Chapter 7, Survey Sampling
Solution 15
(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 $\, \delta \,$
| $\, n=20 \,$ | $\, n=40 \,$ | $\, n=80 \,$ | |
|---|---|---|---|
| $\, l=0.10 \,$ | $\, 211.583 \,$ | $\, 145.650 \,$ | $\, 96.918 \,$ |
| $\, l=0.50 \,$ | $\, 87.4677 \,$ | $\, 60.2 \,$ | $\, 40.06288 \,$ |