The [tex]$P$[/tex]-value of a test is a crucial concept in statistical hypothesis testing. To understand it, we need to focus on how it relates to the interpretation of evidence in the context of our hypotheses.
1. Alternative Hypothesis and Null Hypothesis:
- The null hypothesis (denoted as [tex]\( H_0 \)[/tex]) represents a general or default position, such as "there is no effect" or "there is no difference."
- The alternative hypothesis (denoted as [tex]\( H_a \)[/tex]) represents the position that there is an effect or a difference.
2. Interpreting the P-Value:
- The P-value is defined as the probability of obtaining test results at least as extreme as the results observed, assuming that the null hypothesis [tex]\( H_0 \)[/tex] is true.
- It helps us understand whether the observed data is consistent with the null hypothesis or if it provides enough evidence to consider the alternative hypothesis.
Given these points, let's address the question:
_The [tex]$P$[/tex]-value of a test is the probability of getting evidence for the alternative hypothesis, [tex]\( H_a \)[/tex], as strong or stronger than the observed evidence, when which hypothesis is true?_
From the definition of the P-value, we know that:
- The P-value is calculated under the assumption that the null hypothesis ([tex]\( H_0 \)[/tex]) is true.
- It represents the probability of observing data as extreme as or more extreme than what was observed, under the null hypothesis.
Therefore, the correct statement is:
"The [tex]$P$[/tex]-value of a test is the probability of getting evidence for the alternative hypothesis, [tex]\( H_a \)[/tex], as strong or stronger than the observed evidence, when the null hypothesis, [tex]\( H_0 \)[/tex], is true."
