In the context of hypothesis testing, what does it mean to fail to reject the null hypothesis?

• A. The alternative hypothesis has been proven to be false.
• B. The null hypothesis has been proven to be true.
• C. The test statistic is equal to the critical value, so the null hypothesis must be rejected.
• D. There is insufficient evidence in the sample to conclude that the alternative hypothesis is true at the chosen significance level.

Answer: (D)

Verified step by step guidance

1. Understand that in hypothesis testing, the null hypothesis (\(H_0\)) represents a default or status quo assumption, while the alternative hypothesis (\(H_a\)) represents a claim we want to test against \(H_0\).
2. Recognize that failing to reject the null hypothesis means the data does not provide strong enough evidence to support the alternative hypothesis at the chosen significance level (\(\alpha\)).
3. Note that failing to reject \(H_0\) is not the same as proving \(H_0\) is true; it simply means there is insufficient evidence to conclude \(H_a\) is true based on the sample data.
4. Recall that the decision to reject or fail to reject \(H_0\) depends on comparing the test statistic to critical values or using a p-value relative to \(\alpha\).
5. Therefore, failing to reject the null hypothesis means: 'There is insufficient evidence in the sample to conclude that the alternative hypothesis is true at the chosen significance level.'