Discussion
Consider the following sampling designs : a stratified sample scheme with SRSWOR within each stratum and a SRSWOR. Both have the same sample size. We are interested in estimating the population mean. Let \(V_{\text{prop}}^{2}\) and \(V_{\text{SRS}}^{2}\) be the corresponding variances of the usual estimators of the population mean. Then,
Answer: D
The keyed relation is \(V_{\text{prop}}^{2} \le V_{\text{SRS}}^{2}\).
Ignoring finite-population terms of order \(1/N_{h}\), the variance of the mean under SRSWOR splits into within-stratum and between-stratum parts:
\[ \begin{aligned} &V_{\text{SRS}} - V_{\text{prop}} \\ &\approx \frac{1 - f}{n}\sum_{h} W_{h}(\bar{Y}_{h} - \bar{Y})^{2} \ge 0 \end{aligned} \]
Proportional allocation removes the between-strata part, so it is never worse, and the two are equal when all stratum means coincide. That is why '\(\le\)' rather than strict '\(\lt\)' is the safe statement, and '\(=\)' or '\(\gt\)' is wrong in general. (With very small strata the exact formula can reverse the inequality slightly; the textbook result assumes the strata are not tiny.)
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