Suppose you conduct a two-tailed hypothesis test at the $5\%$ significance level and obtain a test statistic of $1.34$. What is the correct conclusion?

• A. Fail to reject the null hypothesis because $1.34$ does not exceed the critical value for $\alpha =0.05$.
• B. Reject the null hypothesis because $1.34$ is greater than the critical value for $\alpha =0.05$.
• C. Reject the null hypothesis because the test statistic is positive.
• D. The result is inconclusive because the test statistic is always compared to the p-value, not the critical value.

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 5%, so each tail has 2.5%.
2. Determine the critical values for the test statistic at \( \alpha = 0.05 \) in a two-tailed test. These critical values correspond to the z-scores that cut off the lower 2.5% and upper 2.5% of the standard normal distribution.
3. Look up or calculate the critical z-values for \( \alpha = 0.05 \) two-tailed test, which are approximately \( \pm 1.96 \).
4. Compare the absolute value of the test statistic \( |1.34| \) to the critical value \( 1.96 \). If the test statistic is greater than the critical value, reject the null hypothesis; otherwise, fail to reject it.
5. Since \( 1.34 < 1.96 \), conclude that there is not enough evidence to reject the null hypothesis at the 5% significance level.