Sample Size, a Leaky Voters' List and the Demand for Caste-Free Strata
The zilla panchayat of a drought-prone taluk in Kalaburagi district must estimate the proportion of households that segregate wet waste at source. The taluk has 8,000 households. A pilot survey of 120 households found 48 segregating. The panchayat wants a 95 per cent confidence interval with a margin of error of 3 percentage points and has asked for the sample size, first treating the taluk as effectively infinite and then correcting for its actual size. Enumerators plan to draw the sample from the voters' list, although officials know that roughly 600 tenant households of recent migrants do not appear on it. The panchayat is also being pressed to report separate urban (2,000 households) and rural (6,000 households) figures, since pilot proportions there were 0.55 and 0.35 respectively, and to keep the total sample at the size computed.
1. Compute the required sample size with and without the finite population correction, and verify that the chosen n actually delivers the 3 percentage point margin.
Read the worked answer
Pilot proportion p = 48/120 = 0.40, so q = 0.60; z = 1.96 for 95%; e = 0.03.
Infinite population: n₀ = z²pq/e² = (3.8416 × 0.40 × 0.60)/0.0009 = 0.92198/0.0009 = 1024.4 → 1025 households.
With N = 8000: n = n₀/[1 + (n₀ − 1)/N] = 1025/(1 + 1024/8000) = 1025/1.128 = 908.7 → 909 households. The correction saves 116 interviews - worth having, but it does not rescue a bad frame.
Verification with n = 909: SE = √[pq/n × (N − n)/(N − 1)] = √[0.24/909 × 7091/7999] = √(0.000264 × 0.8865) = 0.0153, so margin = 1.96 × 0.0153 = 0.030. Exactly the target.
Always round up: 908 would fall fractionally short, and the FPC must use the finite N, not be applied as a flat 10% haircut.
2. Officials propose to 'just interview more households' to deal with the missing 600 tenant households. Classify this problem and assess whether a larger sample fixes it.
Read the worked answer
The voters' list is the sampling frame, and it omits about 600/8000 = 7.5 per cent of households - tenants and recent migrants, who are systematically different: more mobile, less likely to own bins or compost pits, so probably less likely to segregate. This is frame bias, a non-sampling error, and specifically under-coverage.
Sample size cannot fix it. Sampling error shrinks roughly as 1/√n, but bias does not shrink at all: doubling n gives a tighter interval around a systematically understated proportion. The textbook symptom is telling - the computed margin of 3 points would be honestly achieved and yet the estimate would still be wrong, which is worse than a wide interval because it looks precise.
Remedies to recommend: build the frame by a door-to-door household listing of the migrant colonies and add those units to the list; or use an area frame of blocks; or draw a supplementary sample from the missing stratum and combine the estimates with weights. Also report the coverage rate and the imputed share of uncovered households as a limitation of the estimate.
3. Allocate the 909 households between urban and rural strata in proportion to size, compute the stratified estimate, and justify stratification here.
Read the worked answer
Proportional allocation: n_h = n × N_h/N. Urban: 909 × 2000/8000 = 227.25 → 227. Rural: 909 × 6000/8000 = 681.75 → 682. (227 + 682 = 909.)
Stratified estimate: p_st = (N₁p₁ + N₂p₂)/N = (2000 × 0.55 + 6000 × 0.35)/8000 = (1100 + 2100)/8000 = 3200/8000 = 0.40.
So the headline figure is 40 per cent, the same as the pilot pooled value, but it is now built from two reported stratum figures and cannot be produced by accident from an unlucky draw.
Why stratify: the strata proportions differ by 20 percentage points, so urban and rural households are far from homogeneous. A simple random sample of 909 could, by chance, over- or under-represent the urban fifth and drift from the true value; stratification removes that between-stratum variation from the sampling error and guarantees both groups are represented. It also delivers the urban and rural breakdowns the panchayat has been asked for at no extra fieldwork cost.