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

FOURTH SEMESTER – APRIL 2007
ST 4500 – BASIC SAMPLING THEORY
Date & Time: 21/04/2007 / 9:00 – 12:00 Dept. No. Max. : 100 Marks
SECTION – A
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Answer ALL questions ( 10 x 2 = 20 marks)
 What is meant by Census ? What are the constraints for carrying out a census?
 If T_{1 }and T_{2 } are unbiased for θ, then show that one can construct uncountable number of unbiased estimators for θ.
 Explain how a sample of size `n’ is drawn in SRSWOR using Lottery method.
 In SRSWOR, let y_{i} denote the yvalue of the i^{th} drawn unit. Find the discrete probability distribution of y_{i .}
 In PPS sampling, find the probability of selecting i^{th } population unit in a given draw.
 Show that under SRSWOR is more efficient than under SRSWR.
 Write all possible linear systematic samples , when N = 12 and n = 4.
 Describe Centered Systematic Sampling Scheme.
 Compute the number of units to be sampled from each stratum when there are 4 strata of sizes 40, 30, 60 and 70. The total sample size is 40.
 State V (_{st} ) under proportional allocation for a given sample size.
SECTION – B
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Answer any FIVE questions ( 5 x 8 = 40 marks)
 Show that an estimator can become biased under one sampling scheme even though it is unbiased under another sampling scheme.
 Under usual notations, derive cov (y_{i }, y_{j }) ; i ≠ j , in SRSWOR.
 Using the probability of selecting a subset of the population as a sample, prove that sample mean is unbiased for population mean.
 Show that Lahiri’s method of selection is a PPS selection.
 A population contains 5 units. It is known that
Y_{i }/ P_{i } – Y)^{2} P _{i} = 100 . Under PPSWR, compare
‘ = and ” = .
 Deduct the formula for , V() and v() in SRSWR using the formula for , V() and v() available in PPSWR.
 Describe circular systematic sampling with an example.
 Derive values of n_{h } such that c_{o} + is minimum for a given
value of V (_{st}).
SECTION – C
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Answer any TWO questions ( 2 x 20 = 40 marks)
 ( a ) In SRSWOR, derive V () by considering all possible samples and their
corresponding probabilities. ( 14 )
( b ) Let ν denote the number of distinct units in a simple random with replacement
sample of size 3 drawn from a population containing 4 units. Find P(ν =1) ,
P(ν =2) and P(ν = 3). ( 6 )
 ( a ) In SRSWOR, derive V () using probabilities of inclusion. ( 10 )
( b ) In CSS, assuming linear trend, prove the following :
( i ) The sample mean coincides with population mean when
k is odd.
( ii ) The sample mean is unbiased for population mean when
k is even. ( 10 )
 ( a ) Compare the mean based on distinct units with the sample mean under
( i ) SRSWR ,
( ii ) SRSWOR , taking N = 4 and n = 3. ( 8 )
( b ) Show that s^{2} / n is an unbiased estimator of V() under SRSWR. ( 12 )
 With 2 strata, a sampler would like to have n_{1} = n _{2} for administrative convenience instead of using the values given by the Neyman allocation. If V and V_{opt} denote the variances given by n_{1 }= n_{2 }and the Neyman allocation respectively, show that ^{2} , where
r = as given by Neyman allocation.
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