Budgeting the Survey Strata

A population is split into three strata with population shares $W = (0.2, 0.3, 0.5)$ and within-stratum standard deviations $S = (10, 20, 40)$. Interviewing one unit costs $1$, $4$ and $16$ in strata 1, 2 and 3 respectively. You draw a simple random sample within each stratum and use the stratified estimator $\bar y_{st} = \sum_h W_h \bar y_h$. The total budget is exactly $1000$. Choose the stratum sample sizes (treated as continuous) to minimize $\mathrm{Var}(\bar y_{st})$, ignoring finite-population corrections. Give the minimum variance as a reduced fraction.

• combinations
• combinatorial games
• estimation
• variance

Source: Cost-optimal allocation in stratified sampling (Neyman allocation with unequal costs), as treated in Cochran, Sampling Techniques. Statement written for AxiomIQ.