# Chap11 - Chapter 11 Hypothesis Testing 1 I Inference in...

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1 Chapter 11 Hypothesis Testing

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2 Correct acceptance of H 0 pr(I= H 0 | T=H 0 ) = (1 – α ) Type I Error pr(I= H 1 | T=H 0 ) = α [aka size ] Type II Error pr(I= H 0 | 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 0 : Null Hypothesis H 0 : 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

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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

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6 We usually form only two hypotheses The null hypothesis: H 0 The alternative hypothesis: H 1 In this class, H 0 is always a very specific hypothesis about the precise value of some parameter E.g., we might use: H 0 : 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 0 0 μ μ - = - x

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8 Accept H 0 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 0 ) 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

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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 0 is true) How much of a risk are we willing to take of incorrectly inferring that H 1 is true when H 0 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 0 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

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12 Only α of H 0 ’s distribution (1 – α ) of H 0 ’s distribution μ H0 Accept H 0 Accept H 1 Critical Bound: 1.64, if: 2200 α = .05 n= 1, and H 0 = 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 . | 0 0 = - H q S X pr X H μ ( 29 05 . | 0 = H b X pr

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14 Determining a Critical Bound As with confidence intervals, it is easy to transform our standardization into an
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