STATISTICS
Paper-11
(Time Three ll_ours J (Maximum Mqrks 200 J
INSTRUCTIONS
Candidatei.should attempt Questions 1 and
5 which are compulsory, and any THREE
of the remaining questions, selecting at
least ONE question from each Section.
All questions carry equal marks.
Marks allotted to each part of a question
are indicated against each.
Assume suitable data, if considered
necessary, and indicate the same clearly .
.Answers must be written in ENGLISH only.
Unless otherwise indicated, symbols and
notations have their usual meanings.
o I o
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SECTION-A
(INDUSTRIAL STATISTICS AND
OPTIMIZATION.TECHNIQUES)
1. Attempt any FOUR of the following 4x
(a)Describe the nature of a two person zero sum
game.Define the terms strategy of a
player pay off and a fair game using one
example.
(b)Solve the following linear programming problem
Maximize Z 2x1+3x2
subject to x1
x1
x1+3x2<=9


(c)Explain how a control chart helps to control
the quality of a manufactured product. Describe
the basis of a control chart. Distinguish clearly
between the charts for variables and charts for
attributes.
(d)What is Average Sample Number and
Average Total Inspection Explain the
method of their calculation for single sampling
plan. Why are ASN and ATI calculated?

(e)Let be a Markov chain with the state
space S and the transition probability
matrix
<img src='./qimages/1492-1e.jpg'>
(i)Classify the states and determine the period
of each state.
(ii)find the mean recurrence time of each of the
states.
Explain the different control charts for
attributes used in industrial inspection of
manufactured units.
During an examination of equal length
of cloth, the following are the number of
defects observed:
2.
How will you draw a control chart for ·
the number of defects and comment whether
the process is under control or not? 12

(b)Define the average sample number and
the average outgoing quality in the case
of double sampling inspection and indicate
their usefulness in choosing a sampling
scheme. Describe the general method of
plotting OC curve of double sampling plan. 14

firm is contemplating the introduction of
three products in its three plants B
and C. Only a single product is decided
to be introduced in each of the plants.
The unit cost of producing i-th product in
j-th plant is given in the following matrix of
Table
<img src='./qimages/1492-2c.jpg'>
(i)How should the products be assigned so that
the total unit cost is minimised
(ii)If the quantity of different products to be
produced is given in Table II then what
assignment shall minimise the aggregate
production cost Hence obtain the optimum
total cost.

(iii)It is expected that the selling prices of the
products produced by different plants would
be different as shown in Table III assuming
that the quantities mentioned in Table II would
be produced and sold, how should the
products be assigned to the plants to obtain
maximum profit? Hence obtain the maximum
total profit
<img src='./qimages/1492-2c3.jpg'>

A scientific equipment manufacturing company is
engaged in producing equipment for use of science
laboratories. The processing time for two different
assembly lines is regarded as a random variable
and is described by the following distributions

<img src='./qimages/1492-3a.jpg'>

Using the following random numbers, generate
data on the process times for six units of the item
and compute the expected process time for the
product
3441 7674 4349 4383 8311 1519
For the purpose, read the numbers horizontally,
taking the first two digits for the processing time
on assembly A1 and the last two digits for
processing time on assembly A2 14
(b)What is a statistical software Mention a few
of them. Given a sample of 20 observations
{x1,....,x20} describe input, output and processing
commands to obtain simple arithmetic average and
variance of sample mean. 14
24 hour supermarket has the following minimal
requirements for cashiers
<img src='./qimages/1492-3c.jpg'>
Time of a day assumes 24 hour clock. Period 1
follows immediately after period 6. A cashier works
eight consecutive hours; starting at the beginning
of the one of the six periods. Determine a daily
employee worksheet which satisfies the
requirements with the least number of personnel.
Formulate the problem as a linear programming
problem (L.P.P.). 12
4.(a)The arrivals at a counter in a bank occur in
accordance with a Poisson process at an average
rate of 8 per hour. The duration of service of a
customer has exponential distribution with a mean
of 6 minutes. Find the probability that an arriving
customer: (i)has to wait on arrival (ii)finds
4 customers in the system and has to
spend less than 15 minutes in the bank. Estimate
also the fraction of total time that the counter is
busy. 14
(b)Describe the concepts of a reliability of a
component series system parallel system.
Consider a system consisting of n components such
that the failure of the i th component occurs in
accordance with a Poisson process of intensity ai.

Find the reliability of the system in series system
and parallel system. 14
(c)Show that the expected number of busy
servers in an M c queue in steady state is cp
and that the expected number of idle servers is c
p).
SECTION-B
(QUANTITATIVE ECONOMICS AND
OFFICIAL STATISTICS)
12
5. Attempt any FOUR of the following:- 4xl0=40
(a)Define the time reversal test for index numbers.
Give one example of index number which satisfies
this test and one example which does not:
(b)Define a time series. Mention its important
components with illustrations and describe a
method of smoothing of time series by semi
averages.
(c)What do you mean by fertility of a population?
Define (i)Crude Birth Rate General
Fertility Rate and Total Fertility
Rate. Discuss their relative merits and demerits
as measures of fertility.

studying a pair of related variables, a statistician
summarised the data as follows
<img src='./qimages/1492-5d.jpg'>
Find Sx r and the line of regression of
y on x. Verify that this line passes through the point
of means.
(e)Describe the population projection by vital
statistical method and by fitting mathematical
curves using time series data of populations in
different years.
6.(a)Explain the nature of cyclical variations in a
time series. How do seasonal variations differ from
them Give an outline of the moving average
method of measuring seasonal variations. 12
(b)Give the genesis of Logistic curve
<img src='./qimages/1492-6b.jpg'>
Explain any one method of fitting this curve to
time series data regarding production in various
years. 14
(c)Write a note on the present official statistical
system in India relating to population. What
do you know about the census collected in
2011 with respect to (i)collection of data
by Official Statistical System (ii)processing of
data in different organizations final
publications about the results of Population
Statistics? 14
7.(a)Given the following table for lx, the number of
rabbits living at age complete the life table for
rabbits:
<img src='./qimages/1492-7a.jpg'>
Let R1, R2, R3 denote the three rabbits of age
2 and 3 years respectively. Find the
probability that least one of them will be
alive for one ·year more R1,R2,R3 will be
alive for two years time one of the three is
alive in two years and all will be dead
in two years time. 14
(b)Explain briefly the concepts of reliability and
validity of scores in educational and psychological
experiments. Describe a method of obtaining
the reliability coefficient and the validity
coefficient.
(c)Describe the Indirect method of standardising death
rates.Discuss how you would compare mortality
between(i)two different communities two
different periods of time two different causes
of death (iv)two different occupations. 14
8.(a)Discuss briefly (i)the problems that are involved
in the construction of an index number of prices.

(ii)the importance and limitations of index
numbers (iii)construction of the wholesale price
index numbers. 12
(b)Define (i)the autocorrelation rk of order
k and (ii)second order auto regressive series. For the series determined by
Ut+1=aUt+ where E has zero
mean, find the correlogram if successive values
of E are independent. 14
(c)Describe the problem of multicollinearity
linear regression model with k regressors.
What are the consequences on estimation How
will you remove multi-collinearity and obtain the
best linear regression equation for response
variable? 14