For a 90% confidence interval for the population mean, what is the value of $z_{\frac{\alpha }{2}}$?

• A. $1.28$
• B. $1.96$
• C. $2.33$
• D. $1.645$

Answer: (D)

Verified step by step guidance

1. Understand that the value \( z_{\alpha/2} \) corresponds to the critical z-value that cuts off the upper \( \alpha/2 \) tail of the standard normal distribution, where \( \alpha = 1 - \text{confidence level} \).
2. Calculate \( \alpha \) for a 90% confidence interval: \( \alpha = 1 - 0.90 = 0.10 \).
3. Divide \( \alpha \) by 2 to find the tail probability: \( \alpha/2 = 0.10 / 2 = 0.05 \).
4. Find the z-value such that the area to the right of \( z_{\alpha/2} \) under the standard normal curve is 0.05, or equivalently, the area to the left is 0.95.
5. Use a standard normal distribution table or statistical software to look up the z-value corresponding to a cumulative probability of 0.95, which gives the critical value \( z_{\alpha/2} \) for the 90% confidence interval.