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

(a) Sampling for variables and attributes; estimator of proportion and its variance

Sampling for variables: the characteristic is quantitative, each unit has a measured value, and the target is a mean or total, such as wheat yield per hectare or household monthly expenditure in an NSSO survey. Sampling for attributes: the characteristic is qualitative, each unit is classified as having the attribute or not, and the target is a proportion or count, such as whether a household is literate or a farm is irrigated. Estimation under SRSWOR: let N units contain M with the attribute, so P = M/N and Q = 1 − P. In a sample of size n, let n′ units have it. Step 1: Define yᵢ = 1 if the unit has the attribute and 0 otherwise. Then p = n′/n = ȳ, and E(p) = P, so p is unbiased for P. Step 2: The population variance is S² = NPQ/(N − 1). Step 3: V(p) = (1 − f)S²/n = (N − n)PQ/[(N − 1)n], where f = n/N. Step 4: The sample variance s² = n pq/(n − 1) is unbiased for S², where q = 1 − p. Step 5: An unbiased estimator of V(p) is v(p) = (1 − f)s²/n = (N − n)pq/[N(n − 1)].

(b) Verifying optimum allocation and estimating the variance

Step 1: Optimum (Neyman) allocation for equal costs requires nᵢ proportional to NᵢSᵢ. Here Sᵢ = 6, 7, 9, 12, so NᵢSᵢ = 180, 350, 540, 720, with total 1790. Step 2: The optimum sizes are 50 × NᵢSᵢ/1790 = 5·03, 9·78, 15·08, 20·11, which round to 5, 10, 15, 20, the given allocation. The ratios nᵢ/(NᵢSᵢ) are nearly equal. Step 3: Weights Wᵢ = Nᵢ/200 = 0·15, 0·25, 0·30, 0·30, so ȳ_st = ΣWᵢȳᵢ = 5·25 + 10 + 12 + 16·5 = 43·75. Step 4: v(ȳ_st) = ΣWᵢ²(1/nᵢ − 1/Nᵢ)sᵢ² = 0·135 + 0·245 + 0·3645 + 0·432 = 1·1765, so the standard error is about 1·08. Check: the optimum-allocation formula (ΣWᵢSᵢ)²/n − ΣWᵢSᵢ²/N = 80·10/50 − 85·15/200 = 1·176, matching. The existing allocation is optimum (to rounding), and the estimated variance of the stratified mean is 1·1765.

(c) SRS versus PPS; drawing a PPS sample

Difference: in SRS every unit has equal selection probability 1/N. In PPS, unit i has probability Pᵢ = Xᵢ/X proportional to a size measure Xᵢ (such as area, workers or capital). PPS is suited to skewed populations where a few large units dominate, and the Hansen-Hurwitz estimator is unbiased for the total. (i) Cumulative total method (with replacement): Step 1: Form Cᵢ = X₁ + … + Xᵢ, with Cₙ = X = total. Step 2: Draw a random number R from 1 to X. Step 3: Select the unit i with Cᵢ₋₁ < R ≤ Cᵢ. Step 4: Repeat n times independently; units may repeat. (ii) Lahiri's method: Step 1: Let M be the maximum of Xᵢ (or any number at least as large). Step 2: Draw a unit i at random from 1..N and a number j at random from 1..M. Step 3: If j ≤ Xᵢ, accept unit i; otherwise reject the pair and draw again. Step 4: The accepted unit has probability Xᵢ/X. Repeat until n units are accepted. Lahiri's method avoids computing cumulative totals.