Critical Values. In Exercises 41–44, find the indicated critical value. Round results to two decimal places.

${\text{z}}_{\text{0.10}}$

The critical value is indicated as$z_{0.10}$.

The critical value $z_{\alpha }$at a particular value of$\alpha$ signifies that $\alpha$is the area to the right of $z_{\alpha }$.

Mathematically,

$P\left(z>z_{\alpha }\right)=\alpha$

In this case, for $\alpha =0.10$,$P\left(Z>z_{0.10}\right)=0.10$ .

The notation of the critical value is used $z_{0.10}$to represent the z-score with an area of 0.10 to its right.

Due to the one-to-one correspondence between the area and probability,

$$\begin{gathered} P\left(Z>z_{0.10}\right)=0.10 \\ 1-P\left(Z<z_{0.10}\right)=0.10 \\ P\left(Z>z_{0.10}\right)=0.90 \end{gathered}$$

In the standard normal table, the value closest to 0.90 is 0.8997, and the corresponding row value 1.2 and the column value 0.08 correspond to the z-score of 1.28.

The value of $z_{0.10}=1.28$has an area of 0.10 to its right. So, $z_{0.10}=1.28$corresponds to a cumulative left area of 0.90.

Thus, the critical value is $z_{0.10}=1.28$.