### Chapter 7, Survey Sampling

#### Solution 30

We can compare the variance of the two methods:

$$
\,
\begin{align*}
\Var(R) &= \frac {r(1-r)} {n(2p-1)^2} && \text{From problem-28} \\
\Var(R) &= \frac {r(1-r)} {np^2} && \text{From problem-29}
\end{align*}
\,
$$

Note that in the first case, if $\, p = \frac 1 2 \,$, or even if p is close to $\, \frac 1 2 \,$, then variance is undefined or becomes very large. While in the second case this problem does not arise. Thus the second method is better.

I am not sure how to understand this conceptually that second method is better without looking at the variances.

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