Find the critical value Za/2 that corresponds to the confidence level 81.7%

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!

Learn how to find the critical value Zα/2 for a confidence level of 81.7%. This solution explains the step-by-step process, including calculating alpha and using the Z-table to find the critical Z-value of approximately 1.34. Suitable for college-level statistics students.

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