D-GT-M-TTB
STATISTICS
Paper-11
I Time Allowed Three Hours I I Maximum Marks 200 I
INSTRUCTIONS
Candidates should attempt Question Nos. 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. Answers must be written in ENGLISH only. Assume suitable data, if considered necessary, and indicate the. same clearly. Unless otherwise indicated, sfmbols and notations have their usual meanings. Important Note All parts/ sub-parts of a question being attempted are to be answered contiguously on the answer-book. That is, where a question is being attempted, all its constituent parts/ sub-parts must be answered before moving on to the next question. Pages left blank, if any, in the answer book(sl must be clearly struck out. Answers that follow pages left blank may not be given credit.

Section-A( Industrial Statistics and Optimization Techniques

1. Answer the following 8 x 5=40

What is the problem of duality in linear· programming? For the following linear programming problem
Maximize f 2X 3Y 5Z
subject to the conditions
lOX+ 9Y 10

9X +2Y -4Z 80
X Y Z 0
write the dual problem. Further show that dual of the dual problem is the primal problem.
For a two-person zero-sum game problem, explain the meaning of the following terms
Pure and mixed strategy
Saddle point
Value of the game
The following matrices are payoff matrices for a game problem
p G p

Obtain value of game in each case.
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•
For a Markov chain with two states
and transitional probability matrix
P (Pu p,2
show that the steadyP21
P22
state probabilities will be given by
1t
P·::n n P12
1 2
1 Pu P21 1 P12 P22
Motherboard of a laptop has four
components C and D. 500 number
of motherboards were subjected to
accelerated operational tests. They
contain the following failure data
Component
Number of failures
A
10
B
8
c
40
D
20
Determine the system reliability of the
motherboard.
Show that the following function
represents the failure time density
function
· t 2 I 2 for 0 1
.!.(t2 3{t for 2
2
1 2 2 2
2
t 3(t 3(t for 2 t 3
0 otherwise
Obtain expressions for the probability of
failure within time t and the reliability
for time t.
Compute reliability and hazard rate at
time t 0 · t t 1· t 2 and
t 2. 5.
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2. Explain clearly the working procedure
for a double sampling plan for
attributes, when two points are flxed on
OC curve and plan is designed so as to
minimize AT!.
What are principal and supplementary
OC curves? Also discuss in detail howyou
will draw OC, ASN and AOQ
curves for this plan.
Give the working procedure for the
following multiple sampling plan 14
n2 c2 n3 c3
Ten samples each of size 50 are drawn
from an on-line production process
and the number of defectives found is
recorded as under
12, 4
Draw the chart for defectives
and give your comments. If the process
is not under control, how will you revise
the chart? 12
There are three retail stores S2 and
S3 where the produced units are to be
transported and their requirements are
400, 100 and 500 units respectively.
These units can be transported from
three godowns G" G2 and G3 and the
available units at these godowns are
200, 300 and 500 respectively.
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Unit transportation cost matrix for the
above is given below
s3
s
7 3 4
G2 2 1 3
G3 3 4 6
Determine the optimum allocation for
transporting units from godowns to
retail stores, so that the total cost of
transportation is minimum. 14
3. Explain the meaning of the following
terms
Lead time
Scheduling period
Order level
Buffer stock
Probabilistic demand
Discuss fully order-level-lot size
inventory model, stating the
assumptions underlying it. Obtain
expression for total inventory cost and
determine the optimum values of the
decision variables, so that the total
inventory cost is minimum.
A company produces two products A
and B during a given time period. Each
of these products has to undergo four
different manufacturing operations
12
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namely grinding, turning, assembly
and testing. The manufacturing time
requirements in hours per unit for these
operations are given below
Product
Operation
A B
Grinding 2 4
Turning 6 2
Assembly 12 6
Testing 10 8
The average capacity for these
operations in hours is 60 for grinding,
120 for turning, 400 for assembly and
400 for testing. All units produced are
saleable in the market and profit
contribution is 2 each for product A
and 3 each for product B. Formulate
linear programming problem and
determine the optimum solution
..
graphically to maximize profit. 14
Derive differential-difference equation
for M I M 11 FIFO) queuing system.
State the assumptions underlying it.
Obtain steady-state solution for the
system. Also fmd expressions for-
average queue length;
average number of units m the
system;
idle and busy period for the server. 14
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4. Explain the following
Reliability of a component
Reliability for a series system
Reliability for a parallel system
Reliability bathtub curve
Discuss briefly how
engineering helps in
reliability
business
promotional plans.
A certain system consists of 10
identical units which are connected in
series. It 1s desired that the system
reliability should be 0 · 95. Determine
how good the component should be. 12
Give lucid exposition of any two of the
following 12
Item-by-item sequential sampling
plan
Cumulative sum control charts
Renewal theory
A bakery keeps stocks of popular brand
bread. By past experience, the following
probability distribution for daily
demand is found as under
Daily demand (units) 0 10 20 30 40 50
Probability 0·12 0·08 0·30 0·10 0·20 0·20 .
From the above-
calculate average daily demand for
bread based on the probability
distribution;
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use the following sequence of
random numbers to simulate
demand for next 10 days
45, 26, 32, 37, 59, 88, 07, 15, 72, 04
find out the stock situation for the
next 10 days if the bakery-man
decides to make 40 breads per day;
estimate average daily demand
based on simulated data for the
next 10 days;
fmd out the average daily stock
situation based upon the simulated
data for the next 10 days. 16
Section-B
Quantitative Economics and Official Statistics
5. Answer the following 8x5=40
Write the basic assumptions underlying
k-variate general linear model
with usual notations.
Establish the following for the above
model

V@J cr 2
2 I
8
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What are time and factor reversal tests
of index numbers? Examine whether
the following index numbers satisfy
these tests or not

1

2
where L is Laspeyres's index number
and P is Paasche's index number.
What are the names given to above
and index numbers?
Discuss briefly the role of NSS
organization in developing, collecting
and implementing (analyzing and using)
Indian statistical data.
Explain different components of a time
series giving their use and importance.
Discuss how you will separate trend,
seasonal and random component from a
given time series data.
The following data give production of
sandalwood in a forest area during the
years 1981 to 1987
Year 1981 1982 1983 1984 1985 1986 1987
Sandalwood
metric tons) IS 26 43 64 95 130 171
Fit the following trend model to the
above data
Y =a. Pt yt 2 u
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where y are parameters and U is the
disturbance term.
Give your comments. on the fitted model.
Also estimate production of sandalwood
for the year 1988.
6. What are generalized least squares
estimators? Stating the basic
assumptions underlying this model,
obtain expression for generalized least
squares estimators. Also find the
variance of these estimators and
compare them with Ordinary Least
Squares estimators. 14
For estimating simultaneous linear
system of equations, how would you use
the following methods of estimation and
why? 12
Indirect Least Squares method
Two-Stage Least Squares
method
What are stationary time series?
Explain their importance. Discuss the
application of correlogram analysis and
periodogram analysis for stationary time
series.
7. Explain the nature and applications of
Gompertz curve and modified Gompertz
curve.
Discuss in detail any one method for
fitting these curves to a given
14
population data. 14
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(OJ What are autocorrelated disturbances in
linear models? Explain the effect of the
presence of such disturbances. Describe
test for detecting autocorrelation.
Discuss Cochran-Orcutt interactive
procedure to deal with the problem of
autocorrelation.
Explain the following, giving suitable
14
illustrations 12
Fixed base and chain-based index
numbers
Deflating of index numbers
Splicing of index numbers
8. Explain the following,
illustrations
Crude Birth Rate
General Fertility Rate
giving
Age-Specific Fertility Rate (ASFR)
Total Fertility Rate
Gross Reproduction Rate
Net Reproduction Rate
What is a life table? Explain various
terms and concepts used in life tables
and discuss the importance and uses of
life tables. How would you construct a
12
life table for given population data? 14
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Discuss how official statistics pertaining
to trade, production and prices are
collected m India. Give complete
procedure. Also list some important
publications for these official statistics
with special reference to India. •. 14

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