Part 2 of 5
$z_{\alpha / 2}$ for the $87 \%$ confidence interval
\[
z_{\alpha / 2}=
\]

Step 1 ：The problem is asking for the z-score for a 87% confidence interval. The z-score is a measure of how many standard deviations an element is from the mean. In the context of confidence intervals, the z-score represents the number of standard deviations a data point would be from the mean of a distribution in order to capture the central 87% of the data.

Step 2 ：To calculate this, we first need to calculate the alpha level. The alpha level is 1 minus the confidence level, so in this case it would be \(1 - 0.87 = 0.13\).

Step 3 ：Since we are looking for a two-tailed test (because of the \(\alpha / 2\)), we need to divide the alpha level by 2, which gives us \(0.065\).

Step 4 ：We then need to find the z-score that corresponds to this alpha level. This can be done using a z-table or a statistical function.

Step 5 ：The $z_{\alpha / 2}$ for the $87 \%$ confidence interval is approximately \(\boxed{1.514}\).