Download - TRIAL STPM Mathematics M 2 (SARAWAK) SMK Paku

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TRIAL STPM MATHEMATICS M (SARAWAK) –SMK Paku QA

Section A

[45 marks]

Answer all questions in this section.

1 In a box containing 80 light bulbs, 60 are in good condition. Two bulbs are taken

randomly, one after the other, from the box. Find the probability that only one of

the two bulbs taken is in good condition.

(a) If the first bulb is returned into the box. [3 marks]

(b) If the first bulb is not returned into the box. [3 marks]

2 The table below shows the prices of fish and the quantities of it bought by a

housewife at a market in the year 2011 and 2012.

2011 2012 Fish

Price

(RM per kg)

Quantity

(kg)

Price

(RM per kg)

Quantity

(kg)

Parang 11.00 2 12.00 1

Tenggiri 12.00 2 13.00 2

Bawal Putih 10.00 2 a 1

Kembung 8.00 3 10.00 2

Selar Kuning 4.00 5 5.00 6

(a) If the simple aggregate price index increases by 20 % from 2011 to 2012,

determine the value of a. [3 marks]

(b) Calculate the Laspeyres price index, and comment on the housewife’s

change in expenditure on fish. [3 marks]

3. A random sample of 24 people were asked to record the number of kilometres

they travelled by car in a given week. The distances, to the nearest kilometres, are

shown below.

67 66 85 93 63 44 87 57

72 77 67 48 66 50 52 74

70 70 41 108 58 74 62 68

(a) Construct a stem-and-leaf diagram to represent these data. [2 marks]

(b) Determine the median and interquartile range of this distribution.

[3 marks]

(c) Draw a box-and-wisker plot for this data. [3 marks]

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2

4 The probability density function of the time t, in minutes, of a student spends

every morning waiting for bus to school is given by

otherwise

t

t

t

t

tf ,155

,50

0

,75

1

5

175

2

)( <≤

<≤

−=

(a) Find the cumulative distribution function of T. [4 marks]

(b) Calculate the mean waiting time of the student. [3 marks]

5. The water vapour content, w (in g) per 100 g samples of mud, taken at a depth of

d meters from a river mouth is measured. The results are shown in the table

below.

Depth (d) 0 5 10 15 20 25 30 35

Water vapour content (w) 92 80 57 44 32 22 20 17

(a) Plot these data on a scatter diagram [2 marks]

(b) Calculate the Pearson’s correlation coefficient for the above data.

[4 marks]

(c) Comment on the relationship between the depth of the mud and the water

vapour content. [2 marks]

6. Unemployed school leavers in the Malaysia (figures in thousands) is tabulated

below.

Year Quarter 1 Quarter 2 Quarter 3 Quarter 4

2010 22 12 110 31

2011 21 26 150 70

2012 50 36 146 110

(a) Find the four-quarter centred moving average for the time series.

[3 marks]

(b) Calculate the seasonal variation using an additive model.

[3 marks]

(d) Use this model to predict unemployed school leavers for the first quarter

of the year 2013. [4 marks]

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Section B

[15 marks]

Answer any one question in this section.

7. Everyday, a bus leaves town A for town B at 0800 hours. The time taken for this

journey is recorded for a certain period. The result are shown in the following

table.

Time (minutes) Frequency

2825 ≤≤ x 6

2928 ≤≤ x 12

3029 ≤≤ x 27

3130 ≤≤ x 30

3231 ≤≤ x 18

3332 ≤≤ x 14

3433 ≤≤ x 9

3534 ≤≤ x 4

4035 ≤≤ x 5

a) Draw a histogram to represent these data. [3 marks]

b) Calculate the median, mean and standard deviation for this data.

[5 marks]

c) Hence, calculate the Pearson coefficient of skewness and comment the

skewness of the distribution. [2 marks]

d) Plot a cumulative frequency curve of the above data. Hence, find the

number of times a bus arrives at town B between 0830 hours and 0836

hours. [5 marks]

8 (a) On average, the number of books read by an adult is 5 books per year. Using

the Poisson distribution, find the probability that

(i) an adult reads exactly 3 books per year, [2 marks]

(ii) an adult reads more than 2 books in 5 years. [2 marks]

(b) The marks obtained by the candidates for a paper in an examination are

distributed normally with mean of 50 and standard deviation 10.

(i) If a candidate must obtain 70 marks to score a distinction for the paper,

find the percentage of candidates who obtained distinctions for this paper

in the examination. [3 marks]

(ii) If 70% of the candidates pass the paper, determine the minimum marks

required to get a passing grade for the paper. [3 marks]

(c) In a group of teachers, the expected number of teachers who own Proton cars

is 8 and the variance is 1.6. Find the probability that

(i) a teacher chosen at random owns a Proton car, [3 marks]

(ii) exactly 4 teachers from the group own Proton cars. [2 marks]

END OF QUESTION PAPER

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ANSWER TRIAL STPM MATHEMATICS M (SARAWAK) –SMK Paku QA

1

[2M]

[1M]

[2M]

[1M]

2. a) Simple aggregate price index 1000

×=∑∑

x

xn

12010048101211

5101312=×

++++

++++ a

a =14

b) Laspeyres Price Index 10000

×=∑∑

qp

qp nn

100)5(4)3(8)2(10)2(12)2(11

)5(5)3(10)2(14)2(13)2(12×

++++

++++=

= 120.9

The housewife’s expenditure on fish in year 2012 increased by 20.9% when

compared with 2011.

2M

1M

2M

1M

3a)

stem leaf

4 1 4 8

5 0 2 7 8

6 2 3 6 6 7 7 8

7 0 0 2 4 4 7

8 5 7

9 3

10 8

2M

b)

1M

1. a) P(only one of the two bulbs is in good

condition ,replacing the first bulb)

= 280

20

80

60××

= 0.375

b) P(only one of the two bulbs is in good

condition, with the first bulb not

returned into the box)

= 279

20

80

60××

= 0.3797

[1M]

Key: 4 1 means 41

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7

67

12

)24(2

1Median

=

=

=

th

th

5.57

2

5857

6

)24(4

1Q1

=

+=

=

=

th

th

74

18

)24(4

3Q3

=

=

=

th

th

1M

c.

75.32

)5.5774(2

35.57

)(2

3Qboundary Lower 131

=

−−=

−−= QQ

75.106

)5.5774(2

374

)(2

3Qboundary Upper 131

=

−−=

−−= QQ

Outlier = 108

1 M

2M

4a)

Interquartile range = 13 QQ −

= 74 -57.5 = 16.5

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5.

2M

6 a) Graph Year Quarter Unemployed 4-quarter Centred 4-quarter Deviation

3M

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9

school

leavers

moving

average

moving average

(t)

(y-t)

2010 1 22

2 12

43.75

3 110 43.625 66.375

43.50

4 31 45.25 -14.250

47.00

2011 1 21 52.00 -31.000

57.00

2 26 61.875 -35.875

66.75

3 150 70.375 79.625

74.00

4 70 75.250 -5.250

76.50

2012 1 50 76.000 -26.000

75.50

2 36 80.500 -44.500

85.50

3 146

4 110

b) seasonal variation

Year Quarter 1 Quarter 2 Quarter 3 Quarter 4

2010 66.375 -14.250

2011 -31.000 -35.875 79.625 -5.250

2012 -26.000 -44.500

Unadjusted

seasonal

variation

-28.500 -40.188 73.000 -9.750

Correction

factor

-1.3595 -1.3595 -1.3595 -1.3595

Seasonal

Variation

-27.14 -38.83 74.36 -8.39

C) The least square method

4822 =∑ x 66=∑ x 794182 =∑ y 784=∑ y 4627=∑ xy

12

)66(482

12

)784)(66(4629

2

−

−= = 2.664

1M

[1M]

[2M]

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10

−=

12

66664.2

12

784a =50.68

Regression line, y = 50.68+2.664x

For the first quarter of the year 2013, eT = 50.68+2.664(13) = 85.312

Unemployed school leavers for the first quarter of the year 2013

=85.312+ (-27.14) = 58.172 ≈ 59

Or

Average quarterly increment =7

625.435.80 −=5.2679

For the first quarter of the year 2013, eT = 80.5 + 3 x 5.2679 = 96.304

Unemployed school leavers for the first quarter of the year 2013

= 96.304 + (-27.14)= 69.164 ≈ 69

1M

1M

1M

1M

2M

1M

7a Histogram

b) 5.62)125(2

1

2

1==N

minutes58.30)1(30

45)125(2

1

30

2

1

Median,

=

−

+=

−

+= cf

FN

LMm

2M

x f fx 2fx Cumulative

frequency,F

26.5 6 6

28.5 12 18

29.5 27 45

30.5 30 75

31.5 18 93

32.5 14 107

33.5 9 116

34.5 4 120

37.5 5 125

5.3861=∑ fx 25.1199052 =∑ fx

89.30125

5.3861,Mean ===

∑∑

f

fxx

220.2)89.30(125

25.119905,Deviation Standard 2

2

=−=−= ∑∑∑

∑∑

f

fx

f

fxσ

1M

2M

c. Pearson coefficient of skewness 1M

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11

= deviation standard

median)3(mean −= 4189.0

2.220

)58.303(30.89=

−

The distribution is positively skewed or skewing to the right.

1M

7d)

.

The number of times a bus arrives at town B between 0830 hours and 0836

hours = 122-45 = 77

2M

8a) Let X be the number of books read by an adult

Then, )5(0PX ≈

a)

1404.0

!3

)5()3(

35

=

==−e

XP

b) In 5 years, 2555 =×=λ

1M

1M

1M

[

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12

0.1

!2

)25(

!1

)25(

!0

)25( -1

)]2()1()0([1)2(

225125025

=

++=

=+=+=−=>−−− eee

XPXPXPXP

1M

b)(i)

0.0228

0.9772 -1

2)P(z-1

2)P(z

10

5070)70(

=

=

<=

≥=

−≥=≥ zPXP

Hence, percentage of candidates = 0.0228 x 100% = 2.28%

(ii)

76.44

524.010

50

0.524z

0.70z)P(Z

70.010

50

70.0)(

=

−=−

∴

=

=≤

=

−≥

=≥

x

x

xZP

xXP

Hence, minimum marks required = 45

2M

1M

2M

1M

8c) Expected number, 8)(E == npx

Variance, Var (x) = npq = 1.6

8

6.1=

np

npq

q = 0.2 p = 0.8 n = 10 )8.0,10(≈X

(i) P( a teacher chosen at random owns a proton car) = 0.8

(ii) P( exactly 4 teachers from the group own Proton cars)

= P(X = 4)

= 4104

4

10 )2.0()8.0( −C = 0.005505

2M

1M

2M

Download - TRIAL STPM Mathematics M 2 (SARAWAK) SMK Paku

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