Download LOYOLA COLLEGE (AUTONOMOUS), CHENNAI –600 034 B.Sc., DEGREE EXAMINATION - STATISTICS

01.11.2003
9.00 - 12.00
LOYOLA COLLEGE (AUTONOMOUS), CHENNAI –600 034
B.Sc., DEGREE EXAMINATION - STATISTICS
FOURTH SEMESTER – NOVEMBER 2003
ST 4500 / STA 504 - BASIC SAMPLING THEORY
Max:100 marks
SECTION-A
Answer ALL the questions.
(10x2=20 marks)
1. Explain sampling frame and give two examples.
2. If there are two unbiased estimators for a parameter then show that one can construct,
uncountable number of unbiased estimators.
3. If T is an estimator for  , then show that MSE (T) = V(T) + [B(T)]2 .
4. Explain Lottery method for drawing random numbers.
5. Show that probability of including the ith population unit (i = 1, 2, …, N) when a
n
SRSWOR of size n is drawn from a population containing N units is
.
N
6. Find the probability of selecting ith population unit in cumulative total method.

N n
7. Examine whether the estimator  
 yi is unbiased for the population total under
n i 1
PPSWR.
8. Show that the sample mean y under SRSWOR is more efficient than y under SRSWR.
9. Explain Linear Systematic Sampling Scheme.
10. When do we use Neyman allocation?
SECTION-B
Answer any FIVE questions.
(5x8=40 marks)
11. Examine the validity of the following statement using a proper illustration :
"property of unbiasedness depends on the sampling scheme under use".
1
, r  1, 2, ....., N .
12. Prove that, under usual notations, in SRSWOR, P[yi = r ] 
N
13. What is PPS sampling? Describe cumulative total method?




14. Deduce expressions for  , V(  ) and v (  ) under SRSWR using the expressions for  ,


V(  ) and v (  ) available under PPSWR.
1 L
15. Prove that y st 
 N h y h is an unbiased estimator for population mean Y under
N h 1
stratified random sampling. Derive V ( y st ) .
16. Derive the formula for Neyman allocation.
17. Prove that the sample mean coincide with the population mean in Centered Systematic
Sampling, when there is linear trend in the population.
18. a) List all possible Balanced Systematic Samples if N = 40, n= 8.
b) List all possible Circular Systematic Samples if N = 7, n = 3.
SECTION-C
Answer any TWO questions.
(2x20=40 marks)
19. a) Describe the principal steps involved in the planning and execution of a survey. (14)
b) Let y denote the sample mean of only distinct units under SRSWR. Find E y
 
 
and V y .
(6)
5
20. a) A population contains 5 units and it is known that
 i
  
i 1

i
2

   i  100. Compare


 y1 y 2 
2 y1 1 y 2
with
. Find the values of  for which



2 


3 P1 3 P2
 P1 P2 


y
y
   1  (1 ) 2 is less efficient than  1 .
P1
P2
b) Show that Lahiri' s method of selection is a PPS selection.

1 
1
2
(12)
(8)

21. a) Show that an unbiased estimator of V(  HHE ) is

1
v ( HHE ) 
n (n  1)
2
 yi 

   HHE  .

i 1  Pi

n
(10)
L
b) Derive values of nh such that Co +
C n
h 1
h
h
is minimum for a given value of
V ( y st ) .
(10)
22. a) Compare V ( y SRS ) , Vnopt ( y st ) and V prop ( y st ) assuming Nh is large for all h = 1,2, ….,
L.
(12)
b) A sampler has 2 strata. He believes that S1 and S2 can be taken as equal. For a given
V prop ( y st )
W1 c1  W2 c 2
cost c = c1 n1 + c2 n2, show that
=
.
(8)
Vopt ( y st )
(W1 c1  W2 c 2 ) 2
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