STPM 950/2

Section A [45 marks]

Answer all questions in this section.

1. The masses (in kg) of a group of students are given as follows.

54 68 84 56 68 88 57 72 102

59 70 63 75 64 76 66 81

(a) Display the above data in an ordered stemplot. [2 marks]

(b) Find the median and the interquartile range. [3 marks]

(c) Draw a boxplot to represent the data. [3 marks]

2. Nicole has 25 classmates and she wants to invite 10 of them to her birthday party.

(a) If two of her classmates are not talking to each other and they would not attend the

party together, find the probability that only one of them attends the party. [2 marks]

(b) If Chee Seng, Mun Siong and Willy are good friends and they would attend the

party if and only if they are all invited, find the probability that they don’t attend the party.

[2 marks]

3. X is a continuous random variable with the following probability density function.

otherwise,0

41, xx

a

xf

(a) Determine the value of a. [3 marks]

(b) Find the value of m if 5.0)( mXP . [2 marks]

4. The ABC Electric Company is studying the relationship between kilowatt-hours

(thousands) used and the number of rooms in a private single-family residence. A random

sample of 10 homes yields the following data.

Number of rooms, x 12 9 14 6 10 8 10 10 5 7

Kilowatt-Hours (thousands), y 9 7 10 5 9 6 8 10 4 7

(a) Obtain the equation of the regression line of y on x in the form bxay , where

a and b are given to three decimal places. [6 marks]

(b) Based on your answer in (a), interpret the value of b obtained. [1 mark]

(c) Estimate the number of kilowatt-hours used for a six-room house. [2 marks]

STPM 950/2 2

5. The following table shows the prices (RM per unit) and quantities sold (in thousand

units) for four types of commodities in January 2013 and January 2014.

Commodity

January 2013 January 2014

Price

(RM per unit)

Quantity

(×1000 units)

Price

(RM per unit)

Quantity

(×1000 units)

A 3.00 18 4.00 20

B 5.00 6 5.00 9

C 4.00 20 6.00 18

D 1.00 14 1.50 20

(a) Taking January 2013 as the base period, calculate a simple aggregate quantity

index for January 2014 and comment on your answer. [3 marks]

(b) Taking January 2013 as the base period, calculate

(i) the Laspeyres price index for January 2014; [2 marks]

(ii) the Paasche price index for January 2014. [2 marks]

(c) By comparing Paasche index and Laspeyres index in (b), comment on the change

in consumption patterns of the commodities A, B, C and D. [2 marks]

6. The following table shows the mobile phone sales (in thousand units) of a company for

the years 2011 to 2013.

Quarter

Year 1

st 2

nd 3

rd 4

th

2011 10 4 7 15

2012 14 10 13 20

2013 20 16 19 24

(a) Plot the above data as a time series. [2 marks]

(b) Comment on the basic trend and the seasonal variations. [2 marks]

(c) Calculate the centred 4-quarter moving averages and the adjusted seasonal

variation for each quarter using a multiplicative model. [6 marks]

STPM 950/2

3

Section B [15 marks]

Answer any one question in this section.

7. In Town A, 2% of the population are found to have a particular type of rare disease.

(a) Calculate the probability of finding at least one person with this disease in a

random sample of 6 people. [3 marks]

(b) Use normal distribution as an approximation to estimate the probability of finding

more than 25 people with this disease in a random sample of 1000 people. [4 marks]

Suppose that 20% of those who are infected and 5% of those who are not infected with

this rare disease have a particular skin condition.

(c) Find the probability that a person selected at random from Town A has the

particular skin condition. [2 marks]

(d) Two people from Town A are selected at random. Find the probability that both

of them do not have the particular skin condition. [2 marks]

(e) Determine the smallest sample size that must be taken so that the probability of

including at least one person with the particular skin condition is more than 0.95. [4 marks]

8. The cumulative frequency distribution for the money (in RM) spent by 50 customers at

a grocery store on a particular day is shown below.

Money spent (RM) Number of customers

20 0

40 3

60 9

80 24

100 34

120 46

140 50

(a) Calculate the mean and the standard deviation of the expenditure of these

customers in the grocery store. [5 marks]

(b) Plot a cumulative percentage frequency curve. Hence, estimate the median and

the percentage of customers who spent between RM50 and RM100. [8 marks]

(c) 20% of the customers who spent more than RM x are given a free gift. Estimate

the value of x (correct to the nearest RM1). [2 marks]

STPM 950/2 4

MATHEMATICAL FORMULAE

Summary statistics

For ungrouped data

kth

percentile =

integeran not is 100

if ,

integeran is 100

if ,2

1

nk

rx

nk

rxx

r

rr

Standard deviation =

2

22

xn

x

n

xx

For grouped data

kth

percentile = cf

Fnk

Lk

k

k

1100

Standard deviation =

2

22

xf

fx

f

xxf

Probability distributions

Binomial distribution

xnx ppx

nxXP

1)( , x = 0, 1, 2, …, n.

Poisson distribution

!

)(x

exXP

x , x = 0, 1, 2, …

Correlation and regression

Pearson correlation coefficient

n

yy

n

xx

n

yxxy

r2

2

2

2

Spearman rank correlation coefficient

1

6

12

1

2

nn

d

r

n

i

i

s

Least squares regression line

bxay ,

n

xx

n

yxxy

b2

2

& xbya

STPM 950/2

5

No. Answer Scheme Marks

1 (a)

Stem Leaf

5 4 6 7 9

6 3 4 6 8 8

7 0 2 5 6

8 1 4 8

9

10 2

Key: 8 1 means 81 kg

(b) Median = 68 kg

Interquartile range = 76 – 63

= 13 kg

(c) (Use graph paper)

Scale + label

Q1, Q2, Q3

Outlier + All correct

B1

B1

B1

M1

A1

D1

D1

D1

Total 8 marks

2 (a) P(only one of them attends the party) =

10

25

9

23

1

2

C

CC

= 2

1

(b) P(they don’t attend the party) = 10

25

7

22

3

3

1C

CC

9478.0

115

61

M1

A1

M1

A1

Total 4 marks

x

54 63 68 76 88 95.5 102

Upper

boundary

Outlier Q1 Q2 Q3

STPM 950/2 6

3

otherwise,0

41, xx

a

xf

(a) 1

4

1

dxx

a

2

1

1122

124

1

a

a

xa

(b) 5.01

2

1

1

dxx

m

25.2

5.1

5.01

m

m

m

B1

M1

A1

M1

A1

Total 5 marks

4 (a) 728,895,75,91 2 xyxyx

29189510

759172810

b

= 0.6801

1.96801.05.7 a

= 1.311

The equation is xy 680.0311.1

(b) b = 0.680 means the usage of electricity is expected to increase by

680 kilowatt-hours for every increase of 1 room in a house.

(c) 6680.0311.1ˆ y

= 5.391

The usage of electricity for a six-room house is estimated to be 5.391

thousand kilowatt-hours.

B1

M1

A1

M1

A1

B1

B1

M1

A1

Total 9 marks

STPM 950/2

7

5 (a) Simple aggregate quantity index for January 2014

52.115

1001420618

2018920

The quantities of the commodities sold increased by 15.52% from

January 2013 to January 2014.

(b)

p0 q0 p1 q1 p0q0 p1q0 p1q1 p0q1

A 3.00 18 4.00 20 54.00 72.00 80.00 60.00

B 5.00 6 5.00 9 30.00 30.00 45.00 45.00

C 4.00 20 6.00 18 80.00 120.00 108.00 72.00

D 1.00 14 1.50 20 14.00 21.00 30.00 20.00

178.00 243.00 263.00 197.00

(i) Laspeyres price index for January 2014

52.136

100178

243

(ii) Paasche price index for January 2014

50.133

100197

263

(c) Paasche index is lower than Laspeyres index indicates a trend

towards less expensive goods.

M1

A1

B1

M1

A1

M1

A1

B1B1

Total 9 marks

6 (a)

(b) The time series has an increasing trend.

The sales are seasonal with maximum sales in 4th

quarter and

minimum sales in 2nd

quarter.

D2

B1

B1

STPM 950/2 8

(c)

Year Quarter Y 4 quarter moving average

Centred moving

averages, T

Seasonal variation,

S=Y÷T

2011

1 10

2 4

3 7 9.00 9.5000 0.7368

4 15 10.00 10.7500 1.3953

2012

1 14 11.50 12.2500 1.1429

2 10 13.00 13.6250 0.7339

3 13 14.25 15.0000 0.8667

4 20 15.75 16.5000 1.2121

2013

1 20 17.25 18.0000 1.1111

2 16 18.75 19.2500 0.8312

3 19 19.75

4 24

1 2 3 4

2011 0.7368 1.3953

2012 1.1429 0.7339 0.8667 1.2121

2013 1.1111 0.8312

Mean Seasonal Variation

1.1270 0.7826 0.8018 1.3037 4.0151

Adjusting factor

0.9962 0.9962 0.9962 0.9962

Adjusted Seasonal Variation

1.1227 0.7796 0.7988 1.2987 3.9998

M1A1

(column

5)

M1

A1

M1A1

Total 10 marks

STPM 950/2

9

7 (a) Let X = the number of people with the rare disease

Then, X ~ B (6, 0.02)

P 1X 1 - P 0X

1142.0

98.01 6

(b) 6.19,20,1000 npqnpn

6.19 ,20N~X

5.25P25P XX

1071.0

10706.0

2423.1P

6.19

205.25P

Z

Z

(c) P(a person has the particular skin condition)

053.0

05.098.02.002.0

(d) P(a person does not have the particular skin condition) = 0.947

P(both people do not have the particular skin condition)

8968.0

947.0 2

(e) Let Y = the number of people with the particular skin condition

Then, .0530 ,B~ nY .

01.55947.0ln

05.0ln

05.0947.0

05.00P

95.00P-1

95.01P

n

Y

Y

Y

n

Therefore, the smallest sample size to be taken is 56 people.

B1

M1

A1

B1

B1

M1

A1

M1

A1

M1

A1

B1

M1

M1

A1

Total 15 marks

STPM 950/2 10

8 (a)

x f

30 3

50 6

70 15

90 10

110 12

130 4

Mean expenditure 60.83RM50

4180

Standard deviation 26.8350

385000

= RM26.67

(b)

x c. f. Cumulative % frequency

< 20 0 0

< 40 3 6

< 60 9 18

< 80 24 48

< 100 34 68

< 120 46 92

< 140 50 100

Median = RM82.00

Percentage of customers spent RM50 – RM100 = 68 - 10

= 58%

(c) x = RM110

M1A1

B1M1

A1

B1

D3

M1A1

M1A1

M1A1

Total 15 marks