# Mathematical Statistics and Data Analysis - Solutions

### Chapter 7, Survey Sampling

#### Solution 21

To reduce the width of interval $\, \bar X \pm 1.96\sigma_{\bar X} \,$ by half, we need to reduce the variance, $\, \sigma_{\bar X} \,$ by half. Since $\, \sigma_{\bar X}=\frac {\sigma} {\sqrt n} \,$, it follows that we need to increase $\, \sqrt n \,$ by two times which in turn means that we need to increase $\, n \,$ or the sample the size by four times.

$$\tag*{\blacksquare}$$