LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – STATISTICS

AC 14

FOURTH SEMESTER – APRIL 2006

                                                   ST 4500 – BASIC SAMPLING THEORY

(Also equivalent to STA 504)

 

 

Date & Time : 25-04-2006/9.00-12.00         Dept. No.                                                       Max. : 100 Marks

 

 

SECTION – A

Answer  ALL  questions.  Each  carries TWO  marks.     (10 x 2 =  20 marks)

 

1. Define a population and give two examples.
2. State the difference between parameter and  statistic.
3. If there are two unbiased estimators for a parameter, then show that there are uncountable number of unbiased estimators for that parameter.
4. Explain Lottery method of selecting a  Simple Random Sample.
5. In SRSWOR , find the probability of selecting a given subset consisting of ‘n’ units of the population consisting of  ‘N’ units.
6. In PPS sampling , find the probability of selecting  i^th  population unit when a

simple random without replacement sample of size ‘n’ is drawn from a population

containing ‘N’ units.

1. Show that Hansen – Hurwitz estimator is unbiased for population total.
2. Show that the sample mean is more efficient under SRSWOR  scheme than under SRSWR scheme.
3. In SRSWR , obtain the variance of the sample mean based on only distinct units.
4. Describe Balanced Systematic Sampling Scheme.

 

SECTION – B

Answer any FIVE questions.  Each carries EIGHT marks.      (5 x 8 = 40 marks)

 

1. Illustrate that one can obtain more than one unbiased estimator for a parameter.
2. In SRSWOR , derive the variance of sample mean by using probabilities of inclusion.
3. Deduct the formulae for Ŷ,  V( Ŷ ) and  v(Ŷ ) under SRSWR using the formulae for  Ŷ , V(Ŷ) and v(Ŷ) available under PPSWR.
4. Explain the consequences of  using an LSS scheme  when  ‘N’ is not a  multiple of  ‘n’.
5. In  Stratified Random Sampling , derive the variance and  estimated variance of the sample mean.
6. Derive Neyman allocation formula.  Hence what do we conclude about the size of the sample taken from any stratum?
7. Explain Circular  Systematic Sampling Scheme.  List all possible samples under this scheme when N = 7 and n = 3.
8. In PPSWR , derive the variance of Hansen – Hurwitz  estimator for population total.

SECTION – C

 

Answer any TWO questions.  Each carries TWENTY marks.      (2 x 20 = 40 marks)

 

19(a).  In SRSWOR , show that sample mean is unbiased  for population mean by

using  the probability of selecting a subset of the population as a sample.  (8)

19 (b). Describe Lahiri’s  method  and prove that this method  of selection is a

PPS  selection.  (12)

 

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

( (Y₁ /P₁) –  Y)² P₁ +  …  + ( (Y₅ / P₅) – Y)²P₅  =  100.

Compare  Ŷ₁  =  ((y₁ /P₁) + ( y₂ / P₂ )) / 2   and

Ŷ₂  = (2/3) (y₁ /P₁) + (1/3) ( y₂ / P₂).

Find the values of  α for which

Ŷ_α  =  α (y₁/ P₁)  +  (1 – α ) (y₂ / P₂)  is  less efficient than Ŷ₁ .(12)

20(b).  Verify whether or not the sample mean coincides with the population

mean in Centered Systematic Sampling scheme when the population is

linear. ie Y_i  =   α + βi  ,   i = 1,2,…,N.   (8)

 

21. In a population consisting of linear trend, show that a systematic sample is

less precise than a stratified random sample, with a strata of size 2 k and

2 units chosen per stratum, if n > (4k + 2)/ (k  + 1), when the 1^st    stratum

contains the 1^st   set of  2k units,  the 2^nd stratum contains the 2^nd set of 2k units

in   the population and so on.  (20)

 

22. With 2 strata , a sampler would like to have n₁  =   n₂  for administrative

convenience instead of using the values given by the Neyman Allocation.

If V and Vopt  denote the variances given by n₁  =  n₂    and the Neyman

Allocation respectively, show that (V-Vopt )/Vopt  = ( (r-1) / ( r+1)) ² , where

r = n₁ / n₂  as given by Neyman allocation.   Assume that N₁  and N₂  are large.

(20)

 

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