950/2 STPM

Upper Six

April 2013

MATHEMATICS (M)

PAPER 2

One and a half hours

KOTA KINABALU HIGH SCHOOL

STPM SECOND TERM TRIAL EXAMINATION

Instructions to candidates:

DO NOT OPEN THIS QUESTION PAPER UNTUIL YOU ARE TOLD TO DO SO

Answer all questions in Section A and any one question in Section B. Answers must

written in English.

All necessary working should be shown clearly.

Scientific calculators may be used. Programmable and graphic display calculators are

prohibited.

A list of mathematical formulae, statistical tables and graph papers are provided on

pages 5 and 6 of this question paper.

This question paper consists of 6 printed pages

2

Section A [45 marks]

Answer all questions in this section.

1 The nurse of a company performs a routine health check of 100 workers of the

company. The report on the heart rates of the workers is tabulated as below.

Heart rate (per minute) Number of workers

3

15

28

24

20

7

2

1

(i) Calculate

an estimate of the mean. [2 marks]

(ii) Calculate

the standard deviation of the heart rate of the workers. [3 marks]

(iii) Plot a relative cumulative frequency curve for the above data. Hence, determine

the median and the percentage of workers who heart rates is more than 87 per

minute. [5 marks]

2

If A and B are incidents where .

Calculate

[3 marks]

[2 marks]

Determine if A and B are independent or mutually exclusive. [4 marks]

3 The discrete random variable X represents the number of air-conditioners sold by a

company in a week. X has a Poisson distribution with mean and

Determine the value of [3 marks]

Calculate the probability that five air-conditioners are sold in four weeks.

[2 marks]

4 The data below show the average closing share prices (RM) for the LTR Company for

the first 12 weeks of 2013:

Month Week 1 Week 2 Week 3 Week 4

January

February

March

39.25

38.25

37.00

38.75

38.00

38.25

38.75

37.55

38.50

39.00

37.50

39.74

Find the trend values using the moving average method. [3 marks]

Using additive model, calculate the adjusted seasonal variation for each week. [5

marks]

3

Forecast the average closing for this company’s shares for the fourth week of April

2013. [3 marks]

5 For 10 married couples, the height of the husband (x cm) and the height of the wife

(y cm) are summarized by

and Calculate the coefficient of the correlation, and comment on the

result. [4 marks]

6 The manager of sport shoes company wants to determine the level of its market share

in the shoe market. He collects data on the average prices and the number of pairs of

sports shoes sold (in thousands of pair) in year 2011 and 2012.

(i) Calculate

a weighted prices index of the shoes in year 2012 using 2011 as the base year and

the quantities in 2011 as the weights. [3 marks]

(ii) Calculate

a weighted quantities index of the shoes in year 2012 using 2011 as the base year

and the prices in 2012 as the weights. [3 marks]

2011 2012 Type of shoes

Price Quantity (‘000) Price Quantity (‘000)

Jogging shoes

Badminton shoes

Tennis shoes

28.00

32.00

35.00

320

1500

1000

35.00

43.00

46.00

400

1900

1300

4

5

Section B [15 marks]

Answer any one question in this section.

1 The discrete random variable X takes the value k with the probability

, where c is a constant.

(i) Determine the value of c, [3 marks]

(ii) Construct a probability distribution table of . [2 marks]

(iii) Calculate the mean and variance of X, [4 marks]

(iv) Find the cumulative distribution function, F(x). [3 marks]

(v) Hence, sketch the graph of this function. [3 marks]

2 The following table shows the marks obtained by 10 form six students for the

Mathematics and Economics papers in a test.

Students A B C D E F G H I J

Mathematics (x) 45 23 33 8 43 15 28 39 45 1

Economics (y) 44 19 36 14 34 8 17 26 29 3

(i) Plot the

data in a scatter diagram. [2 marks]

(ii) The

equation of the regression line of y on x is y = a + bx and the regression line of x

on y is x = c +dy, determine the values of a, b, c, d and the equation of both

regression line. [8 marks]

(iii) Estimate the marks for Mathematics paper obtained by a student who score 40

marks for Economics paper, give your answers correct to the nearest 1 mark.

[2 marks]

(iv) Calculate

the coefficient of determination and interpret your answer. [3 marks]

END OF QUESTION PAPER

6

MARKING SCHEME SMK TINGGI KOTA KINABALU, SABAH

Section A:

1

(i) Mean =

= [1 mark]

= 76.35 [1 mark]

(ii) Standard deviation =

= [2 marks]

= 6.92 [1 mark]

(iii) Graph correctly plotted. [3 marks]

Median = 76 minutes [1 mark]

The percentage of workers who heart rates is more than 87 per minute = 93%

[1 mark]

Heart rate (per minute) Number of

workers(f)

Mid point (t)

3 62.5

15 67.5

28 72.5

24 77.5

20 82.5

7 87.5

2 92.5

1 97.5

2 Given .

[1 mark]

=

[1 mark]

7

[1 mark]

[1 mark]

(i) , so event A and B are not mutually exclusive. [2 marks]

Since OR so

event A and B are not are independent. [2 marks]

3 For Poisson distribution,

(i) Var(X) =

[1 mark]

[1 mark]

cannot be negative, so . [1 mark]

(ii) Y : the number of air conditioners sold in 4 weeks

P(X = 5) = [1 mark] = 0.000983 [1 mark]

8

4 (i)

Month Week Prices

(RM) 4 point MA Trend

1 39.25

2 38.75

3 38.75 38.81

January

4 39.00 38.59

1 38.25 38.35

2 38.00 38.01

3 37.55 37.67

February

4 37.50 37.54

1 37.00 37.69

2 38.25 38.09

3 38.50

March

4 39.74

[3 marks]

(ii)

Month Week Prices

(RM) (Y)

Trend

(T) SV

1 39.25

2 38.75

3 38.75 38.81 -0.06

January

4 39.00 38.59 0.41

1 38.25 38.35 -0.10

2 38.00 38.01 -0.01

3 37.55 37.67 -0.12

February

4 37.50 37.54 -0.04

1 37.00 37.69 -0.69

2 38.25 38.09 0.16

3 38.50

March

4 39.74

[2 marks]

Month Week

1

Week

2

Week

3

Week

4

January

February

March

-

-0.10

-0.69

-

-0.01

0.16

-0.06

-0.12

-

0.41

-0.04

-

Average

SV

-0.395 0.075 -0.090 0.370

Adjusted

Factor

-0.01 -0.01 -0.01 -0.01

Adjusted SV -0.385 0.085 -0.080 0.380

[3 marks]

38.9375

38.6875

38.5000

38.2000

37.8250

37.5125

37.5750

37.8125

38.3725

9

(iii) The average closing for this company’s shares for the fourth week of April 2013

= 38.81+ 13 ( ) + 0.380 [2 marks]

= RM 37.85 [1 mark]

5

[2 marks]

r = 0.846 [1 mark]

Strong positive correlation. [1mark]

6

Prices index of the shoes in year 2012 =

[2 marks]

= 132.34 [1 mark]

(i) Quantities index of the shoes in year 2012 =

[2 marks]

= 127.77 [1 mark]

Section B:

7 (i)

[1 mark]

[1 mark]

[1 mark]

(ii)

X = k 0 1 2

P(X = k)

[2 marks]

10

(iii)

=

= [1 mark]

=

= 1 [1 mark]

= [1 mark]

= [1 mark]

(iv) F(x) = P(X x)

F(x) = 0, x < 0

= 1, [3 marks]

(v)

[3 marks]

0

1

P(X=k)

X

11

8 (i) Scatter diagram correctly plotted. [2 marks]

(ii) 6824,10112,8097,230,280 22===== ∑∑ ∑ ∑ ∑ yxxyyx

The equation of the regression line of y on x is y = a + bx;

[1 mark]

= 0.72931 [1 mark]

a =

[1 mark]

The equation of the regression line of y on x is y = 2.58 + 0.729x [1 mark]

The equation of the regression line of y on x is x = c + dy;

[1 mark]

= 1.08018 [1 mark]

c =

[1 mark]

The equation of the regression line of x on y is x = 3.16 + 1.08y [1 mark]

(iii) x = 3.16 + 1.08y

= 3.16 + 1.08 (40) [1 mark]

= 46.36

12

= 47 marks [1 mark]

(iv)

[1 mark]

[1 mark]

78.8% of the variation in y can be explained by x. [1 mark]