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
Answer ALL questions ( 10 x 2 = 20 marks)
- What is meant by Census ? What are the constraints for carrying out a census?
- If T1 and T2 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 yi denote the y-value of the ith drawn unit. Find the discrete probability distribution of yi .
- In PPS sampling, find the probability of selecting ith 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
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 (yi , yj ) ; 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
Yi / Pi – 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 nh such that co + is minimum for a given
value of V (st).
SECTION – C
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 s2 / n is an unbiased estimator of V() under SRSWR. ( 12 )
- With 2 strata, a sampler would like to have n1 = n 2 for administrative convenience instead of using the values given by the Neyman allocation. If V and Vopt denote the variances given by n1 = n2 and the Neyman allocation respectively, show that 2 , where
r = as given by Neyman allocation.
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