Question

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

$z_{0.04}$

Short answer

The indicated critical value of $z_{0.04}$is 1.75.

Step-by-step solution

Given information

The critical value is$z_{0.04}$

Describe the critical value 

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

Mathematically,

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

In this case, $\alpha =0.04$, and $P\left(Z>z_{0.04}\right)=0.04$.

Step 3: Simplify the term to obtain the cumulative probability

For obtaining the cumulative probability, simplify the term as follows.

$$\begin{gathered} P\left(z>z_{0.04}\right)=0.04 \\ 1-P\left(z<z_{0.04}\right)=0.04 \\ P\left(z<z_{0.04}\right)=0.96 \end{gathered}$$

From the standard normal table, the value of 0.96 is closest to 0.9599, and the corresponding row value of 1.7 and column value of 0.05 implies that the z value is 1.75.

Summarize the result

The value of $z_{0.04}=1.75$has an area of 0.04 to its right. So, $z_{0.04}=1.75$corresponds to a cumulative left area of 0.96.

Thus, the critical value is $z_{0.04}=1.75$.