Chapter 7, Survey Sampling

Solution 4

The key here is - a statistic/estimate is better if its variance is small because the smaller the variance better the accuracy of the estimate.

Denoting the variance of the estimate as $\sigma^2_{S_1}$ and $\sigma^2_{S_2}$ for Population 1 and Population 2 and ignoring the finite population correction:

$$ \sigma^2_{S_1} = \frac {\sigma^2_1} {n_1} $$
$$ \sigma^2_{S_2} = \frac {\sigma^2_2} {n_2} = \frac { {(2 \sigma_1)}^2 } {2 n_1} = \frac {2 \sigma_1} {n_1} = 2 \sigma^2_{S_1} $$

Thus variance of the sample mean of second sample is twice the variance of the sample mean of first sample. Or the first sample has small variance - thus first sample is better for estimating the Population mean.

$$\tag*{$\blacksquare$} $$