ASSIN UMP - Chapter 9 Testing Hypothesis and Assesing...

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Chapter 9 Testing Hypothesis and Assesing Goodness of Fit 9.1-9.3 The Neyman-Pearson Paradigm and Neyman Pearson Lemma We begin by reviewing some basic terms in hypothesis testing. Let H 0 denote the null hypothesis to be tested against the alternative hypothesis H A . Definition : Let C be the subset of the sample space which leads to rejection of the hypothesis under consideration. C is called the critical region or rejection region of the test. Based on our analysis of the data and resulting conclusion, there are two types of errors that can be made: type I error is where the null hypothesis is rejected when it is, in fact, true. This is usually denoted by α and called the significance level of the test. type II error is where the null hypothesis is accepted (or not rejected) when it is, in fact, false.
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The probability tha H 0 is rejected when it is false is called the power of the test and is denoted by 1 - β . We will first be concerned with testing a simple null hypothesis H 0 : θ = θ against a simple alternative hypothesis H A : θ = θ . Definition : Let C denote a subset of the sample space. Then C is called a best critical region of size α for testing H 0 : θ = θ against H A : θ = θ if, for every subset A of the sample space for which P [ A : H 0 ] = α , the following are true (a) P [ C ; H 0 ] = α and (b) P [ C ; H 1 ] P [ A ; H 1 ].
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Example 1 Consider a random variable X that has a binomial distribution with n = 5 and p = θ . We wish to test H 0 : θ = 1 / 2 versus H A : θ = 3 / 4, at significance level α = 1 / 32.
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  • Fall '15
  • Rogers
  • Science, Null hypothesis, Hypothesis testing, Statistical hypothesis testing, Statistical power, θ, critical region

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