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.

99.5%

The level of significance is 99.5%.

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.

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 99.5% confidence level,

\(\begin{aligned}{c}\alpha = 0.005\\1 - \frac{\alpha }{2} = 0.9975\end{aligned}\)

To find the z score corresponding the area 0.9975,

In the standard normal table for positive z score, find the value 0.9975, corresponding row value is 2.8, and column values is 0.01, which corresponds to the z-score of 2.81.

Therefore, the critical value for 99.5% level of significance is 2.81.