Statistics II 2012

8/12/2019 Statistics II 2012

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STATISTICS

Paper II

Maximum Marks : 300

ime Allowed : Three Hours

INSTRUCTIONS

Each question is printed both in Hindi and in English.

Answers must be written in the medium specified in the

Admission Certificate issued to you, which must be

staled clearly on the cover of the answer-book in the

space provided for the purpose. No marks will be given

for the answers written in a medium other than that

specified in the Admission Certificate.

Candidates should attempt Questions no. 1 and

5

which are compulsory, and any

three

of the remaining

questions selecting at least

one

question from each

Section.

The number of marks carried by each question is

indicated at the end of the question.

Assume suitable data if considered necessary and

indicate the same clearly.

Symbols I Notations used carry usual meaning, unless

otherwise indicated.

Charts I Figures, wherever required, are to be drawn in

the answer book itself and

not

on separate graph sheet.

IMPORTANT : Whenever a question is being

attempted, all its parts/sub-parts must be attempted

contiguously. This means that before moving on to the

next question to be attempted, candidates must finish

attempting all parts/sub-parts of the previous question

attempted. This is to be strictly followed.

Pages left blank in the answer-book are to be

clearly struck out in ink. Any answers that follow

pages left blank may not be given credit.

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SECTION A

1. (a) Describe single and double sampling plans used in

acceptance sampling. Define the operating

characteristic function ola sampling plan.

(b)

The cost of maintenance of a machine is given as an

increasing function of time and its scrap value is

constant. If time is measured continuously, then

show that the average cost will be minimized by

replacing the machine when the average cost to

date becomes equal to the current maintenance

cost.

(c)

Describe how the control chart based on sample

ranges is constructed.

(d)

Describe Type I and Type II censoring. Discuss the

consequence of such censoring in the analysis of

lifetime data.

(e)

Show that in a linear programming problem, the

feasible solutions form a convex set.

x12=60

2. (a) A person needs at least 10, 12 and 12 units of

chemicals A, B and C respectively for his garden. A

liquid product available in the market contains 5, 2

and 1 units of A, B and C respectively per jar. A dry

product contains 1, 2 and 4 units of A, B and C per

carton. The price of the liquid product is 3 per jar

and that of the dry product is 2 per carton. How

many jars and cartons should be purchased to meet

the requirements and to minimize the cost ?

0

F-DTN-M-TUBE

Contd.]



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F-DTN-M-TUBB



3. (a) A sample of 100 screws was selected on each of

25 successive days in a factory manufacturing

screws, and each screw was examined for defects.

The data for number of unacceptable screws on

different days are given below :

Day

.

No. of

defective

screws

I I

Day

No. of

defective

screws

1

7

14

6

2

4

15

2

3

3

6

9

4

6

7

7

5

4

8

6

6

9

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7

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1

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5

22

7

10

3

23 4

11

7

4

8

12

8

5

6

13

4

Assume that the production process was in control

during the period. Determine the upper and the

lower control limits (UCL, and LCL) based on this

data for the proportion of defective items.

0

F-DTN-M-TUBB

Contd.'







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F-DTN-M-TUBB

Contd.]



(b) Consider a Markov chain with three states 0, 1

and 2. The state 0 is an absorbing state, while

from any of the other two states the chain moves

into any of the three states with equal probability

(i.e. p(a I b) = 1 for a = 0, 1, 2 and b = 1, 2).

3

Starting from either state 1 or 2, let N= the

number of steps or transitions needed before

absorption into state 0 takes place. Derive the

probability distribution of N. What is the expected

number of steps or transitions before absorption

into state 0 ?

30

F-DTN-M-TUBB

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SECTION B

5. (a) For the AR(1) process defined as X

i

= 0.6 Xt

_ +

E t,

where t = I, 2, 3, ..., X

0 = 0 and

F

1 , £

2

, Ea

, ...

are

independent N(0, a

2

) variables, what is the

expression for the spectrum (i.e., spectral

density) ?

(b)

Describe the problem of multi-collinearity in

multiple linear regression. Flow does it affect

parameter estimation ? Describe some solution for

this problem.

(c)

Describe Laspeyres' and Paasche's methods (hr

computing price indices. Discuss how

Marshall — Edgeworth formula leads to a

compromise between the two methods.

(d)

Describe the exponential growth curve. Explain

how one checks whether a given time series has

exponential growth pattern and how one can fit an

exponential trend to a time series.

(e)

What is the percentile score of a student in a

test ? Suppose that the original scores of the

student in a test are approximately normally

distributed. Then what will be the mean and the

variance of the percentile scores ?

x12=60

F-DTN-M-TUBB

2

Contd.]



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F-DTN-M-TUBB

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Contd.]



6.

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y

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i

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j = 1, ...,

f

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2

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least squares estimates of a, a

s and fi

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lbr r, s = 1, ..., I, r s and p, q = 1, . ,J,pxq.

Obtain also the variance expressions for those least

squares estimates.

0

(b) Derive the normal cquations for fitting a linear

trend to time series data. Suppose that a time

series X

t

has been observed at time points

t = + 1, ± 3, + 5,

19, ± 21 and for this series

x t

=

528.2

nd

Xt

= — 1956.4.

t

Determine the linear trend equation that will fit

the data.

0

7.

(a) What is the Logistic curve ? Describe in detail the

method of Pearl and Reed and also Rhodes' method

for fitting a logistic curve to population data.

0

(b) Define Irving Fisher's ideal price index. Explain

why it is called an ideal index by giving relevant

mathematical details. Give an example of

well-known price-index formula that is not ideal

and explain why it is so.

0

F-DTN-M-TUBB

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th e front cover of th is q uestion paper.

Documents

Statistics II 2012