STPM 950/2

TRIAL STPM MATHEMATICS M (SELANGOR) –SMK Seafield, Subang

Section A [45 marks]

Answer all questions in this section.

1. The following table shows the amount of time spent by a random sample of 95 students

on their mobile phones per day.

Time spent per day, t minutes Number of students

50 <≤ t 11

105 <≤ t 18

2010 <≤ t 32

3020 <≤ t 18

5030 <≤ t 12

6050 <≤ t 4

(a) Calculate estimates of the mean, median and standard deviation of the time spent

per day on mobile phones by these students. [7 marks]

(b) Calculate Pearson’s coefficient of skewness. Comment on the shape of the

distribution. [3 marks]

2. A survey of children in a child care center found that 58% of the children are girls.

5.2% of the girls and 7.8% of the boys write using left hands. A child is selected at random.

(a) Find the probability that the child is a girl who writes using left hand. [2 marks]

(b) Find the probability that the child writes using left hand. [2 marks]

(c) If the child writes using left hand, find the probability that the child is a boy.

[2 marks]

3. The probability distribution of a discrete random variable X is given in the following

table.

x −1 0 1 2

( )xXP = 0.15 0.40 2k k

(a) Find the value of the constant k. [2 marks]

(b) Calculate )(XE and )(Var X . [5 marks]

STPM 950/2 2

4. A housing consultant believes that the number of houses sold in a region for a given

year is related to the mortgage rate in that period. He collected the following data.

Year Mortgage interest rate, X Housing sales index, Y

1990 12 80

1991 10 90

1992 8 105

1993 6 115

1994 7 125

1995 8 120

1996 10 115

1997 14 85

1998 13 70

1999 11 80

(a) Plot a scatter diagram for the above data. [2 marks]

(b) State, with a reason, whether the scatter diagram in (a) displays a positive or a

negative correlation. [2 marks]

(c) Calculate the coefficient of determination. Interpret your answer. [4 marks]

5. The following table shows the average prices and daily sales of fish, prawns and crabs

for January and February 2013.

Price (RM / kg) Total Sales (RM) Item

January February January February

Fish 5.68 5.99 198.80 209.65

Prawn 24.50 28.30 220.50 283.00

Crab 31.80 35.40 333.90 424.80

(a) Taking January 2013 as the base period, calculate the simple aggregate price

index for February 2013. Comment on your answer. [3 marks]

(b) Taking January 2013 as the base period and the total sales for February as the

weights, calculate the weighted average of price relatives for February 2013. Comment on

your answer. [3 marks]

STPM 950/2

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6. The following time series plot shows the quarterly revenues (RM’000,000) for a toy

production company from 1997 to 1999.

(a) Comment on the basic trend and the seasonal variations. [2 marks]

(b) State, with a reason, whether an additive or a multiplicative model is more

suitable to be used to decompose the time series.

[2 marks]

(c) The quarterly seasonal variations for the above data are given in the table below.

Quarter 1 2 3 4

Seasonal variation −0.6895 k −0.6427 2.0761

(i) Determine the seasonal variation for the second quarter, k. [2 marks]

(ii) Interpret the seasonal variation for the fourth quarter. [1 mark]

STPM 950/2 4

Section B [15 marks]

Answer any one question in this section.

7. A continuous random variable X is defined by

( ) ( )3

30

0

,0

,3

,1

3

>

≤<

≤

−=>

x

x

x

xkxXP

(a) Show that ( )27

81 =>XP . [3 marks]

(b) Find the cumulative distribution function of X, and sketch its graph. [3 marks]

Three independent observations of X are taken.

(c) Find the probability that at least one of the observations is greater than 1.

[4 marks]

729 independent observations of X are taken.

(d) Using normal distribution as an approximation, find the probability that more

than 229 of the observations are greater than 1. [5 marks]

8. The Ace Appliance Store sells a variety of electronic equipment and home appliances.

The following table shows the quarterly sales (×RM100,000) for the last four years.

Quarter Year

I II III IV

2009 5.3 4.1 6.8 6.7

2010 4.8 3.8 5.6 6.8

2011 4.3 3.8 5.7 6.2

2012 5.6 4.6 6.1 5.3

(a) Calculate the centred four-quarter moving averages. [2 marks]

(b) Using a multiplicative model, calculate the adjusted seasonal variation for each of

the four quarters. [4 marks]

(c) Deseasonalise the data and find the least squares regression equation using the

deseasonalised data. [6 marks]

(d) Forecast the sales of the store for the fourth quarter of year 2013. [3 marks]

STPM 950/2

5

ANSWER TRIAL STPM MATHEMATICS M (SELANGOR)–SMK Seafield, Subang

No. Answer scheme Marks Total

1(a)

(b)

Mean, 87.1895

5.1792==x minutes

Median, 78.151032

295.4710 =×

−+=m

Standard deviation,

25.508312 =∑ fx

38.1395

5.1792

95

25.508312

=

−=s

Pearson’s coefficient of skewness

( )

6928.0

38.13

78.1587.183

=

−=

The distribution is positively skewed.

M1A1

M1A1

B1

M1A1

M1

A1

B1

10 marks

2(a)

(b)

(c)

P(the child is a girl who writes using left hand)

= 0.052 × 0.58

= 0.03016

P(the child writes using left hand)

= 0.03016 + 0.078 × 0.42

=0.06292

P(the child is a boy|the child writes using left hand)

5207.0

06292.0

42.0078.0

=

×=

M1

A1

M1

A1

M1

A1

6 marks

3(a)

(b)

0.15 + 0.40 + 2k + k = 1

k = 0.15

( ) ( ) ( ) ( ) ( )15.0230.0140.0015.01 +++−=XE

= 0.45

( ) ( ) ( ) ( ) ( ) ( )15.0230.0140.0015.01 22222 +++−=XE

= 1.05

( ) ( )245.005.1Var −=X

= 0.8475

M1

A1

M1

A1

B1

M1

A1

7 marks

STPM 950/2 6

4(a)

(b)

(c)

Negative correlation.

As the mortgage interest rate increases the housing sales index

decreases.

∑ ∑∑ ==== 985,1043,99,10 2 yxxn

∑ ∑ == 9355,1005252 xyy

Coefficient of determination,

( ) ( )( )[ ]( ) ( )[ ] ( ) ( )[ ]22

2

2

9851005251099104310

98599935510

−⋅−

−=r

= 0.7136

Hence, 71.36% of the variation in the number of houses sold is

accounted for by the variation in the mortgage interest rate.

D2

B1

B1

B1

M1

M1

A1

B1

9 marks

5(a)

(b)

Simple aggregate price index for February 2013

44.112

10080.3150.2468.5

40.3530.2899.5

=

×++++

=

The price of the seafood has increased by 12.44% from January

to February 2013.

Weighted average of price relatives for February 2013

27.111

10080.42400.28365.209

80.42480.31

40.35283

50.24

30.2865.209

68.5

99.5

=

×++

×+

×+

×=

The price of the seafood has increased by 11.27% from January

to February 2013.

M1

A1

B1

M1

A1

B1

6 marks

6(a)

(b)

(c)(i)

(ii)

The time series has an increasing trend.

The revenue is the highest in the fourth quarter each year but

rather low in the first three quarters.

An additive model is more suitable because the amplitude of

the seasonal variations is almost constant as the trend rises.

7439.0

00761.26427.06895.0

−=

=+−+−

k

k

2.0761 means that the revenue in the fourth quarter is

RM2.0761 million above the trend value.

B1

B1

B1

B1

M1

A1

B1

7 marks

STPM 950/2

7

7(a)

(b)

(c)

(d)

( )

27

1

1033

=

=−

k

k

( ) ( )31327

11 −=>XP

27

8=

( ) ( )

−−=

,1

,327

11

,0

3xxF

3

30

0

>

≤<

≤

x

x

x

Let X = the number of observations greater than 1.

Then,

27

8,3B~X .

( ) ( )011 =−=≥ XPXP

6515.0

27

191

3

=

−=

27

8,729B~X .

Hence, ( )152,216N~ɺX .

( ) ( )5.229229 >=> XPXP

( )1368.0

095.1

152

2165.229

=

>=

−>=

ZP

ZP

M1

M1

A1

B1

D2

B1

B1

M1

A1

B1

B1

B1

M1

A1

15 marks

2

1.5

1

0.5

-0.5

-1 1 2 3 4

0 3

1

( )xFy =

y

x

STPM 950/2 8

8(a)

(b)

Year

Quarter

Sales

(×RM100,000),

Y

4-pt

moving

average

Centred

moving

average,

T

S=Y/T

1 5.3

2 4.1

3 6.8 5.725 5.6625 1.2009 2009

4 6.7 5.600 5.5625 1.2045

1 4.8 5.525 5.3750 0.8930

2 3.8 5.225 5.2375 0.7255

3 5.6 5.250 5.1875 1.0795 2010

4 6.8 5.125 5.1250 1.3268

1 4.3 5.125 5.1375 0.8370

2 3.8 5.150 5.0750 0.7488

3 5.7 5.000 5.1625 1.1041 2011

4 6.2 5.325 5.4250 1.1429

1 5.6 5.525 5.5750 1.0045

2 4.6 5.625 5.5125 0.8345

3 6.1 5.400 2012

4 5.3

Q1 Q2 Q3 Q4

2009 1.2009 1.2045

2010 0.8930 0.7255 1.0795 1.3268

2011 0.8370 0.7488 1.1041 1.1429

2012 1.0045 0.8345

Mean

Seasonal

Variation

0.9115 0.7696 1.1282 1.2247

Adjusting

factor 0.9916 0.9916 0.9916 0.9916

Adjusted

Seasonal

Variation

0.9038 0.7631 1.1187 1.2144

Adjusted seasonal variations are:

Q1 Q2 Q3 Q4

0.904 0.763 1.119 1.214

M1A1

(column

5)

B1

(column

6)

B1

M1

A1

STPM 950/2

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(c)

(d)

t Y S Deseasonalised

Series, y=Y/S

1 5.3 0.9038 5.8641

2 4.1 0.7631 5.3728

3 6.8 1.1187 6.0785

4 6.7 1.2144 5.5171

5 4.8 0.9038 5.3109

6 3.8 0.7631 4.9797

7 5.6 1.1187 5.0058

8 6.8 1.2144 5.5995

9 4.3 0.9038 4.7577

10 3.8 0.7631 4.9797

11 5.7 1.1187 5.0952

12 6.2 1.2144 5.1054

13 5.6 0.9038 6.1961

14 4.6 0.7631 6.0280

15 6.1 1.1187 5.4528

16 5.3 1.2144 4.3643

∑ ∑∑ ==== 7076.85,1496,136,16 2 yttn

∑ = 6733.719ty

( ) ( )( )( ) ( )2136149616

7076.851366733.71916

−

−=b

02600.0−=

( )

−−

=16

13602600.0

16

7076.85a

58.5=

Least squares regression equation is ty 0260.058.5 −= .

4th

quarter of 2013, t = 20.

( )200260.058.5 −=y

= 5.06

2144.106.5ˆ ×=Y

= 6.145

Forecast sales = RM6.145 (×100,000)

M1A1

(column

4)

B1

M1

M1

A1

M1

M1

A1

15 marks