Final answer:
The correct definition of the p-value is the probability of observing a test statistic as extreme as the one obtained, assuming that the null hypothesis is true. Both options (c) and (d) are incorrect interpretations, as they mistakenly describe the p-value as the probability that the null or alternative hypothesis is true, respectively.
Step-by-step explanation:
The p-value in hypothesis testing is a measure of the strength of the evidence against the null hypothesis provided by the sample. It is the probability that the observed value of the test statistic will occur or be more extreme, assuming that the null hypothesis is true. In determining the correctness of the definitions:
- Statement (a) is incorrect because the p-value is not the minimally tolerable type I error; it is a measure of the evidence against the null hypothesis.
- Statement (b) is incorrect in defining the p-value as the minimally tolerable type II error; this mistake conflates the p-value with the concept of power of a test.
- Statement (c) is the incorrect interpretation as it wrongly assumes the p-value reveals the probability that the null hypothesis is true.
- Statement (d) is the incorrect interpretation because the p-value does not measure the probability that the alternative hypothesis is true.
The correct definition of the p-value is that it is the probability of obtaining test results at least as extreme as the results actually observed, under the assumption that the null hypothesis is true. This correct definition aligns with the provided reference that highlights the p-value as an indication of the likelihood of the extreme test statistic's occurrence if the null hypothesis were true.