Significance and critical regions

Objective. Interpret significance levels and rejection regions.

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Learn: the key idea

The significance level is the probability threshold, often 5% or 1%, below which we consider sample results too unlikely to have occurred under H₀, leading us to reject it. The critical region is the set of extreme values of the test statistic for which we reject H₀, found so that its probability is as close as possible to the significance level without exceeding it. For example, at 5% significance for X~B(20,0.5), the lower critical region might be X≤5 since P(X≤5) is just under 0.05. Values outside the critical region mean there is insufficient evidence to reject H₀.

Worked example

At the 5% significance level, explain what the critical region represents in a hypothesis test.

1 $Critical region contains extreme values of the test statistic$
2 $These values have probability close to but not exceeding 5% under H₀$
3 $If the observed value lies in this region, H₀ is rejected$
4 $This indicates significant evidence for H₁$

Answer: The set of extreme outcomes that lead to rejecting H₀

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Common mistake. Students reject H₀ whenever the test statistic is unusual-looking, without comparing its probability to the significance level. Always calculate the actual probability and compare it directly to the stated significance level.

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