Rejecting the Null Hypothesis

B

The alpha risk is the probability of rejecting the null when it is actually true. So if we had rejected the null, we can conclude, given what we know of the alpha, that the null is false.

When conducting hypothesis testing, the null hypothesis (H0) is the statement we assume to be true initially, while the alternative hypothesis (H1) is the statement we are trying to gather evidence in support of. The alpha risk, also known as the significance level, represents the maximum probability of rejecting the null hypothesis when it is actually true.

If we reject the null hypothesis, it means that we have found sufficient evidence to cast doubt on its validity. In this context, the alpha risk refers to the probability of making a Type I error, which occurs when we reject the null hypothesis when it is, in fact, true.

Given the choices provided, we need to select the answer that aligns with rejecting the null hypothesis and considers the alpha risk.

Let's analyze each answer choice:

A. Alternative hypothesis is false: This choice suggests that rejecting the null hypothesis implies that the alternative hypothesis is false. However, this is not accurate. Rejecting the null hypothesis does not necessarily prove the alternative hypothesis to be true or false.

B. Null hypothesis is false: This choice correctly identifies that rejecting the null hypothesis means that it is false. When we have sufficient evidence to reject the null hypothesis, we conclude that it is not supported by the data. This aligns with the purpose of hypothesis testing.

C. Null hypothesis is true: This choice contradicts the very act of rejecting the null hypothesis. If we reject the null hypothesis, it means that we have evidence against its validity, so we cannot conclude that it is true.

D. Both the null hypothesis and the alternative hypothesis are false: This choice suggests that both hypotheses are false. However, rejecting the null hypothesis does not provide information about the truth or falsehood of the alternative hypothesis. It only indicates that the null hypothesis is not supported by the evidence.

E. Both the null hypothesis and the alternative hypothesis are true: This choice implies that both hypotheses are true, which is contradictory. In hypothesis testing, we assume that either the null hypothesis or the alternative hypothesis is true, not both simultaneously.

Considering the explanations above, the correct answer is B. When we reject the null hypothesis, we can conclude that the null hypothesis is false, given the evidence gathered during hypothesis testing.