Critical Region, Critical Values and Significance Level

A critical region, also called the rejection region, is an area under the probability distribution curve that contains all test statistic values indicating the null hypothesis must be rejected. The critical region is demarcated by the critical value and may fall at the right, left, or both tails of the distribution based on the alternative hypothesis direction.

A critical value is calculated using z, t, or chi-square distribution tables at a specific significance level. It is a fixed value for a given sample size and significance level that creates a demarcation between test statistic values suggesting rejection of the null hypothesis and those indicating acceptance.

The significance level, denoted by α and commonly set at 0.05 or 0.01, is the probability that the calculated test statistic will fall in the critical region when the null hypothesis is actually true. It indicates whether evidence for rejecting a true null hypothesis is statistically strong enough.

Sample statistics such as proportion, mean, or standard deviation are converted into test statistics using z, t, or chi-square equations. When a test statistic falls within the critical region, it signals that the null hypothesis should be rejected based on the sample data.

The critical region may be positioned at the right tail, left tail, or both tails of the distribution based on the direction specified in the alternative hypothesis. This directional placement ensures the rejection region aligns with the specific claim being tested against the null hypothesis.

The critical value is based on a pre-decided significance level. The significance level determines where the critical value is positioned on the distribution, which in turn defines the boundaries of the critical region where test statistics trigger rejection of the null hypothesis.

When testing a claim about population proportion, you convert the sample proportion into a test statistic, then compare it against the critical value determined at your chosen significance level. If the test statistic falls in the critical region beyond the critical value, you reject the null hypothesis.