A critical value is a boundary separating results that lead to rejection of the null hypothesis from those that do not. The critical region is the complete set of test-statistic values for which the null hypothesis, , is rejected.
Under , the probability of falling in the critical region is , or as close to as the distribution permits. Therefore, a smaller creates a more extreme rejection boundary and stronger evidence is required.
First choose the significance level, , which is the maximum permitted probability of rejecting when it is true. The alternative hypothesis, , determines where the critical region is placed.
| Alternative hypothesis | Test type | Critical region using |
|---|---|---|
| Right-tailed | ||
| Left-tailed | ||
| Two-tailed | or |
For continuous distributions, the tail area can equal exactly. For discrete binomial or Poisson tests, individual outcomes have fixed probabilities, so the chosen critical region usually maximizes the Type I error probability without exceeding . This distinction is especially important when a critical region is written in terms of integer outcomes.
For a right-tailed test with , the critical value is approximately . Therefore, the critical region is , whose probability under is .
For example, suppose
with known , , and . The corresponding critical sample mean is
Thus, the critical region can be written as . A sample mean in this region provides sufficient evidence to reject .
A common misconception is that a critical value is the same as a p-value. It is not: the critical value is fixed before examining the result, while the p-value is calculated from the observed data.
For AHL 4.18, state , , and ; identify the correct tail; then compare the test statistic with the critical region. If the result is outside the region, write “insufficient evidence to reject ”, never “accept ”, and conclude in context.
