2314 Review Part III - Review p. 20 Hypothesis Tests. We...

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Review p. 20 Hypothesis Tests . We can test hypotheses about any of the population parameters, but for this discussion we use the example of . .. 1. Large Sample Tests about the unknown ( and generally unknowable) mean μ of a population: A claim, called the null hypothesis, is made about μ : 0 0 H : μ = μ . A sample is taken and x is computed. Since the expected value of x is μ , if this sample statistic is "close to" 0 μ , then the null hypothesis is deemed reasonable and not rejected. If x is "far from" the claimed value 0 μ , then 0 H is rejected as unreasonable. "Close to" and "far from" are not precise enough. The decision is, of course, based on probabilities. Since the sample size is large we have a probability model. By the Central Limit Theorem, we know that the distribution of x is normal ! Through this knowledge we can assign probabilities to the 2 kinds of possible error in this decision procedure. The type I error is that of rejecting a true null hypothesis. We call the probability of this error α. The type II error is that of not rejecting a false null hypothesis. Its probability is β . A summary of the decision process: True State of Things 0 H true 0 H false Decision Reject 0 H Type I error No error made Don't Reject 0 H No error made Type II error e Formulating the hypotheses . If the null hypothesis is false, then some alternative, the complementary possibility, must be true. This is called a H , the alternative hypothesis. Thus the 3 possibilities for the hypothesis pairs are: 0 0 0 0 0 0 a 0 a 0 a 0 H : H : H : H : H : H : μ = μ μ μ μ μ μ μ μ < μ μ μ (2-tailed test) ( 1-tailed tests) The question is how, in each case, we should decide which of the 2 possibilities is to be called the null hypothesis and which the alternative. There are a number of considerations. 1. The possibility 0 μ = μ is always included in the null hypothesis. α, the probability of the Type I error, is the easier of the 2 to get a handle on. Analysis of β involves considering a multitude of possible alternative values for μ . In formulating the test procedure we can control α, so the null hypothesis is chosen so that the Type I error is the more serious ( more costly ) of the two. 2. In a research situation, the hypotheses are formulated so that rejection of the null hypothesis supports the research ( new information) conclusion. The null hypothesis is identified with the status quo , the old way of thinking. Since accepting the new information could be costly or disruptive, we reject the null hypothesis in favor of the alternative only if the sample evidence overwhelming suggests that we should.
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Review p. 21 Thus we set α to be
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2314 Review Part III - Review p. 20 Hypothesis Tests. We...

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