02.04.2004
9.00 - 12.00
LOYOLA COLLEGE (AUTONOMOUS), CHENNAI –600 034
B.Sc., DEGREE EXAMINATION - STATISTICS
FOURTH SEMESTER – APRIL 2004
ST-4500- BASIC SAMPLING THEORY
Max:100 marks
SECTION -A
Answer ALL questions. Each carries TWO marks.
(10  2 = 20 marks)
1. Define Sampling frame. Give two examples.
2. If there are two unbiased estimators for a parameter, then show that one can construct
uncountable number of unbiased estimators for that parameter.
3. If T is an estimator for  , then show that MSE (T) = V (T) + B(T )2
4. Explain Lottery method of drawing simple random sample of size n.
5. Show that the probability of selecting a given subset consisting of 'n' units of the
1
population of N units is
.
N
 
n
6. Find the probability of selecting ith population unit in a given draw in PPS sampling.
N
7. Examine whether the estimator Yˆ 
n
n
 yi
is unbiased for the population total under
i 1
PPSWR.
8. Prove that the sample mean is a more efficient estimator of population mean under
SRSWOR than under SRSWR.
9. Explain circular systematic Sampling Scheme.
10. Compute the number of units to be sampled for each stratum under proportional
allocation scheme, when the total sample size is 40 and there are 4 strata of sizes 40, 30,
60, and 70.
SECTION - B
Answer any FIVE questions. Each carries EIGHT marks.
(5  8 = 40 marks)
11. Examine the validity of the following statement using a proper illustration: 'An unbiased
estimator under one method of sampling can become a biased estimator under another
method of sampling'.
12. Show that, in SRSWOR,
2
; i, j  1, 2, ..., n; i  j.
cov (yi , yj) = N 1
13. Prove that sample mean is unbiased for population mean in SRSWOR by using the
probability of selecting a subset of the population as a sample.
14. What is PPS sampling? Describe cumulative total method.
15. Derive the variance of Hansen Hurwitz estimator for population total.
16. Derive E ( ( y v ) and Var ( yv ) , where y v denotes the sample mean based on only distinct
units under SRSWR.
17. Write a descriptive note on centered Systematic Sampling.
18. Derive the formula for n h under Neyman allocation.
SECTION - C
Answer any TWO questions. Each carries TWENTY marks.
(2  20 = 40 marks)
19. a) Derive the variance of sample mean, V( y ) in SRSWOR by using probabilities of
inclusion.
(10)
b) Describe Lahiri's method. Show that Lahiri's method of selection is a PPS selection.
(10)
5
Y
20. (a) A population contains 5 units and it is known that  ( i  Y ) 2 Pi  100.
i 1 Pi
1y y 
2 y1 1 y 2
Compare Yˆ '   1  2  and Yˆ '' 

. Find the values of  for which
2  P1 P2 
3 P1 3 P2
y
y
(10)
Yˆ   1 .  (1   ) 2 is less efficient than Yˆ ' .
P1
P2
b) Derive the expressions for Yˆ , V (Yˆ ) and v ( yˆ ) in SRSWR using the expressions for
Yˆ , V (Yˆ ) and v ( yˆ ) available in PPSWR.
(10)
21. a) Show that when Yi   i (i 1, 2, ..., N ), yˆ BSS  y , irrespective of the random start
'r' .
(10)
b) In Linear systematic Sampling, when N is not a multiple of n, explain the undesirable
situations encountered using suitable illustrations.
(10)
22. a) Compare V ( y SRS ), V n opt ( y st ) and V prop ( y st ) assuming N h is large for all h = 1, 2,
... , L.
(10)
b) A sampler has two strata. He believes that stratum standard deviations are the same.
For a give cost C = C1n1+C2n2, show that
W1 C1  W2 C 2
V prop ( y st )

.
(10)
Vopt ( y st )
(W1 C1  W2 C 2 ) 2
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