4.3.3 Notation for Critical Values

In statistical inference, we will need the values on the horizontal $z$ axis that capture certain small tail areas under the standard normal curve.

notation

$z_{\alpha }$ will denote the value on the $z$ axis for which $\alpha$ of the area under the $z$ curve lies to the right of $z_{\alpha }$ . (See Figure 4.19.)

For example, $z_{.10}$ captures upper-tail area .10, and $z_{.01}$ captures upper-tail area .01 .

Figure 4.19 $z_{\alpha }$ notation Illustrated

Since $\alpha$ of the area under the $z$ curve lies to the right of $z_{\alpha }$, $1-\alpha$ of the area lies to its left. Thus $z_{\alpha }$ is the $100\left(1-\alpha \right)$ th percentile of the standard normal distribution. By symmetry the area under the standard normal curve to the left of $-z_{\alpha }$ is also $\alpha$. The $z_{\alpha }$ ’s are usually referred to as $z$ critical values. Table 4.1 lists the most useful $z$ percentiles and $z_{\alpha }$ values.

Table 4.1 Standard Normal Percentiles and Critical Values

Percentile	90	95	97.5	99	99.5	99.9	99.95
$\alpha$ (upper-tail area)	0.1	0.05	0.025	0.01	0.005	0.001	0.0005
$z_{\alpha }=100(1-\alpha )$ percentile	1.28	1.645	1.96	2.33	2.58	3.08	3.27

EX 4.15