To find \( z_{\alpha/2} \) for a 99% confidence interval, we first need to determine \( \alpha \), the significance level.
For a 99% confidence interval:
\[ \alpha = 1 - 0.99 = 0.01 \]
Now, since \( z_{\alpha/2} \) corresponds to the critical value that leaves \( \alpha/2 \) in each tail of the standard normal distribution, we calculate:
\[ \alpha/2 = 0.01/2 = 0.005 \]
Next, we need to find the z-score that corresponds to the cumulative probability of \( 1 - 0.005 = 0.995 \).
Using standard normal distribution tables or a calculator, the value of \( z_{0.005} \) is approximately:
\[ z_{0.005} \approx 2.576 \]
Thus, \( z_{\alpha/2} \) for a 99% confidence interval is:
\[ z_{\alpha/2} \approx 2.576 \]
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