stpm trial 2009 maths2 q&a (n9)

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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2/14

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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3/14

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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4/14

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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5/14

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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6/14

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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7/14

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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8/14

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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9/14

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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10/14

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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11/14

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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12/14

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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13/14

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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14/14

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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stpm trial 2009 maths2 q&a (n9)

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