Determining the sample size

這節要介紹的是, 當我們給定了誤差範圍想要得到給定的機率時, 所需要的抽樣的樣本數 n 是我們所關心的, 有效樣本數少的話可以節省成本付出.

首先我們先定義或是說給定誤差範圍及滿足的誤差的機會, $\begin{array}{rcl}
d&=&\mbox{the desired error margin} \\
1-\alpha&=&\mbox{the probability associated with the error margin} \\
所以在 $100(1-\alpha)\%$ 的誤差機會下,

$\displaystyle z_{\alpha/2}\frac{\sigma}{\sqrt{n}}=d$
$\displaystyle n=\biggl [ z_{\alpha/2}\frac{\sigma}{d} \biggr ]^2$

To be $100(1-\alpha)\%$ sure that the error of estimation $\vert\overline{X}-\mu\vert$ does not exceed d, the required sample size is

$\displaystyle n=\biggl [ z_{\alpha/2}\frac{\sigma}{d} \biggr ]^2$

If $\sigma$ is completely unknow, a small-scale preliminary sampling is necessary to obtain an estimate of $\sigma$ to be used in the formula to compute n.