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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 > 0, a decrease in the SE cannot change the result. If z > 0,
a left-tailed test will always result in “fail to reject”, regardless of the size of z . Conﬁrm this for
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
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.
- Spring '08