LOYOLA COLLEGE (AUTONOMOUS), CHENNAI –600 034
M.Sc., DEGREE EXAMINATION - STATISTICS
FIRST SEMESTER – NOVEMBER 2004
ST 1805/1802 - SAMPLING THEORY
25.10.2004
Max:100 marks
9.00 - 12.00 Noon
SECTION - A
(10  2 = 20 marks)
Answer ALL the questions
1. Define probability Sampling Design and explain the meaning of probability Sampling.
2. Distinguish between varying size sampling design and fixed size sampling design. Give
an example for each design.
3. Define the following:
a) Inclusion indicator
b) First and Second order inclusion probabilities.
4. Prove the following:
a) Ep [Ii (s)] = i ; i = 1, 2, ..., N
b) Ep [Ii (s) Ij (s)] = ij ; i, j = 1,2, ..., N ; i  j.
5. Show that an unbiased estimator for the population total can be found if an only if the
first order inclusion probabilities are positive for all N units in the population.
6. Derive the formula for i and ij under Simple Random Sampling Design.
7. Describe the Linear Systematic Sampling Scheme and write its probability sampling
design.

8. Derive the approximate bias of the ratio estimator Y R for the population total Y.
9. In cluster sampling, suggest an unbiased estimator for the population total. Write the
variance of the unbiased estimator.
10. Explain Multistage Sampling.
SECTION - B
(5  8 = 40 marks)
Answer any FIVE questions.
11. Show that an estimator can be unbiased under one design but biased under another
design.
12. For any sampling design, prove the following:
N
(i)
 π ij  (n 1) π i
j 1
i
N
(ii)
 (π i π j  π ij   i (1   i ),
i 1, 2, ..., N .
j 1
i
13. Suggest an unit drawing mechanism for simple random sampling design and prove that
the unit drawing mechanism implements the simple random sampling design.
14. Explain Lahiri's method of PPS sampling. Show that Lahiri's method of selection is a
PPS selection method.
15. Write the reason for using Desraj ordered estimator instead of Horwitz - Thompson
estimator under PPSWOR sampling scheme. Also prove that the Desraj ordered
estimator is unbiased for the population total.
1
16. Describe the Random Group Method of Sampling. Find an unbiased estimator of
population total under this method and derive its variance.


17. Compare V ( Y LSS ) and V ( Y SRS ) assuming the population values Yi satisfy
Yi =  + i, i = 1, 2, ...N.
[
18. Explain Warner's randomized response technique for estimating the proportion A of the
persons belonging to group A in a population.
SECTION - C
(2  20 = 40 marks)
Answer any TWO questions
19. a) Under any design P(), derive the variance of Horwitz - Thompson estimator and find
its estimated variance.
(16)
b) Define Midzuno Sampling Design and state the unit drawing mechanism for this
design.
(4)
20. After the decision to take a simple random sample has been made, it was realized that the
value of unit with level 1 would be unusually low and the value of unit with label N
would be unusually high. In such cases, it is decided to use the estimator.

*
Y
Y C
if s  1 , s  N
YC
if s  1 , s  N

Y
otherwise,
where C is a pre-determined constant. Show that

* is unbiased for Y for any C.
(i)
Y

(ii)
Derive V ( * )
Y


(iii)
Find the value of C for which * is more efficient than
.
Y
Y


21. a) Show that Linear Regression estimator Y LR is more efficient than Ratio estimator Y R
unless  = R.
(4)
b) Assuming samples are drawn using SRS in both the phases of double sampling,



suggest X , Y , and X d when
(i)
the second phase sample is a subsample of the first phase sample.
(ii)
the second phase sample is independent of the first phase sample.


22. a) In stratified sampling, deduct Y ST , V( Y ST ) and
(i)
SRS is used in all strata
(ii)
PPSWR sampling is used in all strata.
(16)

( Y ST ) assuming
(12)
b) Obtain the variance of the following:
(i)
Hansen - Horwitz estimator in Double Sampling.

(ii)
Estimator Y TS in Two - stage Sampling.
(8)
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