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Unformatted text preview: § 5 Hypothesis Testing § 5.1 Introduction 5.1.1 A parametric hypothesis H is an assertion about the unknown parameter θ in a parametric model. Example § 5.1.1 (i) Man’s height ∼ N ( μ M ,σ 2 ) Woman’s height ∼ N ( μ W ,σ 2 ) Parameter θ = ( μ M ,μ W ,σ ) H : μ M > μ W (i.e. men are on average taller than women); (ii) P (new medical treatment cures disease) = p P (old medical treatment cures disease) = q θ = ( p,q ) H : p > q (i.e. new treatment is more effective); (iii) stock market undergoes daily change X ∼ N (0 ,σ 2 ) when a political event occurs θ = σ H : σ > 1 (i.e. political event destabilizes the stock market); (iv) A chemical fertilizer is cheap but is potentially harmful to consumers. Outcome of a safety test = X ∼ N ( θ, 1) Acceptable safety standard requires θ < 1 H : θ < 1 (i.e. chemical is safe). 5.1.2 Formulation of hypotheses is subjective, depends on the practical situation as well as the attitude of the person who hypothesizes. 5.1.3 A statistical test of H is a data-based procedure for deciding whether the observed data contain evidence against H . 5.1.4 Sample data are random and may be misleading. Conclusions from hypothesis tests may not always produce the correct answer. Good tests have a bigger chance to give correct answers. 45 § 5.2 Hypotheses 5.2.1 Definition. X (data) ∼ p (probability function) A hypothesis test involves two hypotheses about p : 1. null hypothesis H — • conservative, i.e. not to be rejected unless evidence is strong, • usually takes the form of “no difference”, 2. alternative hypothesis H 1 — • specifies the kind of departure from H of interest to the tester, • preferred to H only if such departure is detected with strong evidence. Based on X , the test helps us assess the “relative” plausibility of H and H 1 . 5.2.2 Example § 5.1.1 (cont’d) (i) H : μ M = μ W (no difference) vs H 1 : μ M > μ W (men being taller). However, if one is only interested in whether men and women have different heights on average, the appropriate hypotheses should be H : μ M = μ W vs H 1 : μ M 6 = μ W . (ii) To protect patients, one might consider H : p = q (no difference) vs H 1 : p > q (new treatment more effective) , i.e. we would not recommend use of new treatment unless strong evidence exists in support of its effectiveness. (iii) H : σ ≤ 1 (small destabilizing effect) vs H 1 : σ > 1 (big destabilizing effect); (iv) To protect the farmer, one may wish to test H : θ < 1 vs H 1 : θ ≥ 1 , so that the cheap fertilizer will be abandoned only if there is strong evidence against its safety. To protect consumers, one may do the opposite, i.e. test H : θ ≥ 1 vs H 1 : θ < 1 , so that the cheap fertilizer will NOT be adopted unless there is strong evidence confirming its safety....
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This note was uploaded on 05/04/2011 for the course STAT 1302 taught by Professor Smslee during the Spring '10 term at HKU.
- Spring '10