When constructing a 93% confidence interval for a population proportion $p$, what value of $z$ should be used as the critical value?

• A. $2.33$
• B. $1.64$
• C. $1.81$
• D. $1.96$

Answer: (C)

Verified step by step guidance

1. Understand that the critical value \( z \) corresponds to the z-score that captures the middle 93% of the standard normal distribution.
2. Calculate the significance level \( \alpha \) as \( 1 - 0.93 = 0.07 \). This means the total area in the two tails of the distribution is 0.07.
3. Since the confidence interval is two-tailed, divide \( \alpha \) by 2 to find the area in each tail: \( \frac{0.07}{2} = 0.035 \).
4. Find the z-score \( z_{\alpha/2} \) such that the area to the right of this z-score is 0.035, or equivalently, the area to the left is \( 1 - 0.035 = 0.965 \).
5. Use a standard normal distribution table or a calculator to find the z-score corresponding to a cumulative probability of 0.965, which will give the critical value \( z \) for the 93% confidence interval.