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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 pvalue
approach. You compare your calculated pvalue 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 pvalue. 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 nitpicking 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
clearcut reason to choose one over the others. But we don’t use α = 0.2 or α = 0.8.
The nice thing about providing pvalues 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 lefttailed test. We can calculate the test statistic: z = 2.0.−1 = 5. That is a
3
pretty big z . (If this was a righttailed test you would reject the null hypothesis.) But this is a lefttailed test. The pvalue 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 lefttailed 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.
 Spring '08
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