Definition of Z sub alpha

Question

In my experience, $Z_{\alpha}$ indicates the critical value where the right-tailed area under a standard normal distribution is $\alpha$, i.e.

$$P(Z > Z_\alpha) = \alpha$$

With this rule, if $\alpha = 0.05$, then

$$Z_{\frac{\alpha}{2}} = Z_{0.025} = 1.96 ~~~\text{and}~~~~ Z_{1-\frac{\alpha}{2}} = Z_{0.975} = -1.96.$$

However, in some internet sources, $Z_{\alpha}$ is defined as the inverse function of the standard normal CDF, i.e.

$$P(Z < Z_\alpha) = \alpha$$

which makes the example above have opposite signs:

$$Z_{\frac{\alpha}{2}} = -1.96 ~~~\text{and}~~~~ Z_{1-\frac{\alpha}{2}} = 1.96.$$

They make me confused about how to use correct symbols in the confidence intervals or test statistics. Is there a precise mathematical definition of $Z_{\alpha}$?

Answer 1

No it is not correct to say that $P(Z<z_{\alpha})=\alpha.$ It is $P(Z<z_{\alpha})=1-\alpha$ which is equivalent to $P(Z>z_{\alpha})=\alpha.$