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Rejection region
Definition
A rejection region is the set of possible test-statistic values or sample outcomes that lead a specified hypothesis test to reject its null hypothesis. It is determined by the test, significance level and alternative hypothesis. The region may lie in one tail, both tails or have another form appropriate to the statistic. It belongs to a decision rule established under a probability model, not to a region of parameter values declared impossible by the observed data.
What it means
The region tells us which outcomes would be sufficiently incompatible with the null model according to the chosen test rule. A critical value marks a boundary in many familiar tests. Choosing a significance level controls the probability of entering the rejection region when the null and relevant assumptions hold, exactly or approximately depending on the procedure. Rejection does not give the probability that the null hypothesis is false, and failing to reject does not establish that it is true.
How it works
Specify the null and alternative hypotheses, test statistic and reference distribution. Choose the significance level before examining the result, accounting for multiple tests or sequential monitoring where relevant. Determine the region whose null probability meets the intended error-rate control. Calculate the observed statistic and check whether it lies in that region. Report the p-value, effect estimate and uncertainty alongside the decision. For discrete statistics, an unrandomised test may have a null rejection probability below the nominal level.
Three practical examples
Example 1: A two-sided normal-mean test rejects for sufficiently large positive or negative standardized deviations. Example 2: A one-sided test for an increased defect rate uses an upper-tail rejection region. Example 3: A goodness-of-fit test rejects for sufficiently large discrepancy between observed and expected counts. Each test defines extremeness differently. A region selected after seeing the direction or magnitude of the result no longer has the straightforward interpretation of the prespecified rule.
Think of it this way
Think of a boundary drawn on a map of outcomes before a study begins. Outcomes beyond it trigger the stated decision. The boundary is calibrated using the null model, but it does not turn a result just outside into certainty or a result just inside into proof of no effect. Nearby outcomes usually contain similar information even when a binary reporting rule labels them differently.
Worked example
Suppose independent normal measurements have known standard deviation two, and a sample of 25 is used to test the null mean 100 against a two-sided alternative at level 0.05. The standard error is 2/square root of 25 = 0.4. The statistic is Z = (sample mean - 100)/0.4. The rejection region is Z below approximately -1.95996 or above 1.95996, corresponding to sample means below 99.2160 or above 100.7840. A sample mean of 100.9 gives Z = 2.25 and falls in the region. A mean of 100.6 gives Z = 1.50 and does not. Under the stated null model, the two tails together have probability 0.05. If the standard deviation were estimated rather than known, this exact normal-reference calculation would need reconsideration.
Illustrative case study
Fictional case study: Cedar Quality plans a two-sided test before collecting a validation sample. After seeing a modest increase, a draft report switches to a one-sided criterion to obtain rejection. The analyst restores the planned rule and presents the estimate and uncertainty without changing the hypothesis retrospectively. This fictional case illustrates preserving the interpretation of a test; it does not claim that one-sided tests are inherently invalid when justified in advance.
Common misconceptions and mistakes
Common mistakes include treating the significance level as the probability that a rejected null is true, equating non-rejection with equivalence, and changing tails after observing results. Repeatedly checking a fixed-threshold test can inflate error rates unless the design accounts for monitoring. Multiple comparisons can also alter the chance of at least one false rejection. A statistically significant result can be too small to matter practically, while a nonsignificant result can remain compatible with important effects.
Business use
Business use: Rejection regions provide reproducible rules for planned statistical tests in quality studies and experiments. Specify the outcome, null, alternative and decision consequences before data analysis. Align the test with the actual question and use appropriate methods for repeated looks or multiple endpoints. Report effect sizes and uncertainty so decisions are not reduced to one threshold. Distinguish statistical rejection from contractual acceptance criteria and from evidence of practical benefit.
People also ask
Is the rejection region chosen after seeing the statistic?
It should follow the prespecified test rule, not a convenient result-dependent choice. Q: Does non-rejection prove no effect? A: No; it may reflect limited information or power. Q: Must the actual null rejection probability equal alpha exactly? A: Not always; discrete or approximate procedures can differ from the nominal level.
Check your understanding
Why are both low and high sample means included in the example's region?
Answer: The alternative is two-sided.
What probability equals 0.05 under the assumptions?
Answer: The probability of a statistic falling in the rejection region when the null mean is 100, not the probability that the null is true after observing a result.
Related terms
- Critical values mark test boundaries
Learn next
Draw one-sided and two-sided rejection regions for a standard-normal statistic. Convert their critical values into sample-mean thresholds for different standard errors. Then examine a discrete binomial test to see why its attainable rejection probability can be below the nominal level. Compare threshold decisions with effect estimates and confidence intervals.
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- Relative standard error - Relative standard error, usually abbreviated RSE, expresses an estimate's standard error relative to the magnitude of that estimate, commonly as a percentage.
Sources
Sources: University of Vermont, Statistical Methods glossary, Rejection region: https://www.uvm.edu/~statdhtx/methods8/Glossary/Glossary.html ; NIST/SEMATECH e-Handbook, What are statistical tests?: https://itl.nist.gov/div898/handbook/prc/section1/prc13.htm . These educational and institutional sources explain null hypotheses, critical values and rejection rules. The normal-mean calculation and fictional quality case are original illustrations.
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