Key Takeaways on Rejection Regions & P-Values
The rejection region (or critical region) is the set of test statistic values for which we reject the null hypothesis. Conversely, the non-rejection region is the area where we do not reject the null hypothesis, generally corresponding to the area under 1-alpha.
The rejection region contains test statistic values that are highly unlikely if the null hypothesis were true. If our result falls here, it's strong evidence against the null. The non-rejection region contains values that are more probable under the null hypothesis.
It means we start by assuming the null hypothesis is correct. We base our statistical model, expectations, and calculations on this "innocent until proven guilty" premise. We then check if our observed data is a good fit for this assumption.
Because the probability of getting such an extreme result is very low if the null hypothesis were true. An extreme value suggests that our initial assumption (the null hypothesis) might be wrong, and another explanation is more likely.
The p-value is the probability of seeing a result at least as extreme as the one you observed, assuming the null is true. A tiny p-value (e.g., 0.01) means your observed data is very rare under the null assumption, thus providing strong evidence to reject that assumption.