Discussion
Double sampling is useful :
Answer: A
Double sampling is useful when complete information on the auxiliary variable is not available.
Ratio and regression estimators need the population mean (or total) of an auxiliary variable \(x\). When it is unknown, double (two-phase) sampling first takes a large, cheap sample to measure \(x\) alone and estimate its mean, then a smaller subsample on which the expensive study variable \(y\) is also measured. The first-phase estimate of \(\bar{X}\) stands in for the unknown population value.
So the purpose is to supply missing auxiliary information, not to improve a ratio or regression estimator that already has it (with the true \(\bar{X}\) known, double sampling only adds variance). Heterogeneity of the population is the motivation for stratification, not for double sampling.
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