Question

Finding Critical Values. In Exercises 5–8, find the critical value \[{{\rm{z}}_{{{\rm{\alpha }} \mathord{\left/

{\vphantom {{\rm{\alpha }} {\rm{2}}}} \right.

\kern-\nulldelimiterspace} {\rm{2}}}}}\]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 \[\alpha \], 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,

\(\begin{aligned}{c}\alpha = 0.01\\1 - \frac{\alpha }{2} = 0.995\end{aligned}\)

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.