Question

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

$z_{0.15}$

Short answer

The indicated critical value of $z_{0.15}$is 1.04.

Step-by-step solution

Given information

The z critical value is $z_{0.15}$.

Describe the critical value 

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

Mathematically,

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

In this case, for $\alpha =0.15$, $P\left(Z>z_{0.15}\right)=0.15$.

Find the critical value

The cumulative probability for the critical value is as follows.

$$\begin{gathered} P\left(z>z_{\alpha }\right)=0.15 \\ P\left(z<z_{\alpha }\right)=1-P\left(z>z_{\alpha }\right) \\ P\left(z<z_{0.15}\right)=1-P\left(z>z_{0.15}\right) \\ =0.85 \end{gathered}$$

From the standard normal table, the area of 0.85 is closest to 0.8508, observed corresponding to the row value 1.0 and the column value 0.04.

The value of $z_{0.15}=1.04$has an area of 0.15 to its left. So, localid="1645696656785" $z_{0.15}=1.04$corresponds to a cumulative left area of 0.85.

Thus, the critical value is localid="1645696673512" $z_{0.15}=1.04$.