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# Intuition when we have more precision we need much

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Unformatted text preview: ﬀerence might be too little and too large, ¯ ¯ depending on how precisely we measure x . ¯ As SE decreases (maybe we have a bigger sample), test statistic will increase magnitude. The eﬀect of this is generally an increase in the chances of a null being rejected. Here is an example: Suppose we have a left-tailed test. Test statistic is z = −1, so the null is not rejected with α = 0.05. If, for some reason, SE(x ) were half of what it was before, the new would have been ¯ z = −2 and the null would be rejected. Intuition: when we have more precision, we need much less discrepancy between x and µ0 to ¯ reject the null. Notice: With a left-tailed test, if z &gt; 0, a decrease in the SE cannot change the result. If z &gt; 0, a left-tailed test will always result in “fail to reject”, regardless of the size of z . Conﬁrm this for exercise. Think about the right-tailed as well. Utku Suleymanoglu (UMich) Hypothesis Testing 20 / 39 What determines test results? Signifance level: 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. As α decreases, rejection region gets smaller. 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 asking a lot of 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. So there is a trade-oﬀ between Type I and Type II error probabilities. This is why we don’t set α = 0.000001. Generally speaking an α = 0.05 is norm. If one needs to be more conservative about rejecting the null (asking for a little bit more evidence), α = 0.01 is the choice. An α = 0.1 is also ok. There is no clear-cut reason...
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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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