Mathematical Statistics and Data Analysis - Solutions

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.

$$\tag*{\blacksquare}$$