trial stpm mathematics m 2 (nsembilan) smk tgdurahseremban
TRANSCRIPT
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