Solution
Budgeting the Survey Strata
Show the problem again
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.
Worked solution
We minimize $\sum_h W_h^2 S_h^2 / n_h$ subject to $\sum_h c_h n_h = B$. A Lagrange multiplier gives $n_h \propto W_h S_h/\sqrt{c_h}$: sample more where a stratum is large and variable, less where it is expensive. At that optimum the minimized variance collapses to $$\mathrm{Var}_{\min} = \frac{\left(\sum_h W_h S_h \sqrt{c_h}\right)^2}{B}.$$ Here $\sum_h W_h S_h\sqrt{c_h} = 0.2\cdot10\cdot1 + 0.3\cdot20\cdot2 + 0.5\cdot40\cdot4 = 2 + 12 + 80 = 94$. So the variance is $94^2/1000 = 8836/1000 =$ **2209/250**.
Source: Cost-optimal allocation in stratified sampling (Neyman allocation with unequal costs), as treated in Cochran, Sampling Techniques. Statement written for AxiomIQ.