LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – STATISTICS

AC 34

SECOND SEMESTER – APRIL 2007

ST 2810 – SAMPLING THEORY

 

 

 

Date & Time: 21/04/2007 / 1:00 – 4:00Dept. No.                                              Max. : 100 Marks

 

 

SECTION – A

——————-

Answer ALL questions                                                     ( 10 x 2 = 20 marks)

1. Define Probability Sampling Design and mention its two types.
2. Give an example for a statistic which is unbiased under a

sampling design.

3. Define ( i )   Inclusion indicator.

( ii ) First order inclusion probability.

4. For any sampling design, find mean and variance of   I _i (s).
5. Prove that an unbiased estimator for the population total can be found iff the first order inclusion probabilities are positive for all the units in

the population.

6. Prove that E _p ( s)  =  S under Simple Random Sampling Design.
7. Define Midzuno Sampling Design. Verify whether or not this design is a probability sampling design.
8. Describe Random Group Method for selecting a sample and write the estimator for population total under this method.
9. List all possible modified systematic samples of size 8 when the population size is 40.
10. Show that _LR is more efficient than _R  unless  β = R.

 

 

 SECTION – B

          ——————-

Answer any FIVE  questions                                            ( 5 x 8 = 40 marks)

 

11. Show that the property of unbiasedness is design dependent.
12. Derive variance of Horwitz – Thompson estimator for population total under any design P .
13. Write the unit drawing mechanism for implementing Simple Random Sampling Design and show that this mechanism implements the design.
14. Show that Lahiri’s method of selection is a PPS selection method.
15. Show that v ( _HT   ) is non-negative under MSD for all “s” receiving positive probabilities.

 

16. Derive V ( _DR   ) for n = 2.
17. Show that the usual expansion estimator is unbiased for the population total in CSS , when there is a linear trend in the population.
18. Derive the approximate Bias and Mean Square Error of the

estimator _R..

 

 SECTION – C

          ——————-

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

 

19. ( a ) Derive  _HT  and  V ( _HT   ) using the formula for  Π _i

and  Π _{i j}    under SRS Design.                                 ( 10 )

         ( b ) Suppose from a sample of n units selected using SRS,  a

sub-sample of   n’   units is selected using SRS and included in

the original sample. Derive the expected value and the

approximate  sampling variance of  ‘ ,  the sample mean based

on  ( n + n’ ) units.                                                   ( 10 )

20. ( a ) Obtain Π _i  and  Π _{i j}    under MSD.                         ( 10 )

( b ) Derive estimated variance of _DR.                                        ( 10 )

21.( a ) Describe Warner’s randomized response technique and explain

the procedure for estimating the proportion Π_A .     ( 10 )

( b ) Deduct  _st  ,V ( _st   )  and  v ( _st   ) under

( i ) SRS    and

( ii )  PPSWR designs.                              ( 10 )

22 ( a ) Derive the formula for n _h  under cost optimum allocation.

( 10 )

( b ) Find the mean and variance of  _TS , the estimator for

population total, under two – stage sampling with SRS in

both  stages.                                                                 ( 10 )

 

 

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