intro.hypothesis.testing

intro.hypothesis.testing - Introduction to Hypothesis...

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Introduction to Hypothesis Testing 1 1 Introductory Examples Example 1. It has been assumed for some time now that a particular bottle-filling process produces bottles with fill amounts that follow a Normal distribution with mean 16 oz. Twenty- five bottles are randomly sampled from a week’s worth of production. The sample average fill amount was 15.7 oz. and the sample standard deviation was 0.5 oz. Do these data give evidence against the hypothesis that the mean fill amount is 16 oz.? Example 2. It is believed that 70% of Americans are in favor of increasing corporate taxes. Twenty Americans are randomly sampled, but only 11 are in favor of increasing corporate taxes. Do these data give evidence against the hypothesis that the population proportion is 70%? 2 Hypotheses, Data, and Statistical Tests It is assumed that X f ( ·| θ ) Ω. Test the statistical hypothesis 2 H 0 : θ ω 0 Ω . That is, use data to try to falsify H 0 . Data: x X IID X f ( ·| θ ) , θ Ω. Test ( C ) : Reject H 0 if and only if ( X C ) is observed, i.e. if and only if x C , where critical region C satisfies max θ ω 0 P ( X C | θ ) = α, and α is small, e.g. α = 0 . 05. 3 Logic of Statistical Tests of Hypotheses We have that H 0 ( X 6∈ C ) , with probability at least 1 - α . (See scratch work in footnote 3 ) In words, if H 0 were true, you would likely (or expect to) see ( X 6∈ C ). If you observe the unexpected ( X C ) then conclude that H 0 is false, but acknowledge that this conclusion might be incorrect. 1 August 19, 2008. Do NOT distribute without express permission of author: Joseph B. Lang 2 Definition: A statistical hypothesis is a statement about a probability or population distribution that may or may not be true. 3 For any θ ω 0 , P ( X 6∈ C | θ ) = 1 - P ( X C | θ ) 1 - max θ 0 ω 0 P ( X C | θ 0 ) = 1 - α .
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2 Note that if α = 0 then logicians would recognize this argument as Modus Tollens. Specifically, Modus Tollens Model: H 0 ( X 6∈ C ) , with probability 1. Observation: ( X C ). Conclusion: Conclude that H 0 is false. In statistics, we use Stochastic Modus Tollens Model: H 0 ( X 6∈ C ) , with probability at least 1 - α . Observation: ( X C ). Conclusion: Conclude that H 0 is false, but acknowledge that there is a chance this conclu- sion is incorrect. 4 Error Types, Size, and Power We have two choices in the hypothesis testing setting: we can choose to reject H 0 or we can choose to not reject H 0 . This means there are two types of errors we can make in hypothesis testing. If H 0 were true, we could incorrectly reject it; this is a Type I Error . If H 0 were false, we could incorrectly fail to reject it; this is a Type II Error . Bayesians base their decision to reject or not reject H 0 on posterior probabilities like P ( H 0 is true | X = x ) and P ( H 0 is false | X = x ). In contrast, Frequentists do not try to compute the likelihood that H 0 is or is not true. Instead they are satisfied measuring operating characteristics of a test such as Test ( C ). Bayesians might ask, “What are the chances that my test leads to the correct conclusion?” 4 Frequentists only ask the hypotheticals, “If H 0
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This note was uploaded on 03/11/2010 for the course STAT 154 taught by Professor Josephblang during the Spring '10 term at University of Iowa.

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intro.hypothesis.testing - Introduction to Hypothesis...

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