# Mathematical Statistics and Data Analysis - Solutions

### Chapter 7, Survey Sampling

#### (b)

We have $\, P(\left\vert \bar X - \mu \right\vert > \delta) \approx l \,$

We shall first find an expression to compute $\, \delta \,$

\, \begin{align*} P(\left\vert \bar X - \mu \right\vert > \delta) &= \\ &= 1 - P(-\delta < \bar X - \mu < \delta) \\ &= 1 - P(\frac {-\delta} {\sigma_{\bar X} } < \frac {\bar X - \mu} {\sigma_{\bar X} } < \frac {\delta} {\sigma_{\bar X} } \\ &= 2\left(1- \Phi\left( \frac {\delta} {\sigma_{\bar X} } \right) \right) && \text{Using CLT. See soln8 for detailed steps} \\ \end{align*} \,

Thus we can compute $\, \delta \,$ as:

\, \begin{align*} &\Rightarrow 2\left(1- \Phi\left( \frac {\delta} {\sigma_{\bar X} } \right) \right) = l \\ &\Rightarrow \Phi\left( \frac{\delta} {\sigma_{\bar X} } \right) = 1 - \frac l 2 \\ &\Rightarrow \delta = \sigma_{\bar X} \times \Phi^{-1}\left( 1 - \frac l 2 \right) \end{align*} \,

$\, \sigma_{\bar X} \,$ can be computed using the formuale $\, \frac {\sigma} {\sqrt{n} } \sqrt{ 1 - \frac{n-1} {N-1} } \,$

We have $\, \sigma = 589.716 \,$ from the example quoted in the problem. To compute $\, \Phi^{-1} \,$, we can use the normal tables given in the end of the book.

Now we just need to insert the values into the formulae to compute $\, \delta \,$

$\, n=20 \,$ $\, n=40 \,$ $\, n=80 \,$
$\, l=0.10 \,$ $\, 211.583 \,$ $\, 145.650 \,$ $\, 96.918 \,$
$\, l=0.50 \,$ $\, 87.4677 \,$ $\, 60.2 \,$ $\, 40.06288 \,$
$$\tag*{\blacksquare}$$