Loyola College M.Sc. Statistics April 2009 Sampling Theory Question Paper PDF Download

      LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – STATISTICS

YB 38

SECOND SEMESTER – April 2009

ST 2813 / 2810 – SAMPLING THEORY

 

 

 

Date & Time: 24/04/2009 / 1:00 – 4:00  Dept. No.                                                     Max. : 100 Marks

 

 

SECTION – A

 

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

 

  1. Define a parameter and a statistic.  Give an example for both.
  2. Give an example for an estimator which is unbiased under a sampling design.
  3. Show  that

(i)   E [ I i (s) ]  =  Π i  ;  i  =  1, 2, …, N,

(ii)  E [ I i (s) I j (s)]  =  Π ij  ;   i , j  =  1, 2, …, N ;   i  ≠ j .

  1. Prove that an unbiased estimator for the population total can be found if and only if the first order inclusion probabilities are positive for all N units in the population.
  2. Prove that E p (  s y ­)  =  S y   under  SRSWOR  Design.
  3. Define Midzuno Sampling Design.  Verify whether or not this design is a probability sampling design.
  4. Describe Random Group Method for selecting a sample and write the estimator for population total under this method.
  5. List all possible Modified Systematic Samples of size 8 when the population size is 40.
  6. Check whether LR is more efficient than   R .
  7. Prove that the Desraj ordered estimator is unbiased for the population total.

 

SECTION – B

 

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

     

  1. Write the unit drawing mechanism for implementing SRSWOR Design and show that this mechanism implements the design.

 

  1. If  T( s, s′ ) is a statistic based on the sets s and s′ which are samples drawn in the first phase  of randomization and the second phase of randomization respectively, then prove that

V( T( s, s′ ) )  =  E1 V2 ( T( s, s′ ) )  +  V1 E2 ( T( s, s′ ) ) ,

where E2 is the expectation taken after fixing the subset s and E1 is the

expectation with respect to the randomization involved in the first phase.

 

  1. Check whether or not LSS is more efficient  than SRS for population with linear trend.

 

  1. Show that the usual expansion estimator is unbiased for the population total in CSS when there is a linear trend in the population.
  2. Check whether the estimated variance v( HT  ) is  non-negative under MSD for all “ s ” receiving positive probabilities.

 

  1. Explain Simmon’s unrelated randomized response model and obtain the estimate of ΠA when ΠY is unknown.

 

  1. Derive the estimated variance of DR.
  2. Derive the formula for n h under Cost Optimum Allocation.

 

SECTION – C 

 

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

 

19 ( a ) Illustrate that an estimator can be unbiased under one design but biased under

another design.                                                                                         ( 10 )

( b )  Derive  HT   and  V (HT ) using the formula for Π i  and  Π ij  under SRSWOR

Design.                                                                                                     ( 10 )

20 ( a ) Describe Warner’s randomized response technique and explain the procedure

For estimating the proportion Π A .                                                         ( 10 )

( b ) Deduce the expressions for   St ,   V (St )   and  v (St ) when samples are

drawn   independently from different strata using    ( i )  SRSWOR,  and

( ii )  PPSWR Designs.                                                                              ( 10 )

  1. Find the expressions for the approximate bias and MSE of the estimator R

and  deduce their expressions under ( i )  SRSWOR,  (ii)  PPSWOR,  and                                        ( iii ) Midzuno Sampling Designs.                                                                 ( 20 )

22 ( a ) Verify whether or not the  Hansen-Hurwitz estimator dhh  under double

sampling is unbiased  for Y and derive its variance.                                 ( 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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