LOYOLA COLLEGE (AUTONOMOUS), CHENNAI –600 034

M.Sc., DEGREE EXAMINATION – STATISTICS

FIRST SEMESTER – NOVEMBER 2003

ST-1802/S717 – SAMPLING THEORY

08.11.2003                                                                                                           Max:100 marks

1.00 – 4.00

 

SECTION-A

 

Answer ALL  the questions.                                                                             (10×2=20 marks)

 

1. Explain probability sampling design.
2. Given N = 5, n = 3, X₂ = 2, X₃ = 3, X = 25. Compute p₂₃ under Midzuno Sampling design.
3. Distinguish between inclusion probabilities and inclusion indicators.
4. List all possible Balanced Systematic Samples when N = 30 and n = 6.
5. Define Des-Raj ordered estimator.
6. Define Horvitz-Thompson estimator.
7. Write a short note on Yates corrected estimator under Linear systematic sampling.
8. Describe two phase sampling.
9. When is stratified sampling used?
10. Define proportional allocation.

 

SECTION-B

 

Answer any FIVE questions.                                                                           (5×8=40 marks)

 

11. Derive variance of Horvitz-Thompson estimator in Yates-Grundy form.
12. Explain Lahiri’s method and show that Lahiri’s method of selection is a probability proportional to size selection method.
13. Write a note on Warner’s model.
14. Explain ratio estimator, also derive the approximate bias and mean square error of the estimator.
15. Compare Linear systematic sampling and simple random sampling in the presence of a linear trend.
16. Develop Hartly-Ross unbiased ratio type estimator.
17. Derive variances and covariances in the two cases of two phase sampling, assuming simple random sampling is used in both the phases of sampling.
18. Describe Two-stage sampling. Give the unbiased estimator and also derive the variance of the unbiased estimator.

 

 

 

 

 

 

 

SECTION-C

Answer any TWO questions.                                                                           (2×20=40 marks)

 

19. a) Derive V() for n = 2.            (15)
20. b) For any fixed size sampling design yielding samples of size n, prove that

(5)

20. a) Describe Midzuno sampling design and show that it is a sampling design. (5)
21. b) Derive the first and second order inclusion probabilities under Midzuno sampling

design.                                                                                                                          (15)

21. a) Develop Yates corrected estimator under linear systematic sampling. (10)
22. b) Suppose from a sample of n units selected with simple random sampling (SRS) a

subsample of n’ units is selected with SRS duplicated and added to the original

sample. Derive the expected value and the approximate sampling variance of , the

sample mean based on the n+n’ units.                                                                         (10)

22. a) Write a note on proportional allocation for a given cost. Also deduct V  under it

assuming SRS is used in all strata.                                                                             (10)

1. b) A sampler has two strata with relative sizes and . He believes that

S₁, S₂ can be taken as equal.  For a given cost C = C₁ n₁ + C₂ n₂,  show that (assuming

N_h is large).

(10)

 

 

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