LOYOLA COLLEGE (AUTONOMOUS), CHENNAI –600 034

B.Sc., DEGREE EXAMINATION – STATISTICS

FOURTH SEMESTER – NOVEMBER 2003

ST  4500 / STA 504 – BASIC SAMPLING THEORY

01.11.2003                                                                                                           Max:100 marks

9.00 – 12.00

SECTION-A

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

 

1. Explain sampling frame and give two examples.
2. If there are two unbiased estimators for a parameter then show that one can construct, uncountable number of unbiased estimators.
3. If T is an estimator for , then show that MSE (T)  =  V(T)  +  [B(T)]² .
4. Explain Lottery method for drawing random numbers.
5. Show that probability of including the i^th population unit (i =  1, 2, …, N) when a SRSWOR of size n is drawn from a population containing N units is .
6. Find the probability of selecting i^th population unit in cumulative total method.
7. Examine whether the estimator is unbiased for the population total under PPSWR.
8. Show that the sample mean under SRSWOR is more efficient than under SRSWR.
9. Explain Linear Systematic Sampling Scheme.
10. When do we use Neyman allocation?

 

SECTION-B

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

 

11. Examine the validity of the following statement using a proper illustration :

“property of unbiasedness depends on the sampling scheme under use”.

12. Prove that, under usual notations, in SRSWOR, P[y_i =
13. What is PPS sampling? Describe cumulative total method?
14. Deduce expressions for , V() and v () under SRSWR using the expressions for , V() and v () available under PPSWR.
15. Prove that is an unbiased estimator for population mean under stratified random sampling.  Derive .
16. Derive the formula for Neyman allocation.
17. Prove that the sample mean coincide with the population mean in Centered Systematic Sampling, when there is linear trend in the population.
18. a) List all possible Balanced Systematic Samples if N = 40,  n= 8.
19. b) List all possible Circular Systematic Samples if N = 7, n = 3.

 

 

 

 

SECTION-C

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

 

19. a) Describe the principal steps involved in the planning and execution of a survey. (14)
20. b) Let denote the sample mean of only distinct units under SRSWR. Find E

and V .                                                                                                     (6)

20. a) A population contains 5 units and it is known that Compare

with .  Find the values of a for which

is less efficient than .                                                 (12)

1. b) Show that Lahiri’ s method of selection is a PPS selection. (8)
2. a) Show that an unbiased estimator of V() is

(10)

1. b) Derive values of n_h such that C_o + is minimum for a given value of

.                                                                                                                                  (10)

22. a) Compare and assuming N_h is large for all h = 1,2, …., L.

(12)

1. b) A sampler has 2 strata. He believes that S₁ and S₂ can be taken as equal. For a given

cost c = c₁ n₁ + c₂ n₂,   show that   =   .                      (8)

 

 

 

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