Question

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

$z_{0.02}$

Short answer

The critical value for $z_{0.02}$is 2.05.

Step-by-step solution

Given information

The critical value is$z_{0.02}$.

Define the critical value 

The critical value $z_{\alpha }$at a particular value of denotes that is the area to the right of $z_{\alpha }$on the standard normal curve.

Thus,

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

In this case of $\alpha =0.02$, $P\left(Z>z_{0.02}\right)=0.02$.

The notation of critical value$z_{0.02}$ is used to represent the z-score with an area of 0.02 to its right.

Find the critical value z0.02

Due to the one-to-one correspondence between the area and probability under the standard normal curve,

$$\begin{gathered} P\left(Z>z_{0.02}\right)=0.02 \\ 1-P\left(Z<z_{0.02}\right)=0.02 \\ P\left(Z<z_{0.02}\right)=0.98 \end{gathered}$$

In the standard normal table, the value closest to 0.98 is 0.9798, and the corresponding row value 2.0 and the column value 0.05 correspond to the z-score of 2.05, which is the critical value ${\text{z}}_{\text{0.02}}$.

The value of $z_{0.02}=2.05$has an area of 0.02 to its right. So, $z_{0.02}=2.05$corresponds to a cumulative left area of 0.98.

Thus, the critical value is $z_{0.02}=2.05$.