LOYOLA COLLEGE (AUTONOMOUS), CHENNAI –600 034
M.Sc., DEGREE EXAMINATION – STATISTICS
FIRST SEMESTER – NOVEMBER 2004
ST 1805/1802 – SAMPLING THEORY
25.10.2004 Max:100 marks
9.00 – 12.00 Noon
SECTION – A
Answer ALL the questions (10 ´ 2 = 20 marks)
- Define probability Sampling Design and explain the meaning of probability Sampling.
- Distinguish between varying size sampling design and fixed size sampling design. Give an example for each design.
- Define the following:
- a) Inclusion indicator
- First and Second order inclusion probabilities.
- Prove the following:
- Ep [Ii (s)] = pi ; i = 1, 2, …, N
- Ep [Ii (s) Ij (s)] = pij ; i, j = 1,2, …, N ; i ¹
- Show that an unbiased estimator for the population total can be found if an only if the first order inclusion probabilities are positive for all N units in the population.
- Derive the formula for pi and pij under Simple Random Sampling Design.
- Describe the Linear Systematic Sampling Scheme and write its probability sampling design.
- Derive the approximate bias of the ratio estimator for the population total Y.
- In cluster sampling, suggest an unbiased estimator for the population total. Write the variance of the unbiased estimator.
- Explain Multistage Sampling.
SECTION – B
Answer any FIVE questions. (5 ´ 8 = 40 marks)
- Show that an estimator can be unbiased under one design but biased under another design.
- For any sampling design, prove the following:
- Suggest an unit drawing mechanism for simple random sampling design and prove that the unit drawing mechanism implements the simple random sampling design.
- Explain Lahiri’s method of PPS sampling. Show that Lahiri’s method of selection is a PPS selection method.
- Write the reason for using Desraj ordered estimator instead of Horwitz – Thompson estimator under PPSWOR sampling scheme. Also prove that the Desraj ordered estimator is unbiased for the population total.
- Describe the Random Group Method of Sampling. Find an unbiased estimator of population total under this method and derive its variance.
- Compare V () and V () assuming the population values Yi satisfy
Yi = a + bi, i = 1, 2, …N.
[
- Explain Warner’s randomized response technique for estimating the proportion pA of the persons belonging to group A in a population.
SECTION – C
Answer any TWO questions (2 ´ 20 = 40 marks)
- a) Under any design P(×), derive the variance of Horwitz – Thompson estimator and find
its estimated variance. (16)
- b) Define Midzuno Sampling Design and state the unit drawing mechanism for this
design. (4)
- After the decision to take a simple random sample has been made, it was realized that the value of unit with level 1 would be unusually low and the value of unit with label N would be unusually high. In such cases, it is decided to use the estimator.
where C is a pre-determined constant. Show that
- is unbiased for for any C.
- Derive V ()
- Find the value of C for which is more efficient than .
- a) Show that Linear Regression estimator is more efficient than Ratio estimator
unless b = R. (4)
- b) Assuming samples are drawn using SRS in both the phases of double sampling,
suggest , and when
- the second phase sample is a subsample of the first phase sample.
- the second phase sample is independent of the first phase sample. (16)
- a) In stratified sampling, deduct , V() and () assuming
- SRS is used in all strata
- PPSWR sampling is used in all strata. (12)
- b) Obtain the variance of the following:
- Hansen – Horwitz estimator in Double Sampling.
- Estimator in Two – stage Sampling. (8)
Latest Govt Job & Exam Updates: