To find the critical value $Z_{\alpha/2}$ corresponding to a confidence level of 81.7%, follow these steps:

1. Confidence level: $81.7\%$
2. Alpha ($\alpha$): The confidence level is $1 - \alpha$. So, $\alpha = 1 - 0.817 = 0.183$.
3. Divide $\alpha$ by 2: $\frac{\alpha}{2} = \frac{0.183}{2} = 0.0915$
4. Find the Z-score: Use the cumulative distribution function (CDF) of the standard normal distribution to find the Z-value that leaves 0.0915 in the tails. Since 0.0915 is in one tail, the area to the left of $Z_{\alpha/2}$ is $1 - 0.0915 = 0.9085$.

Using the standard normal distribution table or a calculator:

$Z_{\alpha/2} \approx 1.34$

Thus, the critical value $Z_{\alpha/2}$ for an 81.7% confidence level is approximately 1.34.

Would you like more details on how to use the Z-table or any further explanation?

Here are 5 related questions to explore this topic further:

1. How does the confidence level affect the width of a confidence interval?
2. What is the relationship between the critical value and the standard error in hypothesis testing?
3. How do you calculate a confidence interval for a population mean using $Z_{\alpha/2}$?
4. Why do we divide $\alpha$ by 2 when determining the critical value for a confidence interval?
5. Can we use the $t$-distribution instead of the $Z$-distribution for small samples?

Tip: When using a Z-table, remember that it gives you the cumulative area to the left of the Z-score. Make sure to subtract when finding upper-tail probabilities!