Final answer:
A small p-value indicates that the null hypothesis is unlikely and should be rejected in favor of the alternative hypothesis, but it does not assess practical significance. It is compared to an alpha value to make a decision on the null hypothesis.
Step-by-step explanation:
A small p-value implies several things in the context of hypothesis testing. Primarily, it suggests that the parameter value indicated by the null hypothesis is highly unlikely, given the observed data. This leads us to two possible conclusions:
- The parameter value specified by the null hypothesis is considered not plausible.
- The null hypothesis should be rejected in favor of the alternative hypothesis.
However, it is important to note that a small p-value does not necessarily mean the difference is practically significant. Practical significance refers to the real-world importance or effect of the observed difference and is not determined by the p-value alone.
The process of hypothesis testing often includes comparing the p-value to a predetermined significance level, traditionally set at 0.05 or even 0.01. If the p-value is less than the chosen significance level (α = 0.05), the decision would be to reject the null hypothesis. On the other hand, if the p-value is not less than the significance level, the appropriate decision is to not reject the null hypothesis.