Find the critical value, $z_{\frac{\alpha}{2}}$, for an 80% confidence interval.

• A. 0.10
• B. 1.282
• C. 0.40
• D. 0.90

Answer: (B)

Verified step by step guidance

1. Identify the confidence level for the interval, which is 80%. This means that the level of significance, \( \alpha \), is 20% or 0.20.
2. Since the confidence interval is two-tailed, divide the level of significance by 2 to find \( \frac{\alpha}{2} \). So, \( \frac{\alpha}{2} = \frac{0.20}{2} = 0.10 \).
3. Determine the critical value \( z_{\frac{\alpha}{2}} \) using the standard normal distribution table or a calculator. This value corresponds to the point where the cumulative probability is \( 1 - \frac{\alpha}{2} = 0.90 \).
4. Look up the cumulative probability of 0.90 in the standard normal distribution table to find the \( z \)-score. This \( z \)-score is the critical value \( z_{\frac{\alpha}{2}} \).
5. Verify the critical value by checking that it matches the expected value for an 80% confidence interval, which is approximately 1.282.