6.3-6.4

6.3-6.4 - (6.3&6.4 ;useandabuse oftests Test as a decision...

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Introduction to inference (6.3 & 6.4) Significance level and power; use and abuse  of tests

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Test as a decision Type I and II errors Significance level Power Use and abuse of tests
Type I and II errors A Type I error is made when we reject the null hypothesis and the null hypothesis is actually true (incorrectly reject a true H 0 ). A Type II error is made when we fail to reject the null hypothesis and the null hypothesis is false (incorrectly keep a false H 0 ).

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Ideally, the probability of incurring either type of error should be kept small. For a given sample size, lowering the probability of type I error would increase the probability of type II error, and vice versa. The common practice is to first control the probability of type I error: choose a small number α, and make sure that the probability of type I error is not more than α. α is called the level of significance . The probability of type II error can be reduced, for example, by using a larger sample.

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H 0 : µ = 0 vs H a : µ > 0, where µ is the mean percent change in bone density. Suppose the significance level is chosen to be 5%. Can an exercise program increase bone density? From previous studies, we assume that σ = 2 for the percent change in bone density. A reasonable decision rule is to reject H 0 when is sufficiently greater than 0, say if > C, where C is to be determined. Thus the possible values of the sample mean are divided into two disjoint sets: those which are greater than C (rejection region) and those which are not (acceptance region). C, called the critical value , depends on the chosen level of significance. C is the value such that if H 0 is true, then the probability of falsely rejecting it is equal to 0.05. In our example, it’s the value such that P ( > C) = 0.05 when µ = 0.
Suppose n = 25. Then C =0.658. All sample averages larger than 0.658 will result in rejecting the null hypothesis.

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6.3-6.4 - (6.3&6.4 ;useandabuse oftests Test as a decision...

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