Section A [45 marks]

Answer all questions in this section.

1

The table below records the weight loss, in kilograms, of each of 150 people attending a

dietary clinic.

Weight loss x (kg)

Number of

people

12 x

01 x 10 x

21 x

32 x 53 x

105 x

3

7 12

19

38 46

25

(a) What do you think the negative values indicate?

(b) Obtain estimates of the values of

(i) the mean, (ii) the standard deviation

(c) Present the data in a cumulative frequency graph, hence, estimate the median.

(d) Find the Pearson’s coefficient of skewness, and comment on your result.

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2 A shop keeper found that 30% of the customers patronized the shop would buy a book. The probability that a customer will buy a pen after he has bought a book is 0.6.

However the probability that a customer will just buy a pen is 0.2. Find the probability

that a customer patronizes the shop will buy (a) a book and a pen,

(b) a book or a pen,

(c) a pen but not a book.

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3 Kathy finds that when she takes a cutting from a particular plant, the probability that it

roots successfully is .3

1

She takes nine cuttings. Find the probability that

(a) more than five cuttings root successfully,

(b) at least one cuttings root successfully.

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4 (a) Eight pairs of values obtained from random observations on two variables x and y

are (4, 63), (2, 89), (5, 58), (3, 73), (4, 72), (5, 48), (3, 75) and (2, 84).

(i) Plot these values on a scatter diagram.

(ii) State, with a reason, whether the scatter diagram in (i) displays a positive or a

negative correlation. (b) The mathematics score (x) and economics score (y) for eight students are

summarized as follows.

2110)( and,2222)(,26762,472,448 22 yyxxxyyx

Find the correlation coefficient for the scores of the two subjects, and comment on

your result.

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5 A company produces three models of computers, model A, B and C. The average selling

price and the number of computers produced and sold for the years 2008 and 2013 are

shown in the following table.

Model A Model B Model C

Year Quantity

(‘000)

Price

(RM’000)

Quantity

(‘000)

Price

(RM’000)

Quantity

(‘000)

Price

(RM’000)

2008

2013

3

12

10

4

12

20

15

8

6

12

16

9

(a) Taking 2008 as the base year, calculate the Laspeyres and Paasche price indices for

the year 2013.

(b) Give a reason why the the Laspeyres and Paasche price indices are different. .

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6 The following data shows the tax (RM’000s) received by the local government.

(a) Calculate the seasonal index for each quarter using an additive model. (b) Obtain a seasonally adjusted time series.

Year Quarter Tax received

(RM thousand)

Centred moving

average

2011

1 2.8

2 4.2

3 3.0 3.68

4 4.6 3.70

2012

1 3.0 3.76

2 4.2 3.88

3 3.5 3.93

4 5.0 3.99

2013

1 3.0 4.06

2 4.7 4.11

3 3.6

4 5.3

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Section B [15 marks]

Answer any one question in this section.

7 A manufacturer stores drums of chemicals. During storage, evaporation takes place. A

random sample of 10 drums was taken and the time in storage, x weeks, and the

evaporation loss, y ml, are shown in the table below.

x 3 5 6 8 10 12 13 15 16 18

y 36 50 53 61 69 79 82 90 88 96

(a) Draw a scatter diagram to represent these data.

(b) Give a reason to support fitting a regression model of the form y = a + bx to these

data. (c) Find, to 2 decimal places, the value of a and the value of b.

(You may use ∑ 𝑥2 = 1352, ∑ 𝑦2 = 53112 𝑎𝑛𝑑 ∑ 𝑥𝑦 = 8354.)

(d) Give an interpretation of the value b.

(e) Using your model, predict the amount of evaporation that would take place after (i) 19 weeks,

(ii) 35 weeks.

Comment on the reliability of each of your predictions.

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8 (a)

The diagrams show the graphs of two functions, g and h. For each of the functions

g and h, give a reason why it cannot be a probability density function.

(b) The distance, in kilometres, travelled in a given time by a cyclist is represented by

the continuous random variable X with probability density function given by

𝑓(𝑥) = {

30

𝑥2 10 ≤ 𝑥 ≤ 15

0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒

(i) Find the cumulative distribution function, F(x). Sketch the graphs of f(x) and F(x).

(ii) Show that E(X) = 30 ln 1.5.

(iii) Find the median of X. Find also the probability that X lies between the median and the mean.

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