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. Increase the sample size or decrease the sample size
• C. Reject the $H_0$ hypothesis or fail to reject the $H_0$ hypothesis
• D. Accept the $H_a$ hypothesis or accept the $H_0$ hypothesis

Answer: (C)

Verified step by step guidance

1. Understand that in hypothesis testing, the goal is to make a decision about the null hypothesis (\(H_0\)) based on sample data.
2. Recognize that the two possible decisions after analyzing the data are: either to reject the null hypothesis or to fail to reject the null hypothesis.
3. Note that we never 'prove' the null hypothesis true or false; instead, we assess whether there is enough evidence to reject it.
4. Understand that 'failing to reject' the null hypothesis means there is not sufficient evidence against it, but it does not confirm it is true.
5. Remember that these decisions are based on comparing a test statistic to a critical value or using a p-value to determine statistical significance.