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1.६.5/11.5.5. Examination, 2024
SGSE-F-STS
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 in 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 Booklet must be
clearly struck off.
Answers must be written in ENGLISH only.
SGSE-F-STS 1
౧1. (a) The following table shows the initial probability of selection (01) of the
seven units of a population U : (0; ; i = 1, 2, 3, 4, 5, 6, 7). Let a random
sample of size 3 be selected from the population using probability
proportional to size without replacement scheme. Let the units selected
in the three successive draws be U7, 03 and Us respectively. Find the
SECTION. A
selection probabilities with which these units were selected :
Population units (U;) رتا 172 Us U4 Us Ug U7
Label (i) 1 2 3 4 5 | 6 | iL
2 ౨౮ 15011 (1౮ 0:20 0-12 0 0-14 | 0-07 | 0-03
selection (p;)
(b) What are ‘Linear Systematic’ and ‘Circular Systematic’ sampling
schemes ? Under what condition is Linear Systematic sampling scheme
applied for selecting a sample of size n from the population of N units ?
Show that Linear Systematic sampling is also a probability sampling
scheme which ensures equal probability of inclusion of each unit of the
population in the sample.
(c) Explain the processes for analysing an economic problem, being an
econometrician following the classical methodology.
Illustrate the proceeding steps with Keynesian theory of consumption.
Q2. (a) The joint pdf of (© Y) is given by
12 Y) = |
Find the regression of Y on X and X on Y. Interpret the obtained results.
(b) Distinguish clearly between fixed base and chain base index numbers,
and point out their relative merits and demerits. Calculate fixed base
|| 0.< 1
0, otherwise గ
index numbers from the chain base index numbers given below :
Year 2005 | 2006 | 2007 | 2008 | 2009 | 2010 | 2011
ee | ७ | 102 | 115 | 135 | 125 | 108 | 95
Number
(c) Discuss the components of national income according to the expenditure
approach. How are these components calculated ?
SGSE-F-STS
2
15
10
15
15
10
౧౩. (৪)
(b)
(c)
౧4. (8)
(b)
(c)
SGSE-F-STS
SECTION B
In a class of 200 students, 15 failed in a paper of Mathematics. If a
random sample of 5 students was selected from the population using
simple random sampling without replacement scheme in order to
estimate the proportion of failed students in the population, find the
sampling variance of the suggested estimator.
What do you mean by interpenetrating sub-sampling ? With what
purposes are they used often ? Explain its use in any sampling scheme.
Define Das and Des Raj estimators of population mean on the basis of a
sample of size n. Taking n = 2, show that Des Raj’s estimator is an
unbiased estimator. Obtain the sampling variance of Des Raj’s estimator
forn=2.
A population is divided into two strata such that the ratio of their sizes,
Nj and Ng is 11 and ratio of their standard deviations, 61 and وہ is 9.
The ratio of the sample sizes for an arbitrary allocation 15 01/09 and that
for Neyman allocation is nj 7 119 and the ratio of 11/12 and 71 /nj, 15 2. If
the objective is to estimate the population mean by taking with
replacement samples from within strata, compare the precision of the
two allocations.
Describe the operational procedure of Lahiri’s method for selecting a
random sample with unequal probability of selection. Show theoretically
that with this procedure the units are selected with probability
proportional to their size. Mention the advantages and disadvantages of
this method.
Give the operational procedure of two-stage sampling with the help of an
example. Suggest an unbiased estimator of population mean.
A population of 6 clusters, each having 6 values of the study variable Y
is given below :
Cluster 7781008 |Cluster Y-values | (पालः Y-values
I 97426. 5555. 61%. ` 3, 82:15: ૪ 2,4685
1 એ 0 ગ ১ मे 0056160 4.1.228
Let the clusters II, IV and VI be selected randomly in the first-stage
sample. Further, let the Y-values (2, 7, 3) of cluster II, (6, 3, 1) of
cluster IV and (7, 4, 3) of cluster VI be selected randomly in the
second-stage sample. Estimate the population mean on the basis of the
suggested estimator.
10
15
15
15
10
15
Q5. (8)
(b)
(c)
Q6. (a)
(b)
(০)
07. (a)
(b)
(c)
SGSE-F-STS
Explain identification problem. What are the conditions for
identification ?
Examine the identifiability of the following supply-demand model
clearly mentioning the assumptions used, if any.
Qa = 30 + 017 + ઈ
છડ =A + 012 + ૪
Given, 68 = જેડ. 15
What is heteroscedasticity ? Explain the consequences of
heteroscedasticity on OLS estimators. Also explain Spearman’s Rank
Correlation test for heteroscedasticity. 15
Discuss the Koyck method of estimating distributed-lag model and also
explain the features of Koyck transformation. 10
Describe the methods used for estimating simultaneous equation
models, including the Two Stage Least Squares (2 SLS) and Three Stage
Least Squares (3 SLS) methods with their advantages and limitations. 15
Explain the von Neumann test for auto-correlation and provide the
conclusions. Draw the relationship between Durbin-Watson (DW)
d-statistic and von Neumann test. 15
What is multicollinearity ? Why does multicollinearity arise ? How is
multicollinearity detected using Farrar-Glauber Test ? 10
(i) What do you understand by seasonal variations in a time series
data? Give example. Explain the method of link relatives of
computing the seasonal indices.
(ii) ` Write down the utility of time series for economic and business
data. 946
What do you mean by income elasticity of demand ? Given family-budget
data, how would you estimate this elasticity ? What adjustments would
you make for variation in the size of the family ? 19
Find the first three autocorrelation coefficients of the following moving
average process :
20 - 10 ಲ್ಯ 0-8 டோர் 0-4 কা 0-2 €¿ 3›
where {e;} are independent and identically distributed, each with normal
distribution having mean 0 and the same variance. 10
4
९8. (a)
(b)
(c)
SGSE-F-STS
(i)
(ii)
What are the assumptions underlying the ARIMA model, and how
do violations of these assumptions affect model performance ?
How do you interpret ACF (Autocorrelation Function) and PACF
(Partial Autocorrelation Function) plots in the context of ARIMA
model ? 8-7
What are the advantages and limitations of using a periodogram for
spectral analysis ? 10
ദ
(ii)
Describe various types of errors that can occur in index numbers.
During a certain period the cost of living index goes up from
110 to 200 and the salary of a worker is also raised from ₹ 500 to
₹ 900. Does the worker really gain and if so, by how much, in real
terms ? 9-6
