Question

Finding Critical Values. In Exercises 5–8, find the critical value that corresponds to the given confidence level.

99%

Short answer

The critical value$z_{\frac{\alpha }{2}}$ for 99% level of confidence is 2.58 (table 2.575).

Step-by-step solution

Given information

The level of significance is 99%.

Describe the concept of critical value

A critical value is a point on the test distribution that is compared to the test statistics to determine whether to reject the null hypothesis. It is denoted by $z_{\frac{\alpha }{2}}$which is equal to z score within the area of $\frac{\alpha }{2}$in the right tail of the standard normal distribution for$\alpha$ level of significance.

Find the critical value

When finding a critical value $z_{\frac{\alpha }{2}}$for a particular value of , note that$\frac{\alpha }{2}$is the cumulative area to the right of $z_{\frac{\alpha }{2}}$which implies that the cumulative area to the left of $z_{\frac{\alpha }{2}}$must be $1-\frac{\alpha }{2}$.

Here, for 99% confidence level,

$$\begin{gathered} \alpha =0.01 \\ 1-\frac{\alpha }{2}=0.995 \end{gathered}$$

To find the z score corresponding the area 0.9950,

In the standard normal table for positive z score, find the value closest to 0.9950, which is 0.9951, corresponding row value is 2.5, and column values is 0.08 this corresponds to the z-score of 2.58.

Therefore, the critical value for 99% level of significance is 2.58.