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Unformatted text preview: Hypotheses and test procedures Tests for population means Pvalues Hypothesis testing Artin Armagan Sta 113 Chapter 8 Devore November 2, 2009 Artin Armagan Hypothesis testing Hypotheses and test procedures Tests for population means Pvalues Table of contents 1 Hypotheses and test procedures 2 Tests for population means Normal known variance Normal unknown variance Large sample tests or CLT to the rescue 3 Pvalues Artin Armagan Hypothesis testing Hypotheses and test procedures Tests for population means Pvalues Hypothesis A hypothesis is an assertion about a parameter. The population mean μ = 2. The proportion of successes p = . 3. The population mean μ > 2. The difference between two population means μ 1 μ 2 = 0. The difference between two population means μ 1 μ 2 > 4. Artin Armagan Hypothesis testing Hypotheses and test procedures Tests for population means Pvalues The null and the alternate Typically there are two types of hypotheses in the hypothesis testing framework: 1 Null hypothesis: what one believes prior to the test. For example H : μ 1 = μ 2 , H : μ 1 = 10 , H : μ 1 μ 2 = 5 . 2 Alternative hypothesis: a hypothesis contradictory to the null H A : μ 1 6 = μ 2 H A : μ 1 > μ 2 . Artin Armagan Hypothesis testing Hypotheses and test procedures Tests for population means Pvalues Two main uses of hypothesis testing The null hypothesis is the one people focus on in classical hypothesis testing and there are two logical constructions of hypothesis testing. Confirming a theory: In physics one may believe that force is equal to mass times acceleration F = ma one can measure for objects of various masses the force and acceleration and use as a null hypothesis and altervative hypothesis H : F ma = 0 H A : F ma 6 = 0 . This confirmation of a hypothesis is quite common and in this case one typically wishes that the null is not rejected or there is strong evidence for the null. Repudiating a control: A more common use of hypothesis testing is to set the null up as a control and show that there is evidence to reject it. An example is that rocket fuel makes cars run faster. In this case one can take speeds of cars spiked with rocket fuel and compute the population mean, μ 1 , and compare this to the population mean of cars without rocket fuel, μ 2 , and use the following null and alternative hypothesis H : μ 1 = μ 2 H A : μ 1 > μ 2 . In this case we would like to reject the null or that there is strong eveidence against the null Artin Armagan Hypothesis testing Hypotheses and test procedures Tests for population means Pvalues Two types of hypotheses Two types of hypotheses are simple and composite: Simple: A simple hypothesis is one where the distribution under the hypothesis is fully specified. For example X 1 ,..., Xn iid ∼ No( μ,σ 2 = 5) and the hypothesis is H : μ = 10 ....
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 Fall '08
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 PValues, Null hypothesis, Hypothesis testing, Statistical hypothesis testing, Artin Armagan

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