Chap11 - 1 Chapter 11 Hypothesis Testing 2 Correct...

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Unformatted text preview: 1 Chapter 11 Hypothesis Testing 2 Correct acceptance of H pr(I= H | T=H ) = (1 – α ) Type I Error pr(I= H 1 | T=H ) = α [aka size ] Type II Error pr(I= H | T=H 1 ) = β Correct acceptance of H 1 pr(I= H 1 | T=H 1 ) = (1 – β ) [aka power ] I : Inference in favor of: H 1 : Alternative Hypothesis H : Null Hypothesis H : Null Hypothesis T : The Truth of the matter: H 1 : Alternative Hypothesis 3 • Before diving into the “nuts and bolts” details, let’s consider the “big picture” of statistical testing – Example from p. 394 4 Components of a statistical test: – Background assumptions – Hypotheses – Test statistic – Determine the critical bound(s) – Interpret/Report the results. 5 1. For the results of a test for the mean of a population to have any meaning, the drawings from the sample must be: – Independent – Identically distributed 2. The underlying population must be approximately normal • This assumption is frequently violated • But when we use and a fairly large n (e.g., n ≥ 20), the CLT usually makes this true. Background Assumptions X 6 • We usually form only two hypotheses – The null hypothesis: H – The alternative hypothesis: H 1 • In this class, H is always a very specific hypothesis about the precise value of some parameter – E.g., we might use: H : μ = 120 Hypotheses 7 Test Statistic • For a test of the mean, we will use , often in its standardized form as a t- statistic – comes from an unbiased and consistent point estimator x n s x s x h x h μ μ- =- x 8 ← → Accept H Accept H 1 Critical Bound: 9 • A critical bound acts as a threshold: – If our data results in a number less than the critical bound, we accept the hypothesis with the smaller mean (in this case, H ) – If our data results in a number greater than or equal to the critical bound, we accept the hypothesis with the larger mean (in this case, H 1 ) Determining a Critical Bound 10 Determining a Critical Bound • We will adopt the common practice of using α to determine our critical boundary α = pr(decide in favor of H 1 | H is true) – How much of a risk are we willing to take of incorrectly inferring that H 1 is true when H actually is? – No matter what we do, we will have to take some chance (i.e., α > 0) 11 Determining a Critical Bound • A common choice of α is .05. – This choice means that we will set our critical bound so that only 5% of the distribution of H is beyond it: • This can be determined using quantiles: • @qnorm(.95) = 1.64 • @qtdist(.95, 19) = 1.73 • @qtdist(.95, @obs(x)-1) = 1.73 95 . ) 64 . 1 ( = Φ df = n – 1 12 Only α of H ’s distribution (1 – α ) of H ’s distribution μ H0 ← → Accept H Accept H 1 Critical Bound: 1.64, if: 2200 α = .05 • n= 1, and • H = N(0, 1) 13 Determining a Critical Bound • Intuitively, we want to find the number b such that: • Similar to confidence intervals, we use α to determine a quantile q which then determines the boundary b 05 . | = ≥- H q S X pr X H μ ( 29 05 . | = ≥ H b X pr 14...
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This note was uploaded on 10/01/2010 for the course ELEC 6111 taught by Professor Brown during the Spring '10 term at E. Illinois.

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Chap11 - 1 Chapter 11 Hypothesis Testing 2 Correct...

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