1

TRIAL STPM MATHEMATICS M (JOHOR) –SMK TunHO

Section A [ 45 marks]

Answer all questions in this section

1. Tan and Lim, solicitors, monitored the time spent on consultations with a random sample of 120 of their clients. The times, to the nearest minute, are summarised in the following table :

Time ( minutes) Number of clients

10-14 15-19 20-24 25-29 30-34 35-44 45-59 60-89 90-119

2 5 17 33 27 25 7 3 1

Total 120

a) calculate the median and quartiles of this distribution. [ 5 marks] b) comment on the skewness of the distribution. [ 1 mark ] c) calculate the mean of the above data [ 2 marks] d) the solicitors are undecided whether to use median or mean to summarise the data. State, giving a reason, which you would recommend them to use. [ 2 marks]

2. The events A and B are such that P(A B ) = 0.4, 0.12B)P(A ,25.0)( =∩=ABP

a) calculate the value of P(B). [ 2 marks] b) Give a reason why A and B are not independent. [ 1 mark ]

c) calculate the value of ) ( 'BAP ∩ [ 3 marks]

3. The probability that New Year’s Day is on Sunday in a randomly chosen year is 8

1.

a) 12 years are chosen randomly. Find the probability that at least 3 of these years have New Year’s day on a Sunday. [ 4 marks]

b) 60 years are chosen randomly. Use a suitable approximation to find the probability that more than 6 of these years have New Year’s Day on a Sunday. [ 5 marks]

4. The data of 10 paired observations are summarised as follows:

∑∑∑ ∑ ∑ ===== 120726.xy ,59920y ,300831x 600,y ,1480 22x

Find the equation of the least square line in the form of y = a + bx, where a and b

are constants. [ 3 marks]

2

5.

2010 2011 Sauce

Price (RM per unit) Quantity / unit Price (RM per unit) Quantity / unit

Premium Standard Budget

23.60

15.80

10.50

35 45 50

25.20

17.10

11.30

22 38 60

The above table shows the price and quantity of 3 types of sauce sold by a

supermarket for the years 2010 and 2011. By using 2010 as the base year, find the a) simple aggregate price index and interpret. [ 3 marks] b) Laspeyres price index and Paasche quantity index. [ 4 marks]

6. The following table shows the fixed deposits is in a finance company in the years 2009 to 2012.

Fixed deposits ( RM’ 000 ) Quarter Year 1 2 3 4

2009 13 20 35 22

2010 16 28 45 24

2011 20 32 48 32

2012 25 36 55 38

a) Plot a time series, and comment on the trend of the time series. [ 4 marks] b) State whether it is appropriate to use the linear regression method to forecast the amount of fixed deposits. Give a reason. [ 2 marks]

c) calculate the centred four-quarter moving averages for this time series. [ 4 marks]

3

Section B [ 15 marks]

Answer ONLY one question in this section 7. The continuous random variable, t hours, the time taken to perform a particular task, has

the probability density function

otherwise 0

1.0t 0.6 t)-9c(1

0.6t0 10

)(

2

<≤

<≤

=

ct

tf

Where c is a constant. a) find the value of c and sketch the graph of this distribution. [ 7 marks] b) Find the expected time ( in minutes). [ 4 marks] c) determine the probability that the time will be more than 48 minutes. [ 4 marks] 8. A study comparing the amount if advertising time on TV per week for a product and the

number of sales per week for the same product was conducted. The result over 8 weeks are given below:

Advertising time ( minutes), x

10 12 14 15 16 17 22 20

Sales (thousands), y

2.3 2.8 3.1 k 2.9 3.2 5.0 4.0

a) find the coordinates of the point though which the regression line y on x and that of x on y both pass. Give your answer in terms of k. [ 3 marks]

b) given that the regression line of y on x is y = 0.197x + 0.184, find k to the nearest integer. Hence, find the Pearson correlation coefficient r between advertising time and sales per week. [ 7 marks]

c) plot a scatter diagram of y against x.. [ 2 marks] d) state whether it is appropriate to estimate the sales per week if the advertising time on TV is increased to 30 minutes per week. Give a reason. [ 2 marks]

e) What is the effect on r if the advertising time was in hours instead of minutes. [ 1 mark]

4

MARKING SCHEME

PRA PENTAKSIRAN PENGGAL 2 TAHUN 2013

SMK TUN HUSSEIN ONN, KLUANG, JOHOR.

PRA U ATAS

MATHEMATICS M ( PAPER 2)

NO SCHEME MARKS

1.

[10] a) )5(27

57605.29

2

−+=Q

= 30.06 minutes

)5(33

24305.24

1

−+=Q

= 25.41 minutes

)10(25

84905.34

3

−+=Q

= 36.9 minutes

b) positively skewed / skewed to the right

c)

120

)1(5.104)3(5.74)7(52)25(5.39)27(32)33(27)17(22)5(17)2(12 ++++++++

=mean

=120

5.3917

=32.65 minutes

d) Median.

Distribution is skewed

M1

A1

M1(Q1@

Q3)

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2.

[6]

a) 4.0)(

12.0=

BP

P(B) = 0.3

M1

A1

5

b) )()( BPABP ≠

c) 25.0)(

12.0=

AP

P(A) = 0.48

)()()( ' BAPAPBAP ∩−=∩

= 0.48 – 0.12

= 0.36

A1

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3.

[9]

a) X : number of years have New Year’s Day on Sunday

)8

1 ,12(~ BX

)2()1()0(1)3( =−=−=−=≥ XPXPXPXP

102

2

12111

1

12120

0

12 )8

7()

8

1()

8

7()

8

1()

8

7()

8

1(1 CCC −−−=

= 1- 0.2014 – 0.3453 – 0.2713

= 0.182

b) 562.2or 5625.6)8

7)(

8

1(60 and 5.7)

8

1(60 ===== npqnpqnp

6.5625) ,5.7(~ NX

P( X> 6) = )5.6( ≥XP

= )5625.6

5.75.6(

−≥ZP

= )3904.0( −≥ZP

= )3904.0( ≤ZP

= 0.6517

B1(or

implied)

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A1

B1(seen

or implied

M1(conti..

Correction

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4.

[3]

10

1480300831

10

)600)(1480(120726

2

−

−=b

= 0.3903

)10

1480(3903.0

10

600−=a

= 2.2356

2356.23903.0 += xy

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A1

6

5.

[7] a) 100

5.108.156.23

3.111.172.25X

++

++

= 1009.49

6.53X

= 107.4

The price has increased by 7.4% in the year 2011 as compared to year

2010.

b) 100)50(5.10)45(8.15)35(6.23

)50(3.11)45(1.17)35(2.25XLp

++

++=

= 1002062

5.2216X

= 107.5

100)50(3.11)45(1.17)35(2.25

)60(3.11)38(1.17)22(2.25XPq

++

++=

= 1005.2216

2.1882X

= 84.92

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6.

[10]

a) scale, label , points connected

14≥ points correct

all correct

Increasing trend or

Maximum on 3rd quarter of each year

b) Yes.

The trend is linear

c) 4-quarter moving average :

22.5 23.25 25.25 27.75 28.25 29.25 30.25

31 33 34.25 35.25 37 38.5

Centred 4-quarter moving average:

22.875 24.25 26.5 28 28.75 29.75 30.625

32 33.625 34.75 36.125 37.75

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B1

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7

7.

[15] a) 1 t)dt-9c(1 10ct

6.0

0

1

0.6

2 =+∫ ∫dt

12

99

3

101

6.0

26.0

0

3

=

−+

ctct

ct

1)62.14.5()2

99(72.0 =−−−+ cc

ccc

1.44c = 1

c = 36

25 = 0.694

f(t)

b) dtttdtttE )4

25

4

25(

18

125 )( 2

1

6.0

36.0

0−+= ∫∫

=

6.0

0

4

)4(18

125

t +

1

6.0

32

12

25

8

25

− tt

= ))12

)6.0(25(

8

)6.0(25()

12

25

8

25()0)6.0(

72

125(

324 −−−+−

= 0.225 + 45.0125.124

25+−

= 0.592 hour

= 35.52 minutes

c) 48 minutes = 0.8 hour

P (T> 0.8) = dtt)1(4

251

8.0−∫

=

1

8.0

2

8

25

4

25

− tt

= 2)8.0(8

25)8.0(

4

25

8

25

4

25+−−

= 0.125or 8

1

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D1(label&

shape)

D1(points

0.6 and 1

or 2.5 )

D1(all

correct)

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A1

B1(or imp.

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2.5

0.6 1

t

8

8.

[15] a) 75.15

8

126==x

8

452.39.21.38.23.2 +++++++=

ky

= 8

3.23 k+

( 15.75, )8

3.23 k+

b) y = 0.197x+0.184

184.0)75.15(197.08

3.23+=

+ k

k = 2.994

= 3.0

x y x2 y2 xy

10

12

14

15

16

17

22

20

2.3

2.8

3.1

3.0

2.9

3.2

5.0

4.0

100

144

196

225

256

289

484

400

5.29

7.84

9.61

9.00

8.41

10.24

25

16

23

33.6

43.4

45

46.4

54.4

110

80

435.8xy 39.91y 2094x 26.3y ,126 22 =∑=∑=∑=∑=∑ x

8

)3.26(39.91)(

8

)126(2094(

8

)3.26(1268.435

22

−−

−=r

= 0..9287

c) scale and label

correct points

d) No, cannot estimate,

beyond the range of the sample data

e) Change in the unit of measurement has no effect on the value of r

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