LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – STATISTICS

AC 14

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)

 

1. What is meant by Census ? What are the constraints for carrying out a census?
2. If T₁ and T₂  are unbiased for θ, then show that one can construct uncountable number of unbiased estimators for θ.
3. Explain how a sample of size `n’ is drawn in SRSWOR using Lottery method.
4. In SRSWOR, let y_i denote the y-value of the i^th drawn unit. Find the discrete probability distribution of y_{i .}
5. In PPS sampling, find the probability of selecting   i^th   population unit in a given draw.
6. Show that under SRSWOR is more efficient than  under SRSWR.
7. Write all possible linear systematic samples , when N = 12 and  n = 4.
8. Describe Centered Systematic Sampling Scheme.
9. 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.
10. State V (_st ) under proportional allocation for a given sample size.

 

SECTION – B

——————-

Answer any FIVE questions                                            ( 5 x 8 = 40 marks)

 

11. Show that an estimator can become biased under one sampling scheme even though it is unbiased under another sampling scheme.
12. Under usual notations, derive cov (y_i , y_j ) ; i ≠ j , in SRSWOR.
13. Using the probability of selecting a subset of the population as a sample, prove that sample mean is unbiased for population mean.
14. Show that Lahiri’s method of selection is a PPS selection.
15. A population contains 5 units. It is known that

Y_i / P_i   –  Y)² P _i  =  100 .  Under PPSWR, compare

 

‘   =        and    ”   =   .

 

16. Deduct the formula for ,  V()   and  v() in SRSWR using the formula  for  ,  V()  and   v() available in PPSWR.
17. Describe circular systematic sampling with an example.
18. Derive values of n_h  such that c_o  +  is minimum for a given

value of  V (_st).

 

SECTION – C

——————-

Answer any TWO questions                                            ( 2 x 20 = 40 marks)

 

19. ( 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 )

20. ( 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 )

21. ( 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 )

 

22. 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 V_opt 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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