(a) Discuss the difference between sampling for variables and sampling for attributes with examples. For a qualitative…

Sampling for variables and attributes. Sampling for variables treats the study characteristic as quantitative, so each unit has a measured value and the target is a mean or total. The distinction is practical: variable sampling needs measurement scales, while attribute sampling needs clear classification rules. Indian examples are per-hectare wheat yield, household monthly expenditure measured by NSSO, or factory output. Sampling for attributes treats the characteristic as qualitative or dichotomous, so each unit is classified as possessing or not possessing the attribute and the target is a proportion. Examples are whether a household is literate in the Census of India, whether a farm has irrigation, or whether a worker is employed. Variable sampling usually estimates ȳ or total Nȳ, while attribute sampling estimates P or NP. Let the population contain N units, M of which possess the attribute, so P=M/N. In a simple random sample without replacement of size n, let n' be the number possessing the attribute. Then p̂=n'/n, and E(p̂)=P, so p̂ is unbiased. For the binary variable yᵢ=1 if the unit has the attribute and 0 otherwise, the population variance is S²=N/(N-1)P(1-P). Therefore V(p̂)=(1-f)S²/n=(N-n)/(N-1)·P(1-P)/n. Since the sample variance s²=n/(n-1)p̂(1-p̂) is unbiased for S², an unbiased estimator of V(p̂) is v(p̂)=(N-n)/(N n)s²=(N-n)/(N(n-1))p̂(1-p̂).

Optimum allocation and variance. For equal costs, Neyman allocation requires nᵢ/n=N_iSᵢ/ΣN_jSⱼ; if costs differ, the condition becomes nᵢ∝N_iSᵢ/√cᵢ. Here Sᵢ=√(sᵢ²), so Sᵢ are 6, 7, 9 and 12. The products N_iSᵢ are 180, 350, 540 and 720, with total 1790. The optimum sample sizes are 50×180/1790≈5.03, 50×350/1790≈9.78, 50×540/1790≈15.08 and 50×720/1790≈20.11, which round to 5, 10, 15 and 20. The ratios nᵢ/(N_iSᵢ) are also nearly equal, confirming the given allocation as optimum. The stratum weights are Wᵢ=Nᵢ/200, namely 0.15, 0.25, 0.30 and 0.30. The stratified mean is ȳ_st=ΣWᵢȳᵢ=0.15×35+0.25×40+0.30×40+0.30×55=43.75. The estimated variance of this mean under the allocation is V̂(ȳ_st)=ΣWᵢ²(Nᵢ-nᵢ)/(Nᵢ nᵢ)sᵢ². The four terms are 0.15²×25/150×36=0.135, 0.25²×40/500×49=0.245, 0.30²×45/900×81=0.3645 and 0.30²×40/1200×144=0.432. Hence V̂(ȳ_st)=1.1765. This estimate uses the stratum sample variances and finite population corrections, and gives the standard error basis for ȳ_st.

SRS and PPS. Simple random sampling gives every unit the same probability of selection; it is easy to implement and unbiased, but it can be inefficient when units differ greatly in size or contribution. SRS is appropriate when the frame is homogeneous; PPS is appropriate when the frame is of establishments and size is known. PPS sampling gives each unit a selection probability proportional to an auxiliary size measure Xᵢ, such as area under crop, registered capital, or number of workers. It is therefore suitable for skewed populations such as large industrial units or large agricultural holdings, where a few large units dominate the total. For the usual PPS with replacement, let X=ΣXᵢ. In the cumulative total method, form cumulative totals Cᵢ=Σⱼ₌₁ⁱ Xⱼ. Draw n independent random numbers R₁,...,Rₙ from 1 to X; for each Rₖ select the unit i satisfying Cᵢ₋₁<Rₖ≤Cᵢ. Repetitions are allowed, so the same unit may be selected more than once. In Lahiri’s method, let M=max Xᵢ. Draw a random unit index i from 1 to N and a random number j from 1 to M; if j≤Xᵢ accept unit i, otherwise reject the pair and draw again. Conditional on acceptance, the probability of selecting unit i is Xᵢ/X, so each accepted draw is PPS. Repeat independently n times to obtain n draws. If a PPS without replacement sample is required, one should use systematic PPSWOR or a sequential PPSWOR method rather than merely discarding repetitions. Thus, the choice among SRS, stratified optimum allocation and PPS depends on whether the variable is quantitative or qualitative, how strata differ, and whether size measures explain variability.