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MODULE 5 THE WHOLE MODULE

MODULE 5 THE WHOLE MODULE - 5 Hypothesis Testing Dave...

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5. Hypothesis Testing Dave Goldsman Georgia Institute of Technology, Atlanta, GA, USA 8/21/10 Goldsman 8/21/10 1 / 64
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Outline 1 Introduction to Hypothesis Testing 2 Normal Mean Tests (variance known) Simple Hypothesis Test Test Design Two-Sample Normal Mean Tests 3 Normal Mean Tests (variance unknown) Simple Hypothesis Test Two-Sample Normal Mean Tests Pooled t -Test Approximate t -Test Paired t -Test 4 Other Hypothesis Tests Normal Variance Test Two-Sample Test for Equal Variances Bernoulli Proportion Test Goldsman 8/21/10 2 / 64
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Introduction to Hypothesis Testing Introduction to Hypothesis Testing General Approach 1. State a hypothesis. 2. Select a test statistic (to test whether or not the hypothesis is true). 3. Evaluate (calculate) the test statistic. 4. Interpret results — does the test statistic allow you to reject your hypothesis? Details follow. . . Goldsman 8/21/10 3 / 64
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Introduction to Hypothesis Testing 1. Hypotheses are simply statements or claims about parameter values. You perform a hypothesis test to prove or disprove the claim. Set up a null hypothesis ( H 0 ) and an alternative hypothesis ( H 1 ) to cover the entire parameter space. The null hyp sort of represents the “status quo”. Example: We claim that the mean weight of a filled package of chicken is μ 0 ounces. (We specify μ 0 .) H 0 : μ = μ 0 H 1 : μ negationslash = μ 0 This is a two-sided test . We’ll reject the claim if ˆ μ (an estimator of μ ) is “too high” or “too small”. Goldsman 8/21/10 4 / 64
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Introduction to Hypothesis Testing Example: We claim that a certain brand of tires lasts for at least a mean of μ 0 miles. (We specify μ 0 .) H 0 : μ μ 0 H 1 : μ < μ 0 This is a one-sided test . We’ll reject the claim if ˆ μ is “too small”. Example: We claim that emissions from a certain type of car do not exceed a mean of μ 0 ppm. (We specify μ 0 .) H 0 : μ μ 0 H 1 : μ > μ 0 This is a one-sided test . We’ll reject the claim if ˆ μ is “too large”. Goldsman 8/21/10 5 / 64
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Introduction to Hypothesis Testing Idea: H 0 is the old, conservative “status quo”. H 1 is the new, radical hypothesis. Although you may not be toooo sure about the truth of H 0 , you won’t reject it in favor of H 1 unless you see substantial evidence in support of H 1 . “Innocent until proven guilty.” If you get substantial evidence supporting H 1 , you’ll decide to reject H 0 . Otherwise, you “fail to reject” H 0 . (This roughly means that you grudgingly accept H 0 .) Goldsman 8/21/10 6 / 64
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Introduction to Hypothesis Testing 2. Select a test statistic (to test if H 0 is true). For instance, we could compare an estimator ˆ μ with μ 0 . The comparison is accomplished using a known sampling distribution (aka “test statistic”), e.g., z obs = ¯ X μ 0 σ/ n (if σ 2 is known) or t obs = ¯ X μ 0 S/ n (if σ 2 is unknown) Lots more details later.
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