Which of the following is a correct interpretation of a $p$-value that is not very small in the context of hypothesis testing?

• A. The probability that the $\mathrm{null}$ hypothesis is true is high.
• B. The $\mathrm{null}$ hypothesis is proven to be true.
• C. The $\mathrm{alternative}$ hypothesis is more likely to be true.
• D. There is insufficient evidence to reject the $\mathrm{null}$ hypothesis.

Answer: (D)

Verified step by step guidance

1. Understand that a p-value in hypothesis testing measures the probability of obtaining test results at least as extreme as the observed results, assuming the null hypothesis is true.
2. Recognize that a p-value that is not very small means the observed data is not unusual under the null hypothesis, so there is no strong evidence against it.
3. Recall that the p-value does not give the probability that the null hypothesis is true or false; it only assesses the compatibility of the data with the null hypothesis.
4. Interpret a non-small p-value as an indication that there is insufficient evidence to reject the null hypothesis, rather than proving it true or confirming the alternative hypothesis.
5. Conclude that the correct interpretation is: 'There is insufficient evidence to reject the null hypothesis.'