The critical value \( z_{\alpha / 2} \) corresponds to the confidence level of 99%. To find this value, we can use the standard normal distribution table or a calculator.
Step 1: Determine the significance level (\( \alpha \)) associated with the confidence level. The confidence level is 99%, so the significance level is \( 1 - 0.99 = 0.01 \).
Step 2: Find the area in both tails of the standard normal distribution that corresponds to the significance level. Since we want a two-tailed test, we divide the significance level by 2: \( 0.01 / 2 = 0.005 \).
Step 3: Look up the area of 0.005 in the standard normal distribution table or use a calculator to find the \( z \)-score that corresponds to this area.
By using the standard normal distribution table or a calculator, we find that the \( z \)-score corresponding to an area of 0.005 is approximately -2.58.
Therefore, the critical value \( z_{\alpha / 2} \) that corresponds to a 99% confidence level is approximately -2.58.