Modern Mathematical Statistics with Applications by Jay L. Devore & Kenneth N. Berk

Author:Jay L. Devore & Kenneth N. Berk
Language: eng
Format: epub
Publisher: Springer New York, New York, NY

2.A rejection region, the set of all test statistic values for which H 0 will be rejected

The null hypothesis will then be rejected if and only if the observed or computed test statistic value falls in the rejection region.

As another example, suppose a cigarette manufacturer claims that the average nicotine content μ of brand B cigarettes is (at most) 1.5 mg. It would be unwise to reject the manufacturer’s claim without strong contradictory evidence, so an appropriate problem formulation is to test H 0: μ = 1.5 versus H a: μ > 1.5. Consider a decision rule based on analyzing a random sample of 32 cigarettes. Let denote the sample average nicotine content. If H 0 is true, , whereas if H 0 is false, we expect to exceed 1.5. Strong evidence against H 0 is provided by a value that considerably exceeds 1.5. Thus we might use as a test statistic along with the rejection region .

In both the circuit board and nicotine examples, the choice of test statistic and form of the rejection region make sense intuitively. However, the choice of cutoff value used to specify the rejection region is somewhat arbitrary. Instead of rejecting H 0: p = .10 in favor of H a: p < .10 when x ≤ 15, we could use the rejection region x ≤ 14. For this region, H 0 would not be rejected if 15 defective boards are observed, whereas this occurrence would lead to rejection of H 0 if the initially suggested region is employed. Similarly, the rejection region might be used in the nicotine problem in place of the region .

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