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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI600 034.
B.Sc. DEGREE EXAMINATION  STATISTICS
FOURTH SEMESTER  APRIL 2003
ST 4500/ STA 504 BASIC SAMPLING THEORY
11.04.2003
9.00  12.00
Max: 100 Marks
SECTION  A
(10  2 = 20 Marks)
Answer ALL the questions. Each carries TWO marks.
01. Define a population and mention the assumptions made regarding population size
in sampling Theory.
02. Explain ‘Sampling frame’ of a population.
03. What are the constraints for carrying out a census?
04. Distinguish between a statistic and a parameter.
05. Derive the expression for the mean square error of an estimator T in estimating 
in terms of the variance and bias of the estimator T.
06. In SRSWOR, find the probability of selecting rth population unit in ith draw.
07. Explain the cumulative total method of selecting a PPS sample.
08. Write all possible linear systematic samples of size n = 6 given N = 24.
09. Explain stratified random sampling.
10. Name any three allocation schemes used in stratified sampling.
SECTION B
Answer any FIVE questions. Each carries EIGHT marks.
(5  8 = 40 Marks)
11. Consider a population containing 4 units with values 7,4,10,5. Two units are
drawn one by one without replacement with equal probabilities. Verify whether
or not the statistic T =(3y1+ 4y2) /5 is unbiased for the population mean.
12. Show that, under usual notations, in SRSWOR
 2
cov yi , y j  
; i, j  1,2,..., n ; i  j.
N 1
13. Show that Lahiri’s method of selection is a PPS selection.
N n
ˆ
14. Examine whether the estimator Y   y i is unbiased for the population total
n i 1
under PPSWR, where yi is the value of ith drawn unit (i = 1,2,….,n).
15. Deduce Yˆ , V (Yˆ ) and  Yˆ is SRSWR using expressions for
Yˆ , V (Yˆ ) and  Yˆ available in PPSWR.


16. Let y  denote the sample mean of only distinct units under SRSWR.
 
 
Find E y  and V y .
17. Show that in Balanced systematic sampling, the sample mean coincides with
population mean in the presence of linear trend.
18. In stratified sampling, under proportional allocation, derive V y st and find
 
 
an unbiased estimator of V y st .
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SECTION  C
Answer any TWO questions. Each carries TWENTY marks.
(2  20 = 40 Marks)
19. (a) Illustrate that an estimator T can become biased under a sampling scheme
even though T is unbiased under another sampling scheme..
(8)
2
 Yi

  Y  Pi  100.

(b) A population contains 5 units and it is known that i 1  Pi
(12)

5
 y1 y 2 
ˆ " 2 y1  1 y 2 .
   with Y 
3 P1 3 P2
 P1 P2 
Y
y
(ii) Find the values of  for which Yˆ   1  (1   ) 2 is less efficient than Yˆ ' .(10)
P1
P2
20. (a) consider the following data:
Under the usual notations, (i) compare yˆ ' 
Plot number
1
2
3
4
5
6
No. of plants
20
32
14
19
22
7
1
2
No. of roses
56
65
30
47
37
28


Assume that a PPSWR sample of size 2 is drawn. Compute V YˆHHE for the
 
data given. Compare it with V ŶSRS .

(10)

(b) Derive an unbiased estimator of V YˆHHE in PPSWR scheme.
(10)
21. (a) Describe centered systematic sampling.
(b) Explain in detail the principal steps involved in the planning and
execution of a sample survey.
(5)
(15)
22 (a) If Co is the overhead cost and Ch is the cost of collecting information per
L
unit in stratum h, then the total cost of the survey is given by Co+  C h nh .
h 1
In optimum allocation, determine the values of n1, n2,……,nL for which
V y st is minimum subject to a fixed total cost.
(12)
 
(b)
     
Show that if Yi =  + i, i =1,2,..,N, then V y st  V y sys  y SRS ,
where a stratified sample of size n is drawn as follows:
Divide the population into n groups of k units each. And draw 1 unit from
each group where the first stratum contains the first k units, the second
stratum contains the next k units and so on.
(8)
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