# Mathematical Statistics and Data Analysis - Solutions

### Chapter 7, Survey Sampling

#### Solution 7

Ignoring the finite population correction: ${\sigma}^2_{\bar X} = \frac { {\sigma}^2 } {n}$.

Since it is a dichotomous case i.e. value of each member can either be 1 or 0 based on if this member/family is below the poverty level or not respectively, we have ${\sigma}^2 = p(1 - p)$ where p denotes the fraction of population below the poverty line.

This give us:

${\sigma}^2_{\bar X} = \frac { p(1-p) } {n}$.

We know $p = 0.15$ and ${\sigma}^2_{\bar X} = {(0.02)}^2$.

This gives us $n = \frac { p(1-p) } { {\sigma}^2_{\bar X} } = 318.75$

$$\tag*{\blacksquare}$$