(a) Discuss the difference between sampling for variables and sampling for attributes with examples. For a qualitative characteristic, find an unbiased estimator of population proportion along with its variance when sample is drawn by simple random s

Question

(a) Discuss the difference between sampling for variables and sampling for attributes with examples. For a qualitative characteristic, find an unbiased estimator of population proportion along with its variance when sample is drawn by simple random sampling without replacement. Also obtain an unbiased estimator of this variance. 20

(b) The table given below gives the population and sample sizes, stratum means and variance of a stratified random sample of size 50. Symbols used have their usual meanings.

| Stratum Number | Nᵢ | nᵢ | ȳᵢ | sᵢ² |
|---|---|---|---|---|
| 1 | 30 | 5 | 35 | 36 |
| 2 | 50 | 10 | 40 | 49 |
| 3 | 60 | 15 | 40 | 81 |
| 4 | 60 | 20 | 55 | 144 |

Verify that the existing allocation is optimum for given 4 strata. Also calculate the estimate of population variance under this allocation. 15

(c) Differentiate between Simple Random Sampling and Probability Proportional to Size Sampling. How will you draw a PPS sample of size n from a population of size N (n < N) by (i) Cumulative Total Method and (ii) Lahri's Method ? Explain. 15

UPSC Answer Check · Accepted Answer

Begin with a clear conceptual distinction in part (a) between variables (quantitative) and attributes (qualitative) with Indian examples like agricultural yield vs literacy status. Derive the unbiased estimator p̂ = n'/n for population proportion and its variance V(p̂) = (N-n)/(N-1) · p(1-p)/n, then obtain unbiased estimator v(p̂). For part (b), verify Neyman optimum allocation by checking if nᵢ ∝ NᵢSᵢ/√cᵢ (assuming equal costs), then compute V(ȳ_st). For part (c), contrast SRS with PPS on selection probability basis, then detail both Cumulative Total and Lahri's methods with numerical illustration. Allocate approximately 40% time to part (a), 30% each to (b) and (c) based on marks distribution.
- Part (a): Clear distinction between sampling for variables (measurable quantities like income, yield) vs attributes (dichotomous characteristics like employment status, disease presence) with appropriate Indian examples
- Part (a): Derivation of unbiased estimator p̂ = n'/n for population proportion P, its variance V(p̂) = (N-n)/(N-1) · P(1-P)/n under SRSWOR, and unbiased estimator of variance v(p̂) = (N-n)/(N-1) · p̂(1-p̂)/(n-1)
- Part (b): Verification of Neyman optimum allocation condition nᵢ/n = NᵢSᵢ/ΣNⱼSⱼ using given data; calculation showing existing allocation matches or approximates this ratio
- Part (b): Computation of stratified mean estimate ȳ_st = ΣWᵢȳᵢ where Wᵢ = Nᵢ/N, and population variance estimate V(ȳ_st) = ΣWᵢ²(Nᵢ-nᵢ)/(Nᵢnᵢ) · sᵢ²
- Part (c): Systematic comparison of SRS (equal probability) vs PPS (probability ∝ size) on grounds of efficiency, especially for skewed populations like industrial output or agricultural holdings
- Part (c): Step-wise description of Cumulative Total Method: list cumulative totals, select random numbers between 1 and ΣXᵢ, identify selected units
- Part (c): Step-wise description of Lahri's Method: select random number i from 1 to N and random number j from 1 to M (M=max size), accept if j ≤ Xᵢ, else reject and repeat

Dimension	Weight	Max marks	Excellent	Average	Poor
Setup correctness	20%	10	Correctly identifies all parameters for each sub-part: for (a) defines population proportion P and sample proportion p̂; for (b) computes N=200, stratum weights Wᵢ, and sets up Neyman allocation verification framework; for (c) clearly defines size measures Xᵢ and selection probabilities for both PPS methods	Identifies most parameters but makes minor errors in defining finite population correction or stratum weights; incomplete setup for one sub-part	Confuses variables vs attributes, misidentifies stratum parameters, or fails to define size measures for PPS; major conceptual errors in problem setup
Method choice	20%	10	Selects appropriate methodological frameworks: hypergeometric/SRSWOR theory for (a), Neyman optimum allocation formula for (b), and correct algorithmic descriptions for both PPS methods in (c); justifies why each method is suitable	Uses correct general methods but misses nuances like finite population correction in (a) or assumes equal costs without mention in (b); describes one PPS method adequately but other superficially	Applies wrong methods (e.g., SRSWR instead of SRSWOR, proportional instead of optimum allocation); confuses cumulative total with Lahri's method or describes incorrect algorithms
Computation accuracy	20%	10	All calculations precise: for (a) correct variance formula with FPC; for (b) accurate verification that nᵢ ∝ Nᵢsᵢ (computing N₁s₁=180, N₂s₂=350, N₃s₃=540, N₄s₄=720, ratios 5:10:15:20 match 180:350:540:720 approximately), correct V(ȳ_st) = ΣWᵢ²(Nᵢ-nᵢ)sᵢ²/(Nᵢnᵢ) = 0.0025×30 + 0.00196×35 + 0.0015×36 + 0.001×54 = 0.075+0.0686+0.054+0.054 ≈ 0.2516; for (c) numerical illustration of both methods	Correct approach but arithmetic errors in variance calculation or allocation verification; partial numerical demonstration for PPS methods	Major computational errors: incorrect stratum variance formula, wrong weights, or failure to perform any numerical verification; no worked example for PPS methods
Interpretation	20%	10	Interprets results meaningfully: explains why optimum allocation improves precision over proportional allocation in (b); discusses efficiency gains of PPS over SRS for skewed populations (e.g., Indian industrial survey where few large units dominate); notes practical constraints like cost and feasibility in allocation decisions	Provides basic interpretation without depth; mentions efficiency but doesn't relate to specific context or data characteristics; limited discussion of practical implications	No interpretation of numerical results; fails to explain why methods differ in efficiency or when to prefer one over another; purely mechanical presentation
Final answer & units	20%	10	All final answers clearly stated with appropriate units/notations: variance expressions in (a), explicit conclusion that allocation is approximately optimum with calculated V(ȳ_st) ≈ 0.252 in (b), and complete algorithmic summaries for both PPS methods in (c); proper mathematical notation throughout	Final answers present but poorly organized or missing units; incomplete presentation of one sub-part's conclusions; some notational inconsistencies	Missing final answers for one or more sub-parts; no clear conclusion on optimum allocation; algorithms described without final summary; messy or illegible presentation

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