For a two-tailed hypothesis test at the $5\%$ significance level using the standard normal distribution, what are the critical value(s) for $z$?

• A. $-1.96$ and $1.96$
• B. $1.96$
• C. $-1.65$ and $1.65$
• D. $1.65$

Answer: (A)

Verified step by step guidance

1. Identify that the problem involves a two-tailed hypothesis test at the 5% significance level, which means the total area in both tails of the standard normal distribution is 0.05.
2. Since the test is two-tailed, split the significance level equally between the two tails: each tail will have an area of 0.025 (i.e., 0.05 / 2).
3. Use the standard normal distribution table (Z-table) or a statistical software to find the z-value that corresponds to the cumulative probability of 0.025 in the lower tail and 0.975 in the upper tail.
4. The critical values are the z-scores that mark the boundaries of the rejection regions: one negative z-value for the lower tail and one positive z-value for the upper tail.
5. Express the critical values as \(-z_{\alpha/2}\) and \(z_{\alpha/2}\), where \(\alpha = 0.05\), so the critical values are \(-z_{0.025}\) and \(z_{0.025}\).