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Unformatted text preview: ) Hypothesis Testing 18 / 37 Testing Hypothesis about the Population Mean: σ known Universally: Level of significance: α is our choice as researchers. Think about the p-value approach. You compare your calculated p-value with different α’s. If p = 0.04, you reject the null with α = 0.05, but not if α = 0.01. To reject a null with α = 0.01 or α = 0.001, you need a really small p-value. So as α decreases, you ask for more and more evidence against the null hypothesis to be able to reject it. A small α choice means you have a small probability of rejecting a true hypothesis (Type I error, executing the innocent). But a small α is also nit-picking about the evidence and not rejecting H0 most of the time. So maybe you are also not rejecting some false hypotheses: probability of committing Type II error increases. Generally speaking an α = 0.05 is norm. If one needs to be more conservative about rejecting the null, α = 0.01 is the choice. An α = 0.1 is also ok. There is no clear-cut reason to choose one over the others. But we don’t use α = 0.2 or α = 0.8. The nice thing about providing p-values is that you allow the readers to pick their own α’s and arrive at their own conclusions quickly. Utku Suleymanoglu (UMich) Hypothesis Testing 19 / 37 Testing Hypothesis about the Population Mean: σ known Example Remember we have x = 2.5 and σ = 1.5 and n = 25. Suppose now that you have to test ¯ H0 : µ ≥ 1 with the alternative hypothesis H1 : µ < 1. 5 This is a left-tailed test. We can calculate the test statistic: z = 2.0.−1 = 5. That is a 3 pretty big z . (If this was a right-tailed test you would reject the null hypothesis.) But this is a left-tailed test. The p-value is calculated as the left tail probability. P (Z ≤ 5) ≈ 1. That is bigger than any α imaginable. So you fail to reject the null. (Critical value is negative for left-tailed tests, so a positive test statistic cannot be in the rejection region) Even if the test results seems obvious, we still test it properly. And we still don’t call it “accept”: P (Z ≤ 5) = 1, it is P (Z ≤ 5) ≈ 1 And this is an example why a higher test statistic does not necessarily mean right away that you are more likely to reject the null. Generally, a big test statistic in the direction of the tail of the test is likely to reject the null hypothesis. Utku Suleymanoglu (UMich) Hypothesis Testing 20 / 37 Testing Hypothesis about the Population Mean: σ known σ is known: Two Tailed Tests Now we will discuss a slightly different type of test. The difference is in the null and alternative hyp...
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This note was uploaded on 03/17/2014 for the course ECON 404 taught by Professor Staff during the Spring '08 term at University of Michigan.

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