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Unformatted text preview: ∙ Suppose productivity is expensive to measure, so the firm chooses a relatively small sample of workers. The random sample is X i : i 1,..., n , where X i is the change in worker i ’s productivity. 7 ∙ Here the null hypothesis might be that there was no effect on productivity and the alternative that there was some effect (positive or negative): H : and the alternative hypothesis is H 1 : ≠ ∙ Of course, the firm is interested in whether 0 or 0 if H is false. 8 ∙ We will use data and test statistics to test the null hypothesis against the stated alternative. Not surprisingly, the test statistics are usually related to the point estimators we have covered. 9 2 . Stating Null and Alternative Hypotheses ∙ In a general setting, let be a p 1 vector of parameters taking values in the parameter space Θ ⊂ p . The null hypothesis is stated generally as H : ∈ Θ H 1 : ∈ Θ 1 where Θ and Θ 1 are subsets of Θ . ∙ We do not allow H and H 1 to both be true, and so Θ ∩ Θ 1 . 10 ∙ It is sometimes useful for Θ and Θ 1 not to exhaust all the possibilities, but in most cases one effectively takes the alternative to simply be that the null is false: Θ 1 Θ − Θ DEFINITION : (i) The null hypothesis is simple if it can be stated as H : for some ∈ Θ . That is, Θ (a single point). (ii) The null hypothesis is composite if Θ consists of two or more elements. 11 ∙ The same definitions hold for the alternative. The alternative is almost always composite. In rare cases a test is developed against a simple alternative. ∙ We choose the null hypothesis to be the one for which we should require “substantial” evidence before we overturn it. So, in evaluating a job training programming where X is the change in annual earnings and E X , the null hypothesis is that the program was ineffective: H : ≤ 0 (composite) or H : 0 (simple) 12 ∙ In the election example, the null hypothesis is that the official results are correct, and then we require substantial evidence to overturn those results. ∙ The null hypothesis plays the role of the defendent in a legal system that presumes innocence unless guilt can be established beyond a “reasonable doubt.” 13 3 . Type I and Type II Error ∙ We will come up with rules, based on observing a sample of data, that lead us to reject the null hypothesis in favor of the alternative, or fail to reject. ∙ In hypothesis testing, there are two mistakes we can make. A Type I error is when we reject the null hypothesis when it is true. This is considered the more serious error and traditional testing approaches make the chance of a Type I error “small.” 14 ∙ A Type II error occurs when we fail to reject H when it is in fact false. As we will see, it is always possible to make such an error if we want to limit the possibility of a type I error....
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 Fall '12
 Jeff
 Normal Distribution, Null hypothesis, Statistical hypothesis testing, alternative hypotheses

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