\( z_{\alpha / 2} \) for the \( 90 \% \) confidence interval

For a \( 90\% \) confidence interval, the \( z_{\alpha / 2} \) value is approximately \( 1.645 \). This value indicates that if you were to take many samples, about \( 90\% \) of the intervals you calculate using this \( z \)-score would capture the true population parameter. It's the critical value that corresponds to the upper \( 5\% \) tail in a standard normal distribution. In real-world application, knowing the \( z_{\alpha / 2} \) allows you to calculate margin of errors and construct confidence intervals for sample means. For example, if you're estimating a poll's result, using \( 1.645 \) enables you to convey the range in which the true percentage of support lies, providing a clearer picture for decision-makers or stakeholders!