Discussion
Let us consider the two sampling strategies, both based on a sample of size \(n\) : I. SRSWOR II post-stratification. To estimate the population total :
Answer: A
The answer is post-stratification is better than SRSWOR if the stratum sizes are large and errors in \(W_{h}\) can be ignored.
In post-stratification the sample is drawn by SRSWOR and then classified into strata with known weights \(W_{h}\). When the expected sample size in each stratum is large, its variance is approximately that of proportional stratified sampling, plus a small term of order \(1/n^{2}\) from the random stratum sample sizes. Stratification with proportional allocation is never worse than SRS, so under these conditions post-stratification is at least as good as SRSWOR.
Neither method is better always: with small strata the random sample sizes can make post-stratification worse. A large \(n\) alone is not the stated condition; it is the stratum sizes and the accuracy of the weights that matter.
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