LOYOLA COLLEGE (AUTONOMOUS), CHENNAI –600 034
M.Sc., DEGREE EXAMINATION – STATISTICS
FIRST SEMESTER – NOVEMBER 2003
ST-1802/S717 – SAMPLING THEORY
08.11.2003 Max:100 marks
1.00 – 4.00
SECTION-A
Answer ALL the questions. (10×2=20 marks)
- Explain probability sampling design.
- Given N = 5, n = 3, X2 = 2, X3 = 3, X = 25. Compute p23 under Midzuno Sampling design.
- Distinguish between inclusion probabilities and inclusion indicators.
- List all possible Balanced Systematic Samples when N = 30 and n = 6.
- Define Des-Raj ordered estimator.
- Define Horvitz-Thompson estimator.
- Write a short note on Yates corrected estimator under Linear systematic sampling.
- Describe two phase sampling.
- When is stratified sampling used?
- Define proportional allocation.
SECTION-B
Answer any FIVE questions. (5×8=40 marks)
- Derive variance of Horvitz-Thompson estimator in Yates-Grundy form.
- Explain Lahiri’s method and show that Lahiri’s method of selection is a probability proportional to size selection method.
- Write a note on Warner’s model.
- Explain ratio estimator, also derive the approximate bias and mean square error of the estimator.
- Compare Linear systematic sampling and simple random sampling in the presence of a linear trend.
- Develop Hartly-Ross unbiased ratio type estimator.
- Derive variances and covariances in the two cases of two phase sampling, assuming simple random sampling is used in both the phases of sampling.
- Describe Two-stage sampling. Give the unbiased estimator and also derive the variance of the unbiased estimator.
SECTION-C
Answer any TWO questions. (2×20=40 marks)
- a) Derive V() for n = 2. (15)
- b) For any fixed size sampling design yielding samples of size n, prove that
(5)
- a) Describe Midzuno sampling design and show that it is a sampling design. (5)
- b) Derive the first and second order inclusion probabilities under Midzuno sampling
design. (15)
- a) Develop Yates corrected estimator under linear systematic sampling. (10)
- b) Suppose from a sample of n units selected with simple random sampling (SRS) a
subsample of n’ units is selected with SRS duplicated and added to the original
sample. Derive the expected value and the approximate sampling variance of , the
sample mean based on the n+n’ units. (10)
- a) Write a note on proportional allocation for a given cost. Also deduct V under it
assuming SRS is used in all strata. (10)
- b) A sampler has two strata with relative sizes and . He believes that
S1, S2 can be taken as equal. For a given cost C = C1 n1 + C2 n2, show that (assuming
Nh is large).
(10)
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