Finding Critical Values. In Exercises 5–8, find the critical value 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 thatis 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,

$$\begin{gathered} \alpha =0.005 \\ 1-\frac{\alpha }{2}=0.9975 \end{gathered}$$

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.