In the context of hypothesis testing, what are the two possible decisions you can make after analyzing the sample data?

• A. Accept the $H$_a (alternative hypothesis) or accept the $H$₀ (null hypothesis)
• B. Prove the $H$₀ (null hypothesis) is true or prove it is false
• C. Reject the $H$₀ (null hypothesis) or fail to reject the $H$₀
• D. Increase the sample size or decrease the sample size

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 (\(H_a\)). The null hypothesis usually represents a statement of no effect or no difference.
2. After collecting and analyzing sample data, the goal is to make a decision about the null hypothesis based on the evidence provided by the data.
3. The two possible decisions are: either to reject the null hypothesis if the data provides sufficient evidence against it, or to fail to reject the null hypothesis if the data does not provide strong enough evidence.
4. Note that we never 'accept' the null hypothesis outright; instead, we say we 'fail to reject' it because the data might not be strong enough to conclude it is false.
5. Therefore, the correct phrasing of the two possible decisions in hypothesis testing is: 'Reject the null hypothesis' or 'Fail to reject the null hypothesis.'