Suppose you conduct a two-tailed hypothesis test at the $5\%$ significance level and obtain a test statistic of $1.77$. What is the correct conclusion?

• A. Fail to reject the null hypothesis because $1.77$ does not exceed the critical value for $\alpha =0.05$.
• B. The result is inconclusive because the test statistic is exactly equal to the critical value.
• C. Reject the null hypothesis because the test statistic is positive.
• D. Reject the null hypothesis because $1.77$ is greater than the critical value for $\alpha =0.05$.

Answer: (A)

Verified step by step guidance

1. Identify the significance level \( \alpha = 0.05 \) for a two-tailed test. This means the total area in both tails of the distribution is 0.05, so each tail has an area of \( 0.025 \).
2. Determine the critical values for the test statistic corresponding to \( \alpha/2 = 0.025 \) in each tail. For a standard normal distribution, these critical values are \( \pm z_{\alpha/2} \), where \( z_{\alpha/2} \) is the z-score that leaves 2.5% in the upper tail.
3. Look up or calculate the critical z-value for \( \alpha = 0.05 \) two-tailed test, which is approximately \( \pm 1.96 \). This means the rejection regions are \( z < -1.96 \) or \( z > 1.96 \).
4. Compare the test statistic \( 1.77 \) to the critical values \( \pm 1.96 \). Since \( 1.77 \) is less than \( 1.96 \), it does not fall into the rejection region.
5. Conclude that you fail to reject the null hypothesis because the test statistic does not exceed the critical value for the given significance level.