Discussion
The regression estimator of the sample mean based on a SRSWOR :
Answer: D
The regression estimator is better than the sample mean if \(n\) is sufficiently large.
In large samples its variance is approximately \(V(\bar{y})(1 - \rho^{2})\), never larger than that of \(\bar{y}\), and smaller whenever the study and auxiliary variables are correlated. In small samples the regression estimator is biased and its gain is not guaranteed.
So it is not 'always' better (A), it does not have the same efficiency except when \(\rho = 0\) (B), and it is not worse (C) in large samples.
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