LOYOLA  COLLEGE (AUTONOMOUS), CHENNAI-600 034.

B.Sc. DEGREE  EXAMINATION  – STATISTICS

FOURTH SEMESTER  – APRIL 2003

ST   4500/ sta  504   BASIC SAMPLING   THEORY

11.04.2003

9.00 – 12.00                                                                                           Max: 100 Marks

SECTION – A                                  (10 ´ 2 = 20 Marks)

Answer ALL the questions.  Each carries TWO marks.

1. Define a population and mention the assumptions made regarding population size in sampling Theory.
2. Explain ‘Sampling frame’ of a population.
3. What are the constraints for carrying out a census?
4. Distinguish between a statistic and a parameter.
5. Derive the expression for the mean square error of an estimator T in estimating q in terms of the variance and bias of the estimator T.
6. In SRSWOR, find the probability of selecting r^th population unit in i^th
7. Explain the cumulative total method of selecting a PPS sample.
8. Write all possible linear systematic samples of size n = 6 given N = 24.
9. Explain stratified random sampling.
10. Name any three allocation schemes used in stratified sampling.

 

 

SECTION –B                                                (5 ´ 8 = 40 Marks)

Answer any FIVE questions.  Each carries EIGHT marks.

 

11. Consider a population containing 4 units with values 7,4,10,5. Two units are drawn one by one without replacement with equal probabilities.  Verify whether or not the statistic T =(3y₁+ 4y₂) /5 is unbiased for the population mean.
12. Show that, under usual notations, in SRSWOR

 

13. Show that Lahiri’s method of selection is a PPS selection.
14. Examine whether the estimator is unbiased for the population total under PPSWR, where y_i is the value of i^th drawn unit (i = 1,2,….,n).
15. Deduce is SRSWR using expressions for available in PPSWR.
16. Letdenote the sample mean of only distinct units under SRSWR.

Find

17. Show that in Balanced systematic sampling, the sample mean coincides with

population mean in the presence of linear trend.

18. In stratified sampling, under proportional allocation, derive Vand find

an unbiased estimator of V.

 

 

 

 

 

 

SECTION – C                                (2 ´ 20 = 40 Marks)

Answer any TWO questions.  Each carries TWENTY marks.

 

19. (a) Illustrate that an estimator T can become biased under a sampling scheme

even though T is unbiased under another sampling scheme..                         (8)

(b) A population contains 5 units and it is known that   (12)

Under the usual notations, (i) compare  with

(ii) Find the values of a for which  is less efficient than.(10)

20. (a) consider the following data:

 

Plot number	No. of plants	No. of roses
1	20	56
2	32	65
3	14	30
4	19	47
5	22	37
6	7	28

 

Assume that a PPSWR sample of size 2 is drawn.  Compute Vfor the

data given.  Compare it with V.                                                       (10)

• Derive an unbiased estimator of Vin PPSWR scheme.                  (10)

 

21. (a) Describe centered systematic sampling.                                                 (5)

• Explain in detail the principal steps involved in the planning and

execution of  a sample survey.                                                               (15)

 

• (a) If  C_o is the overhead cost and  C_h is the cost of collecting  information per

unit in stratum h, then the total  cost of the survey is given by C_o+

In optimum allocation, determine the values of n₁, n₂,……,n_L  for which

V  is minimum subject to a fixed total cost.                                          (12)

• Show that if Y_i = a + b_i, i =1,2,..,N, then ,

where a stratified sample of size n is drawn as follows:

Divide the population into n groups of k units each.  And draw 1 unit from each group where the first stratum contains the first k units, the second

stratum contains the next k units and so on.                                             (8)

 

 

 

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