Final answer:
The critical value zα/2 needed for a 99.6% confidence interval is approximately 2.878, which is the z-score that leaves 0.002 of the distribution's area in each tail.
Step-by-step explanation:
To find the critical value zα/2 needed to construct a 99.6% confidence interval, you need to determine the z-score that leaves a total area of 0.4% in the two tails of the standard normal distribution, because the confidence level of 99.6% means that we want 99.6% of the distribution's area to be in the center (CL = 1 - α - hence α = 0.004) and only 0.4% of the area to be split between the two tails.
To find the critical value, you'll divide the total area in the tails by two, resulting in 0.002 for each tail, since we are dealing with a two-tailed test.
Therefore, you are looking for z0.002. By using a z-score table, a calculator, or a computer, you can find that the z-score that corresponds to an area of 0.998 (to the left of the z-score) or 0.002 to the right is approximately 2.878. This z-score is your critical value zα/2.
To summarize, the critical value zα/2 needed to construct a 99.6% confidence interval is approximately 2.878.