Suppose you conduct a hypothesis test and obtain a test statistic of $2.23$. If you are using a significance level of $0.05$ and a two-tailed test, what is the most likely conclusion?

• A. Accept the null hypothesis because the test statistic is positive.
• B. There is not enough information to make a decision.
• C. Fail to reject the null hypothesis because the test statistic is not extreme enough.
• D. Reject the null hypothesis because the test statistic falls in the critical region.

Answer: (D)

Verified step by step guidance

1. Identify the null hypothesis (H0) and the alternative hypothesis (H1) for the test. Since it is a two-tailed test, H1 typically states that the parameter is not equal to the null value.
2. Determine the significance level \( \alpha = 0.05 \) and understand that for a two-tailed test, this significance level is split between the two tails of the distribution, so each tail has an area of \( \frac{0.05}{2} = 0.025 \).
3. Find the critical values corresponding to the significance level for a two-tailed test. These critical values are the points beyond which the test statistic would lead to rejection of the null hypothesis. For example, if using a standard normal distribution, find \( z_{\alpha/2} \) such that \( P(Z > z_{\alpha/2}) = 0.025 \).
4. Compare the calculated test statistic (2.23) to the critical values. If the test statistic is greater than the positive critical value or less than the negative critical value, it falls in the rejection region.
5. Based on this comparison, conclude whether to reject or fail to reject the null hypothesis. Since the test statistic 2.23 is likely beyond the critical value for \( \alpha = 0.05 \) in a two-tailed test, the conclusion is to reject the null hypothesis.