kepong baru 2014 (a)

CONFIDENTIAL*

STPM Trial 950/2 [Turn over

*This question paper is CONFIDENTIAL until the examination is over. CONFIDENTIAL*

MARKING SCHEME MATHS (M) 950/2 2014

1 In an agricultural experiment, the gains in mass (in kilogram) of 100 cows during a certain

period were recorded as follows:

Gain in mass

(kg) Frequency

5 – 9 4

10 – 14 12

15 – 19 29

20 – 24 32

25 – 29 13

30 – 34 7

35 – 39 3

(a) Calculate the mean and standard deviation of the gains in mass of 100 cows.

[5 marks]

(b) Find the percentage of the gains in mass of cows in the range of one standard

deviation from the mean. [4 marks]

Solution:

1a.

b.

Mean = = 20.55 kg

Standard Deviation = 2)55.20(

100

46545

= 6.5687 kg

= 6.569 kg or 6.57 kg

)5687.655.20(1x ( 13.981 , 27.119 )

055.69)135

5.24119.27(3229)12

5

981.135.14(

% of the gains in mass of cows =

= 69.005% or 69.0% or 69.01%

M1 A1

B1 M1

A1

M1

M1 M1

A1

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2

2 A container contains 5 green wooden blocks, 7 red wooden blocks and 8 blue wooden blocks.

The blocks are identical except for their colours. John randomly selects wooden blocks from the

container until a green block is selected. If John does not select a green block, he will put it back into

the container and repeat the process until a green block is obtained.

(a) Show that the probability that the second piece selected is red is 21

80 [2 marks]

(b) Find the probability that either the second piece selected is red or the fourth piece selected is

green. [6 marks]

Solution

2 (a) P(A) = P(second block is red)

= P(non-green, red)

= M1

= A1

(b) P(B) = P(fourth is green)

= P(G’G’G’G)

= M1

= A1

M1

A1

M1

A1

CONFIDENTIAL*

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3

3 A student takes a certain route to travel to school each day. The travel times (in minutes) for

the route can be assumed to be normally distributed with mean 23 and standard deviation 4.

(a) Find the probability that a journey will take between 20 and 25 minutes. [3 marks]

(b) If the student begins his journey at 7.05 a.m. each day and the first class starts at 7.30

a.m., find the probability he will be late to school for 2 out of 5 consecutive days. [5 marks]

Solution 3 (a) X = Travel times X ~N(23, 42)

4

2325

4

2320P)2520(P ZX

)5.075.0(P Z

)75.0(P)5.0(P

= 0.69146 – 0.22663

46483.0

(b)

4

2325P)25(P ZX

)5.0(P Z

30854.0

Y = Number of days a student late to school

P(the student is late for 2 out of 5 consecutive days)

32 )69146.0()30854.0)(2

5(

= 0.31471985

= 0.3147 or 0.315

M1

M1 A1 M1

A1

B1

M1

A1

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4

4 A biscuit company has carried out a survey on the age of its customers and the weekly

consumption of its new product of biscuits. The table below shows the weekly consumption, y of the

company product and the age, x of the customer.

x (year) 11 13 19 26 32 38 46 51 58 62

y (gram per week) 42 48 60 70 66 62 49 34 16 8

(a) Determine the least square regression line of on x [5 marks]

(b) Find the non linear correlation curve between x and y. [2 marks]

(c) Determine the consumption of the biscuits for a customer of age 65. Justify your

answer. [1 mark]

Solution

4 (a)

x 11 13 19 26 32 38 46 51 58 62

3.8182 3.6923 3.1579 2.6923 2.0625 1.6316 1.0652 0.6667 0.27586 0.12903

M1

Let u = x and v =

n = 10 , Σ u = 356 Σ u 2= 1,5720 Σ v = 19.19159 Σ v

2 = 54.02041445 Σ uv = 454.99991

B1

M1

= -0.07491453

a =

= 4.586113289 M1

The least square regression line is A1

(b) From

Then

M1

The non linear correlation curve is A1

(c) When x = 65, weekly consumption cannot be estimated because it is outside the range. B1

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5

5 The following table shows the price and quantity for three different brands of 1 kg packed

milk powder sold at a hypermarket.

Brand of Milk Powder

Price (RM) Quantity (Thousands)

2010 2012 2010 2012

A 20 30 30 40

B 30 36 20 15

C 20 30 30 40

Taking 2010 as the base year, calculate

(a) the Paasche quantity index for the year 2012. [2 marks]

(b) the Laspeyres quantity index for the year 2013 if the quantity of milk powder brand A,

brand B and brand C sold increased by 5%, 10% and 15% respectively compared to the year 2012.

[4 marks]

Solutions:

5 (a) The Paasche’s quantity index = 1000

n

nn

pq

pq

M1

= 116.67 A1

(b) The Laspeyres’ quantity index = 10000

0

pq

pqn

The quantity of milk powder sold in the year 2013:

Brand A = 42 000 kg

Brand B = 16 500 kg M1 A1

Brand C = 46 000 kg

M1

= 125.28 A1

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6

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

company from 1997 to 1999.

(a) Comment on the trend. [1 mark]

(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]

Solution

6(a)

(b)

(c)(i)

(ii)

The time series has an increasing trend.

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

B2

M1

A1

B1

6 marks

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7

Section B [15 marks]

Answer any one question in this section.

7 (a) A discrete random variable X has the probability distribution function

, where c is a constant.

(i) Determine the value of the constant c. [2 marks]

(ii) Calculate the mean and variance of X. [5 marks]

(b) A continuous random variable X has the cumulative distribution function as follows.

2

0, 1

( ) , 1 2

1, 2

x

F x ax bx x

x

(i) Show that 1

6a and find the value of b. [3 marks]

(ii) Find 1

( 1 )2

P X . [2 marks]

(iii) Find the value of m such that 1

( )2

P X m . [3 marks]

Solution

7

(a) (i)

M1

A1

(ii)

X = k 0 1 2

P(X = k)

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8

= M1

= A1

= M1

= 1

= M1

= A1

(b) (i) 2( ) , 1 2F x ax bx x

( 1) 0F a b b a

and F(2) = 2a – 4b = 1 M1

2 4 1a a

1

6a M1

1

6b A1

(ii) 1 1

( 1 ) 1 ( 1 )2 2

P X P X

= 21 3 1 3 31 [ ( ) ]

6 2 6 2 8 or 0.375 M1 A1

(iii) 1

( )2

P X m

21 1 1

6 6 2m m M1

2 3 0m m

M1

A1

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9

8 The table below shows the income tax collected (in million RM) quarterly and the centred

moving averages from year 2011 to year 2013.

Year Quarter Income tax (in million RM) Centred moving average

1 12

2011 2 25

3 20 29.125

4 55 31.625

1 21 34.750

2012 2 36 38.375

3 34 42.000

4 70 45.500

1 35 48.000

2013 2 50 51.875

3 40

4 95

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

four quarters. Write down your answers correct to three decimal places. [4 marks]

(b) Obtain a seasonally adjusted time series. [2 marks]

(c) Find the equation of the trend line using the least squares method. [4 marks]

(d) Hence, predict the income tax collected in the first quarter of the year 2014 such that it

exceeds the current trend by 10%. [5 marks]

Solution

8 (a)

Quarter Income tax

Y(in million

RM)

Centred

moving

average, T

Y/T

1 12

2 25

3 20 29.125 0.686695

4 55 31.625 1.739130

1 21 34.750 0.604317

2 36 38.375 0.938111

3 34 42.000 0.809524

4 70 45.500 1.538462

1 35 48.000 0.729167

2 50 51.875 0.963855

3 40

4 95

M1

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10

(b)

Quarter t Income tax

Y(in million

RM)

Seasonal

Index, S S

Y

1 1 12 0.666 18.02

2 2 25 0.950 26.32

3 3 20 0.747 26.77

4 4 55 1.637 33.60

1 5 21 0.666 31.53

2 6 36 0.950 37.89

3 7 34 0.747 45.52

4 8 70 1.637 42.76

1 9 35 0.666 52.55

2 10 50 0.950 52.63

3 11 40 0.747 53.55

4 12 95 1.637 58.03

M1 A1

Quarter

Year

1

2

3

4

2011 - - 0.686695 1.739130

2012 0.604317 0.938111 0.809524 1.538462

2013 0.729167 0.963855 - -

Mean 0.666742 0.950983 0.7481095 1.638796

Adjusting

Factor

0.998844 0.998844 0.998844 0.998844

Adjusted

Seasonal

Variation

0.665971

= 0.666

(3 d.p.)

0.949884

= 0.950

(3 d.p.)

0.747245

= 0.747

(3 d.p.)

1.636902

= 1.637

(3 d.p.)

M1

M1

A1

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11

(c) The equation of the regression line of y on t is y = a + bt

B1 M1

= 3.504440559

a =

M1

The equation of the regression line of y on x is y = 17.15 + 3.504t A1

(d) When t = 13, y = 17.15 + 3.504(13) B1 M1

= 62.702

Yest = 62.702 x 0.666 M1

= 41.759532

The income tax collected in the first quarter of the year 2014 such that it

exceeds the current trends by 10% = 41.759532 x 1.1 M1

= RM 45.9354852 million A1

kepong baru 2014 (a)

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