1 mark]
6. (a) Let A = all are boys
B = all are same sex
)
(
)
(
)
/
(
B
P
B
A
P
B
A
P
=
=
7
5
14
10
3
4
3
5
3
5
=
=
+
C
C
C
..[2 marks]
b. (i)
)
(
)
(
)
/
(
B
P
B
A
P
B
A
P
=
=
7
.
0
3
.
0
=
7
3
..[2 marks]
(ii)
-
+
=
)
(
)
(
)
(
B
P
A
P
B
A
P
EMBED Equation.3
)
(
B
A
P
= 0.5 + 0.7 - 0.3 = 0.9
..[2 marks]
35
.
0
7
.
0
3
.
0
)
(
)
(
=
=
B
P
A
P
Since
)
(
B
A
P
EMBED Equation.3
)
(
)
(
B
P
A
P
Event A and B are not independent...[2 marks]
7.(i) p =
1000
45
= 0.045
Let X = number of flower vases that crack in a sample of 5.
Then
X
~B(5,0.045)..[1 mark]
(
)
2
X
P
= P(X=0) + P(X=1) + P(X=2)
=
5
0
0
5
955
.
0
045
.
0
C
+
4
1
1
5
955
.
0
045
.
0
C
+
3
2
2
5
955
.
0
045
.
0
C
= 0.7944 + 0.1872 + 0.0176
= 0.9992..[4 marks]
(ii)
X
~B(n, 0.045) ..[1 mark]
P(X=0) =
n
n
C
955
.
0
045
.
0
0
0
= 0.1
n
955
.
0
= 0.1
n =
955
.
0
log
1
.
0
log
= 50..[3 marks]
(iii) n =100, p = 0.045
045
.
0
100
=
=
np
m
= 4.5..[1 mark]
955
.
0
045
.
0
100
2
=
=
npq
s
= 4.2975..[1 mark]
Then
X
~N(4.5, 4.2975),
2975
.
4
5
.
4
-
=
X
Z
~N(0, 1)..[1 mark]
(
)
3
>
X
P
=
(
)
5
.
3
X
P
=
-
2975
.
4
5
.
4
5
.
3
Z
P
=
(
)
4824
.
0
-
Z
P
=
(
)
4824
.
0
Z
P
= 0.6851..[3 marks]
8
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
n = 8
=
148
x
=
2780
2
x
=
6
.
23
y
=
3
.
71
2
y
=
4
.
444
xy
(b)
(
)
(
)
-
-
-
=
n
y
y
n
x
x
n
y
x
xy
r
2
2
2
2
8
6
.
23
3
.
71
8
148
2780
8
)
6
.
23
)(
148
(
4
.
444
2
2
-
-
-
=
r
= 0.9286..[3 marks]
This value shows a strong positive correlation between x and y...[1 mark]
c) y = a + bx
(
)
-
-
=
n
x
x
n
y
x
xy
b
2
2
=
8
148
2780
8
)
6
.
23
)(
148
(
4
.
444
2
-
-
= 0.1857..[2 marks]
x
b
a
y
+
=
-
=
8
148
1857
.
0
8
6
.
23
a
= -0.4885..[1 mark]
Equation: y = -0.49 + 0.19x..[1 mark]
Regression line ..[2 marks]
When x = 15, y = -0.49 + 0.19(15) = 2.36
x = 20, y = -0.49 + 0.19(20) = 3.31
(d) The value of y may not exist since x = 40 outside the range of recorded values.
..[2 marks]
STPM 950/2[Turn over
CONFIDENTIAL*
This question paper is CONFIDENTIAL until the examination is over.
STPM 950/2
CONFIDENTIAL*
*This question paper is CONFIDENTIAL until the examination is over.