LOYOLA COLLEGE (AUTONOMOUS), CHENNAI –600 034

M.Sc., DEGREE EXAMINATION – STATISTICS

FIRST SEMESTER – NOVEMBER 2004

ST 1805/1802 – SAMPLING THEORY

25.10.2004                                                                                                           Max:100 marks

9.00 – 12.00 Noon

 

SECTION – A

 

Answer ALL the questions                                                                            (10 ´ 2 = 20 marks)

 

1. Define probability Sampling Design and explain the meaning of probability Sampling.
2. Distinguish between varying size sampling design and fixed size sampling design. Give an example for each design.
3. Define the following:
4. a) Inclusion indicator
5. First and Second order inclusion probabilities.
6. Prove the following:
7. E_p [I_i (s)] = p_i ; i = 1, 2, …, N
8. E_p [I_i (s) I_j (s)] = p_ij ; i, j = 1,2, …, N ; i ¹
9. Show that an unbiased estimator for the population total can be found if an only if the first order inclusion probabilities are positive for all N units in the population.
10. Derive the formula for p_i and p_ij under Simple Random Sampling Design.
11. Describe the Linear Systematic Sampling Scheme and write its probability sampling design.
12. Derive the approximate bias of the ratio estimator for the population total Y.
13. In cluster sampling, suggest an unbiased estimator for the population total. Write the variance of the unbiased estimator.
14. Explain Multistage Sampling.

 

SECTION – B

 

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

 

11. Show that an estimator can be unbiased under one design but biased under another design.

 

12. For any sampling design, prove the following:

 

13. Suggest an unit drawing mechanism for simple random sampling design and prove that the unit drawing mechanism implements the simple random sampling design.

 

14. Explain Lahiri’s method of PPS sampling. Show that Lahiri’s method of selection is a PPS selection method.

 

15. Write the reason for using Desraj ordered estimator instead of Horwitz – Thompson estimator under PPSWOR sampling scheme. Also prove that the Desraj ordered estimator is unbiased for the population total.
16. Describe the Random Group Method of Sampling. Find an unbiased estimator of population total under this method and derive its variance.
17. Compare V () and V () assuming the population values Y_i satisfy

Y_i = a + bi, i = 1, 2, …N.

[

18. Explain Warner’s randomized response technique for estimating the proportion p_A of the persons belonging to group A in a population.

 

SECTION – C

 

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

 

19. a) Under any design P(×), derive the variance of Horwitz – Thompson estimator and find

its estimated variance.                                                                                                (16)

1. b) Define Midzuno Sampling Design and state the unit drawing mechanism for this

design.                                                                                                                          (4)

 

20. After the decision to take a simple random sample has been made, it was realized that the value of unit with level 1 would be unusually low and the value of unit with label N would be unusually high. In such cases, it is decided to use the estimator.

 

 

 

 

 

where C is a pre-determined constant.  Show that

• is unbiased for  for any C.
• Derive V ()
• Find the value of C for which is more efficient than .

 

21. a) Show that Linear Regression estimator is more efficient than Ratio estimator

unless b = R.                                                                                                                (4)

 

1. b) Assuming samples are drawn using SRS in both the phases of double sampling,

suggest , and  when

• the second phase sample is a subsample of the first phase sample.
• the second phase sample is independent of the first phase sample. (16)

 

22. a) In stratified sampling, deduct , V() and () assuming

• SRS is used in all strata
• PPSWR sampling is used in all strata.            (12)

 

1. b) Obtain the variance of the following:

• Hansen – Horwitz estimator in Double Sampling.
• Estimator in Two – stage Sampling.                                            (8)

 

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