TRIAL STPM Mathematics M TgDurahSEREMBAN

This question paper consists of 5 printed pages.

Section A [45 marks]

Answer all questions in this section.

1. The electricity bills (in RM) of 60 houses in a town for a certain month are summarized in

the table below

Electricity bill (in

RM) Number of houses

60 < x 80 3

80 < x 100 10

100 < x 120 14

120 < x 140 20

140 < x 160 9

160 < x 200 4

(a) Display the data on a histogram. [3 marks]

(b) Calculate the mean and mode of the electricity bills. [4 marks]

(c) Hence, comment on the type of distribution displayed by the data. [2 marks]

2. A survey was carried out among the Form 6 students in a school. The percentage of

students studying Economics is 25% and the percentage of students studying Mathematics

is 50% . Out of those studying Economics, 60% of them are studying Mathematics.

A student is selected at random. Find the probability that

(a) the student is studying both Mathematics and Economics [4 marks]

(b) the students is studying Economics but not Mathematics. [3 marks]

3. A discrete random variable X has the following probability distribution function

, 2, 12

1, 1,2

12

0 ,

mx

x

xf x x

otherwise

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

(b) Find Var (X) [4 marks]

(d) If Y = 2X + 3, calculate the standard deviation of Y [3 marks]

2

4. The marks for Paper 1, x, and Paper 2, y , of a Statistics test obtained by eight students are

summarized as follows.

423x , 470y , 2 24479x ,

2 29450y and 26520xy

Calculate

(a) the Pearson’s correlation coefficient and comment on your answer. [3 marks]

(b) the coefficient of determination and explain briefly on the result obtained. [2 marks]

5. The following table shows the price per unit and the total sales of three brands of

electrical products sold by an electrical company.

Electrical Item

Prices per unit(RM) Total Sale ( RM 103)

2008 2010 2008 2010

X 80 85 24.8 24.65

Y 150 156 63.0 71.76

Z 1350 1420 243.0 305.30

(a) By using 2008 as the base year, calculate the Laspeyres quantity index and Paasche

price index for the year 2010 [5 marks]

(b) State with reasons whether price index or quantity index gives a clearer picture on

the growth of the electrical company. [2 marks]

6. The following data shows the number of tourist in the years 2008, 2009 and 2010

Year 2008 2009 2010

Quarter 1 2 3 4 1 2 3 4 1 2 3 4

Number of tourist

(Thousands) 25 46 37 24 30 53 40 29 37 58 45 36

(a) Plot the time series and comment on the appropriateness of a linear trend. [4 marks]

(b) Find the seasonal variation for each quarter by using the additive model. [4 marks]

3

Section B [15 marks]

Answer any one question in this section.

7. Box X contains 4 blue marbles, 2 red marbles and 6 green marbles. Box Y contains

3 blue marbles and 2 green marbles.

A game is played that involves rolling a fair die followed by drawing a marble from either

box X or box Y. If the number shown on the die is greater than 4, then a marble is drawn

from box X, and if otherwise, a marble is drawn from box Y.

(a) Find the probability that

(i) a blue marble is drawn

(ii) the number shown on the die is greater than 4 given that a blue marble is drawn.

(iii) a blue or a red marble is drawn. [10 marks]

(b) The same game is played but with two marbles drawn from either box X or box Y.

Find the probability that both marbles are blue if the first marble is drawn

(i) with replacement

(ii) without replacement. [5 marks]

8. The table shows the advertising expenses and sales revenues of a company selling

computers for the first seven months of 2009.

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

(b) Find the equation of the least square regression line y = a + bx , where a and b are

constants. Interpret the regression coefficient, b, obtained. [7 marks]

(c) Estimate the sale revenue when RM 5.6 million is spent advertising [2 marks]

(d) Determine the Spearman’s coefficient of rank correlation between the advertising

expenses and sales revenues. Interpret your answer. [4 marks]

Advertising Expense

(RM million), x 3 6 5 4 6 10 9

Sales Revenue

(RM million), y 18 20 21 25 26 28 29

4

MATHEMATICAL FORMULAE

Summary Statistics

For ungrouped data

1

2 100

100

r r

th

r

x x kif r n is an integer

k percentilek

x if r n is not an integer

Standard Deviation

2 2

2x x xx

n n

For grouped data

1100

n

kth

k

k

kF

k percentile L cf

Standard Deviation

2 2

2f x x fxx

f f

Probability Distribution

Binomial Distribution

1 , 0,1,2,...,n xx

nP X x p p x n

x

Poisson Distribution

, 0,1,2,...,!

xeP X x x

x

Correlation and regression

Pearson correlation coefficient

Spearman rank correlation coefficient

Least squares regression line

y = a + bx ; ,

x yxy

nr

x yx n y

n n

2 2

2 2

2

1

2

6

11

n

i

is

d

rn n

x yxy

nb

xx

n

2

2

a y bx

5

END OF QUESTION PAPER

MARKING SCHEME

Section A

1. The electricity bills (in RM) of 60 houses in a town for a certain month are summarized in

the table below

Electricity bill (in

RM) Number of houses

60 < x 80 3

80 < x 100 10

100 < x 120 14

120 < x 140 20

140 < x 160 9

160 < x 200 4

(a) Display the data on a histogram. [3 marks]

(b) Calculate the mean and mode of the electricity bills. [4 marks]

(c) Hence, comment on the type of distribution displayed by the data. [2 marks]

|

60

|

80

|

100

|

120

|

140

|

160

|

180

|

200

Electricity Bills (RM)

Frequency

5

10

15

20

D1 – Uniform scale,

axes labelled correctly

D1 – Correct bar height

D1 – Correct boundaries used

6

(b)

Electricity bill (in

RM) Midpt,m f fm

60 < x 80 70 3 210

80 < x 100 90 10 900

100 < x 120 110 14 1540

120 < x 140 130 20 2600

140 < x 160 150 9 1350

160 < x 200 180 4 720

Total 60 7320

Mean 7320

12260

RM M1A1

Modal Class = 120 < x 140

Mode 1

1 2

6120 20 127.06

6 11

dL c RM

d d

M1 A1

(c) Negatively skewed. B1

since mean < mode OR mean – mode = -5.06 < 0 B1

2. A survey was carried out among the Form 6 students in a school. The percentage of

students studying Economics and Mathematics are 25% and 50% respectively. Out of

those studying Economics, 60% of them are studying Mathematics.

A student is selected at random. Find the probability that

(a) the student is studying both Mathematics and Economics [4 marks]

(b) the students is studying Economics but not Mathematics. [3 marks]

Scheme

Let E = Event of getting a student studying Economics

M = Event of getting a student studying Mathematics

(a) Given P(E) = 0.25 , P(M) = 0.50 and P(M|E) =0.60

|P M E P M E P E

= 0.60 0.25

= 0.15 M1 A1

(b) 'P E P E M P E M

0.15P E M P M E

'P E M P E P E M B1 or equivalent

= 0.25 – 0.15 M1

B1 – Stating the probabilities correctly

OR can be seen in tree diagram

B1 – Writing the correct

mathematical statement

7

= 0.1 A1

3. A discrete random variable X has the following probability distribution function

, 2, 12

1, 1,2

12

0 ,

mx

x

xf x x

otherwise

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

(b) Find Var (X) [4 marks]

(d) If Y = 2X + 3, calculate the standard deviation of Y [3 marks]

Scheme

(a) 1 1 1 2

12 2 1 2 12 12

m m

M1

1 11

4 3 6 4

7 51

12 12

m m

m

1m A1

(b)

x -2 -1 1 2

f(x) 1

4

1

3

1

6

1

4

1 1 1 1 1

2 1 1 24 3 6 4 6

E X xf x

B1

2 2 2 22 2 1 1 1 1 5

2 1 1 24 3 6 4 2

E X x f x

B1

22Var X E X E X

25 1 17 89

2 // // 2.4722 6 36 36

M1A1

(c ) Y = 2X + 3

Var Y = 22 Var (X) 89 89

436 9

M1

8

Std. Dev of Y 89

3.1459

M1A1

4. The marks for Paper 1, x, and Paper 2, y , of a Statistics test obtained by eight students are

summarized as follows.

423x , 470y , 2 24479x ,

2 29450y and 26520xy

Calculate

(a) the Pearson’s correlation coefficient and comment on your answer. [3 marks]

(b) the coefficient of determination and explain briefly on the result obtained. [2 marks]

Scheme

(a)

2 22 2

n xy x yr

n x x n y y

2 2

8 26520 423 470

8 24479 423 8 29450 470

r

=0.8469 M1A1

There is a definite positive correlation between the marks obtained for Paper 1and Paper 2.

Student with high marks in Paper 1 is most likely to have high marks in Paper 2. B1

(b) Coefficient of determination = r2 = 0.7172 A1

The proportion of change in the Paper 2 marks that is attributable to the

Paper 1 marks is 71.72% B1

5. The following table shows the price per unit and the total sales of three brands of

electrical products sold by an electrical company.

Electrical Item

Prices per unit(RM) Total Sale ( RM 103)

2008 2010 2008 2010

X 80 85 24.8 24.65

Y 150 156 63.0 71.76

Z 1350 1420 243.0 305.30

(a) By using 2008 as the base year, calculate the Laspeyres quantity index and Paasche

price index for the year 2010 [5 marks]

(b) State with reasons whether price index or quantity index gives a clearer picture on

the growth of the electrical company. [2 marks]

Scheme

Electrical

Item

Prices per unit(RM) Total Sale ( RM 103) q0 qn

2008, p0 2010, pn 2008, p0q0 2010, pnqn

X 80 85 24.8 24.65 310 290

Y 150 156 63.0 71.76 420 460

Z 1350 1420 243.0 305.30 180 215

B1 – values of q0 and qn

9

Laspeyres quantity index0

0 0

100n p

p

q

q

290 80 460 150 215 1350100

310 80 420 150 180 1350

115.61

M1A1

Paasche price index0

100n n

n

q

q

p

p

385 290 156 460 1420 215 24.65 71.76 305.30 10100 100

80 290 150 460 1350 215 80 290 150 460 1350 215OR

105.04 M1 A1

Quantity index gives a clearer picture on the growth of the electrical company since it

enable comparison to be made by fixing the price (weights) in 2008 B1B1

6. The following data shows the number of tourist in the years 2008, 2009 and 2010

Year 2008 2009 2010

Quarter 1 2 3 4 1 2 3 4 1 2 3 4

Number of tourist

(Thousands) 25 46 37 24 30 53 40 29 37 58 45 36

(a) Plot the time series and comment on the appropriateness of a linear trend. [4 marks]

(b) Find the seasonal variation for each quarter by using the additive model. [4 marks]

Scheme

Number of

tourist (‘000)

20

40

60

x

|

Q1

|

Q2

| | | | | | | | | |

x

x

x

x

x

x

x

x

x

x

x

D1 – Uniform scale,

axes labelled correctly

D1 – All points plotted correctly

D1 – Straight lines joining all the points

Linear trend is appropriate for the given time series

data because there is a clear increasing trend. B1

10

Year

Quarter

Number of

Tourist Y

Centred 4-quarter moving average

(T) Y - T

2008

1 25 2 46 3 37 33.625 3.375 4 24 35.125 -11.125

2009

1 30 36.375 -6.375 2 53 37.375 15.625 3 40 38.875 1.125 4 29 40.375 -11.375

2010

1 37 41.625 -4.625 2 58 43.125 14.875 3 45 4 36

Year Quarter

1 2 3 4

2008 - - 3.375 -11.125

2009 -6.375 15.625 1.125 -11.375

2010 -4.625 14.875 - -

Total -11.00 30.50 4.50 -22.50

Mean -5.50 15.25 2.25 -11.25 0.075

(−) Adjustment 0.1875 0.1875 0.1875 0.1875

Seasonal Variation -5.6875 15.0625 2.0625 -11.4375 0

Year

M1: Find the 4-centred moving average

M1: Find the mean

A1: Adjusting Factor

A1: Correct values of the seasonal variation

11

Section B 7. Box X contains 4 blue marbles, 2 red marbles and 6 green marbles. Box Y contains

3 blue marbles and 2 green marbles.

A game is played that involves rolling a fair die followed by drawing a marble from either

box X or box Y. If the number shown on the die is greater than 4, then a marble is drawn

from box X, and if otherwise, a marble is drawn from box Y.

(c) Find the probability that

(i) a blue marble is drawn

(ii) the number shown on the die is greater than 4 given that a blue marble is drawn.

(iii) a blue or a red marble is drawn. [10 marks]

(d) The same game is played but with two marbles drawn from either box X or box Y.

Find the probability that both marbles are blue if the first marble is drawn

(i) with replacement

(ii) without replacement. [5 marks]

Scheme

Let F = Event that the outcomes of rolling a fair die is greater than 4

F’= Event that the outcomes of rolling a fair die is less than or equal to 4

(a)

(i) P(B) = P(FB) + P(F’B) B1 (ii)

|

P F BP F B

P B

B1

= 2 4 4 3

6 12 6 5 M1

2 4

6 1223

45

M1

23

// 0.511145

A1 5

23 A1

(iii ) P B R P B P R B1

23 2 2

45 6 12

M1

17

30

A1

F

F’

B

R

G

B

G

B1 Tree diagram

with correct

probabilities

(can be implied)

12

(b)(i)

2 'P Blues P F B B P F B B

2 4 4 4 3 3

6 12 12 6 5 5 M1

187

675 A1

(ii)

2 'P Blues P F B B P F B B

2 4 3 4 3 2

6 12 11 6 5 4 M1

38

165 A1

B1 Tree diagram

with correct

probabilities

(can be implied)

F’

2

6

4

6

F

B

R

G

B

G

3

5

2

5

B

R

G

4

12

2

12

6

12

B

G

3

5

2

5

F’

2

6

4

6

F

B

R

G

B

G

3

5

2

5

B

R

G

3

11

B

G

2

4

13

8. The table shows the advertising expenses and sales revenues of a company selling

computers for the first seven months of 2009.

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

(f) Find the equation of the least square regression line y = a + bx , where a and b are

constants. Interpret the regression coefficient, b, obtained. [7 marks]

(g) Estimate the sale revenue when RM 5.6 million is spent advertising [2 marks]

(h) Determine the Spearman’s coefficient of rank correlation between the advertising

expenses and sales revenues. Interpret your answer. [4 marks]

Scheme

Advertising Expense

(RM million), x 3 6 5 4 6 10 9

Sales Revenue

(RM million), y 18 20 21 25 26 28 29

Sales Revenues,y

(RM million)

10

20

30

40

x

x

x x

x

x x

D1 – Uniform scale, axes labelled

D1 – Point plotted correctly

14

x y x2 xy

3 18 9 54

6 20 36 120

5 21 25 105

4 25 16 100

6 26 36 156

10 28 100 280

9 29 81 261

43x 167y 2 303x 1076xy

B1 - both B1 B1

2

7 1076 43 1671.2904

7 303 43b

M1

167 43

1.2904 15.93047 7

a

M1

15.9304 1.2904y x A1

The value of regression b indicates that the sales revenue is expected to increase by

RM 1.2904 million for every RM one million spend in advertising. B1

(c ) When x = 5.6 , 15.9304 1.2904 5.6 23.15664y M1

The sales revenue is RM 23.16 million A1

(d)

x Rank y Rank d d2

3 7 18 7 0 0

6 3.5 20 6 -2.5 6.25

5 5 21 5 0 0

4 6 25 4 2 4

6 3.5 26 3 0.5 0.25

10 1 28 2 -1 1

9 2 29 1 1 1

2 12.5d

B1 – both ranks B1

2

6 12.51 0.7768

7 7 1sr

A1

A definite positive correlation. Increase in advertising expenses will definitely result in an

increase in sales revenues. B1

Advertising Expenses,x

(RM million)

|

8

|

2

|

4

|

6

|

10 0