In the context of hypothesis testing, what are the two possible decisions you can make after performing the test?

• A. Increase the sample size or decrease the sample size
• B. Prove the $H_0$ is true or prove it is false
• C. Reject the $H_0$ or fail to reject the $H_0$
• D. Accept the $H_0$ or accept the $H_1$

Answer: (C)

Verified step by step guidance

1. Understand that in hypothesis testing, we start with a null hypothesis (\(H_0\)) and an alternative hypothesis (\(H_a\)).
2. After collecting data and performing the test, we use a test statistic and compare it to a critical value or use a p-value to make a decision.
3. The two possible decisions are: either to reject the null hypothesis (\(H_0\)) if the evidence is strong enough, or to fail to reject the null hypothesis if the evidence is not sufficient.
4. Note that we never 'accept' the null hypothesis; failing to reject it means we do not have enough evidence against it.
5. Therefore, the correct phrasing of the decisions after hypothesis testing is: 'Reject the null hypothesis' or 'Fail to reject the null hypothesis.'