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

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

Answer: (B)

Verified step by step guidance

1. Understand that in hypothesis testing, we start with a null hypothesis (\(H_0\)) which represents a default or status quo assumption, and an alternative hypothesis (\(H_a\)) which represents a new claim or effect we want to test.
2. After collecting and analyzing sample data, the goal is to decide whether there is enough evidence to reject the null hypothesis or not.
3. The two possible decisions are: either to reject the null hypothesis if the data provides strong evidence against it, or to fail to reject the null hypothesis if the evidence is insufficient.
4. Note that we never 'prove' the null hypothesis true; failing to reject it simply means we do not have enough evidence against it based on the sample data.
5. Therefore, the correct phrasing of the decisions in hypothesis testing is: 'Reject the null hypothesis' or 'Fail to reject the null hypothesis.'