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LE.S./1.5.8. Examination, 2026 fae DETACHABLE
STATISTICS
Paper III
Time Allowed : Three Hours Maximum Marks : 200
QUESTION PAPER SPECIFIC INSTRUCTIONS
Please read each of the following instructions carefully before attempting questions.
There are EIGHT questions divided under TWO Sections.
Candidate has to attempt FIVE questions in all.
Both the questions in Section ‘A’ are compulsory.
Out of the SIX questions in Section ‘B’, any THREE questions are to be attempted.
The number of marks carried by a question/part is indicated against it.
Unless otherwise mentioned, symbols and notations have their usual standard meanings.
Assume suitable data, if necessary, and indicate the same clearly.
Attempts of questions shall be counted in sequential order. Unless struck off, attempt of a question
shall be counted even if attempted partly.
Any page or portion of the page left blank in the Question-cum-Answer (QCA) Booklet must
be clearly struck off.
Answers must be written in ENGLISH only.
1 SNCS-C-STS
1.(2)
1.0)
1.()
2.(a)
SECTION ‘A’
From a simple random sample of size n taken without replacement from N units,
a simple random sample of size 72] is duplicated without replacement and added
to the original sample. Show that the mean based on (7771) units is an unbiased
estimate of population mean. Obtain the variance of this estimate. 15
Obtain Generalised Least Square Estimator (GLSE) of in,
Y=XB+U , withE(UU')=0'Q ,
71 10% kxl 120 nxn
© is a positive definite matrix. Also derive variance. covariance matrix of this GLSE
and write down its unbiased estimator. 15
Explain price elasticities of supply and demand. For the demand function
2-0 , C>0, a>0
Obtain price elasticity of demand and comment on its shape. What happens when
a=1? 10
In systematic sampling when N = nK, prove that the variance of the systematic
sample is given by
(-1)5 _(n=1) 2
12) = N n wsy
where
K Ah h
2 _ 1 _yp?2 2 = 1 _ )?
Hence, show that systematic sampling is more efficient as compared to simple
random sampling without replacement if within systematic sample variation is more
as compared to the overall variation of population, that is, 55 10+5
SNCS-C-STS 2
2.(b)
2.(c)
3.(a)
3.(b)
Consider the model
Y=XB+U, — with E(UU'=07)).
: , 270 240 , 319
Given, (X' =(70 0) and (X n=(452) .
obtain the ordinary least square estimator (OLSE) of f and obtain the variances and
covariances between its components. 10
State the methods for estimating seasonal variation in time series. Compare “Ratio
to trend” and “Ratio to moving average” methods.
Assuming that the trend is absent, determine seasonal indices for the following data
and interpret your result.
Year Ist 2nd 3rd 4th
Quarter | Quarter | Quarter | Quarter
2007 30 40 36 34
2008 34 52 50 44
2009 40 58 54 48
2010 54 76 68 62
2011 80 92 86 82 15
SECTION ‘B’
Differentiate between ordered and unordered estimators in pps sampling without
replacement. Write down the Des Raj Estimator, its mean and variance for estimating
population total when a sample of size 2 is drawn from population of size N by
ppswor. 15
Writing down the assumptions usually made in general linear regression model
Y=XB+U, where U~N, (0, 071).
Obtain the maximum likelihood estimator (mle) of o?. Also examine its
unbiasedness. 15
3 SNCS-C-STS
3.(c)
4.(a)
4.(b)
4.(c)
The Ministry of Labour retail price index was 104-6 given for a certain time period.
The price indices in July 2004 were as follows :
Rent and Rates 65, Clothing 220, Fuel and Light 110, Miscellaneous 125.
Given that the weights of different items in the group were as follows :
Food 60, Rent and rates 16, Clothing 12, Fuel and Light 8 and Miscellaneous 4.
What was the price index of the food group ? 10
From a list of 3042 names and addresses, a simple random sample of 200 names,
taken without replacement, showed on investigation 18 wrong addresses. Estimate
the total number of addresses needing corrections in the list and find the standard
error of this estimate. 10
State the rank and order conditions of identification for structural form of the system.
Discuss the identification of the following Keynesian model of income deter-
mination :
C= % + HY, + OT + U
1, = Bot BY +V
T=h+VY+W
Y=C+L+G, >
where C,, J,, 7, and Y, are endogenous variables and Y,, and G, are predetermined
variables. 15
Explain the concept of unit root test. How unit root test is used while applying
Dickey Fuller (DF) test for testing stationarity of a given time series. Interpret the
following models of time series y, for testing stationarity of the time series.
(i) Sf = 0-00576 y, ,
t = (5-7980)
(ii) $* = 28-2054 — 0-00136 y,,
t= (11576) (-0-2191)
(iii) 3 = 190-3857 + 1-4776r — 0-0603 ¥,_;
t = (1-8389) (1-6109) (-1-6252)
where y; = (v, — ¥,_,) and t = (t) value are the critical values for the models.
(The tabulated values at 5% for
(i) there is no constant is —1-9434
(ii) there is a constant is —2-8947
(iii) there is a constant and trend is —3-4614) 15
SNCS-C-STS 4
5.(a)
5.(b)
5.(c)
6.(a)
If variates y,, x; are measured on each unit of a simple random sample of size n
(assumed to be large) then show that variance of R=~ is approximately
III
N
» —Rx.)?
af XO x;)
R="? W=1
where R=2 is the ratio of population means and f= N is the size of
X é
population. 15
Considering the model
Y= a+BX,+U,,
use the Durbin-Watson test for autocorrelation, with the following observations on
Y and X and give your conclusions :
X:{ 14} 2 3 | 4/5} 6] 7
Y:/ 2] 2 2 1 3 | 5] 6
(Given : the values of d, and d, on 5% level of significance with n = 7 and one
explanatory variable are d, = 0-700 and d,, = 1-356) 10
Describe autocorrelation function and correlogram. If a given time series of profit
consists of 90 observations then while doing correlogram what is appropriate lag
length to be taken and why ? Explain the role of correlogram in testing stationarity
of given-time series. 15
If there are two strata and @ be the ratio of actual (=) and (=) obtained through
2 2
Neyman Allocation in stratified random sampling, then ignoring f.p.c, show that
whatever be the values of N,, N, S; and S, the ratio
VO)
is never less than —4? = where V(¥,)y and V(j,,) represent the variances of
(1+9)°
sample mean under Neyman Allocation and actual allocation respectively. 15
5 SNCS-C-STS
6.(b) Obtain the Least Square Estimator of 8 in General Linear Model (GLM)
Y=XB+U
nxl nxk —
1
with usual assumptions, when linear restrictions are imposed in the form
r=R B
Gl gxk py > 7 and R are known constants.
Also obtain variance-covariance matrix of this estimator. 15
6.(c) Describe the scope and problems which generally occur in the construction of
consumer price index numbers. 10
7.(a) What is the main drawback of Cumulative Total Method for drawing a pps sample ?
Explain Lahiri’s Method of drawing a pps sample.
Select a pps sample of size 4 by Lahiri’s Method from the population given below
if random numbers selected are (8, 16), (10, 13), (4, 26), (5, 35), (7, 26) 6+4
Serial Number of unit | Size of the unit
1 50
2 30
3 45
4 25
5 40
6 26
7 44
8 35
9 28
10 27
7b) Explain the concept of heteroscedastic disturbances in General Linear Regression
Model (GLRM). Also explain their consequences. 15
SNCS-C-STS 6
7.(c)
8.(a)
8.(b)
8.(c)
Differentiate between fixed base and chain base indices. In which type of situations
chain base indices are derived ? Write merits and demerits of chain base indices.
For the following series of index numbers of price of four commodities, calculate the
chain index number and fixed base index number. Write your conclusion. 15
Price in rupees
Commodity
2007 | 2008 | 2009 2010 | 2011
A 2 3 4 3 7
B 3 6 9 1 3
Cc 4 12 20 8 19
D 5 7 18 11 22
In case of cluster sampling with N clusters, each of size M, n(<N) clusters are
selected using simple random sampling without replacement (SRSWOR). Write
down an unbiased estimator (¥,) for estimating population mean (Y) in this case.
Obtain its variance and show that
v5) =(4-4)s3
_ 2 _ —2
where sf = WM =DSE— NDS; Is
What is the nature of autocorrelation disturbances in General Linear Regression
Model (GLRM) ? Also state the consequences of autocorrelated disturbances on
GLRM. 10
Explain the following terms using proper examples :
(i) Random walk with Drift
(ii) Trend stationary time series
(iii) Difference stationary time series
(iv) Deterministic trend
(v) Time series integrated of order d 15
7 SNCS-C-STS
