LOYOLA  COLLEGE (AUTONOMOUS), CHENNAI-600 034.

M.Sc. DEGREE  EXAMINATION  – STATISTICS

FIRST SEMESTER  – APRIL 2003

ST  1802/ S  717   SAMPLING  THEORY

08.04.2003

1.00 – 4.00                                                                                             Max: 100 Marks

SECTION – A                                (10 ´ 2 = 20 Marks)

Answer ALL the questions.  Each carries two marks

1. Let the sampling design be

 

               If N=3 then what is the value of p₆₈?

 

2. Given a fixed size sampling design yielding sample of size 5, what is the value

of ?

3. Under what condition the mean square error of an estimator becomes its variance?
4. List all possible balanced systematic samples of size 4 when N = 12.
5. Under usual notations order V_SRS, V_SYS,V_STR assuming the presence of linear trend.
6. Name any two methods of PPS selection.
7. When N=16 and n = 4, what will be your choice for random group sizes in random group method? Give reason.
8. Define ratio estimator for the population total.
9. Name any two randomised response techniques.
10. Explain the term: Optimum allocation.

 

SECTION  – B                                              (5 ´ 8 = 40 Marks)

Answer any FIVE.  Each carries eight marks.

11. Show that under SRS,

 

where

12. Explain any one method of PPS selection in detail with a supportive example.
13. Show that under balanced systematic sampling, the expansion estimator coincides with the population total in the presence of linear trend.
14. Derive the mean square error of and obtain the condition under which is more

efficient than .

 

 

 

 

 

15. Explain the usefulness of two phase sampling in pps sampling.
16. Describe in detail any one method of Randomised Response technique.
17. Derive under Neyman allocation.
18. Verify the following relations with an example

 

(Proof should not be given)

 

Section – C                                    (2 ´ 20 =20 Marks)

Answer any TWO questions.  Each carries twenty marks.

 

19. Describe random group method. Suggest an unbiased estimator for the

population total and derive its variance.

20. Derive the first and second order inclusion probabilities in Midzeno sampling and

show that the Yates-Grundy estimator is nonnegative

21. Develop Yates-corrected estimator.
22. (a)  Describe how double sampling is employed in ratio estimation              (10)

 

(b)  Write a descriptive note on two stage sampling.                                     (10)

 

 

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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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