Hypothesistesting-2 - MAT 2379 Introduction to...

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MAT 2379 - Introduction to Biostatistics Hypothesis Testing Professor: Termeh Kousha Summer 2015 1
1 Hypothesis testing, type I and type II er- rors Hypothesis testing is a procedure that leads us to decide if experimental data supports a hypothesis concerning population(s) parameter(s). We will con- sider hypotheses concerning a population mean μ or a population proportion p . Stating the Hypotheses: Often the researcher would to verify a change in the unknown parameter under new experimental conditions. For exam- ple, a manufacturer of a new fiberglass tire claims that the mean life of the new tires are greater than the mean life of tires using the old manufacturing process. The previous mean life was 65 , 000 km. Let μ denote the mean life of the new tires. The no change hypothesis (that we will call the null hypothesis ) is H 0 : μ = 65 , 000 and the claim or research hypothesis (that we will call the alternative hypothesis ) is H 1 : μ > 65 , 000. We want to test H 0 : μ = 65 , 000 against H 1 : μ > 65 , 000 . Now we consider an example involving a proportion. Suppose that we would like to test the hypothesis that the proportion of defective items pro- duced at a particular plant is p = 2%. Then, we would test H 0 : p = 0 . 02 against H 1 : p 6 = 0 . 02 . 2
Null Hypothesis: The null hypothesis will always be a simple state- ment concerning the unknown parameter θ . That is, it is a statement of the form θ = θ 0 , where θ 0 is some real number. For example, H 0 : μ = 65 , 000 or H 0 : p = 0 . 02. The value of the parameter in the null hypothesis will be the boundary value of the parameter from the alternative hypothesis. Alternative Hypothesis: The alternative hypothesis will be a composite statement concerning θ . It is often the research hypothesis, i.e. the hypothe- sis that we would like to support with the data. We will consider three types of alternatives: ( θ is the unknown parameter and θ 0 is some real number) H 1 : θ < θ 0 is a left-sided alternative; H 1 : θ > θ 0 is a right-sided alternative; H 1 : θ 6 = θ 0 is a two-sided alternative . Definitions: A test statistic is a statistic that is used to test hypotheses. The critical region of the test statistic is a set of possible values of the test statistic such that if the observed of the test statistic falls in the critical region we will reject H 0 and accept H 1 . 3
Type I and II errors If we reject H 0 when H 0 is true, we say that we have committed an error of type I and α = P (type I error) = P ( reject H 0 when H 0 is true) If the observed value of the test statistic does not fall in the critical region, then we fail to reject H 0 . If we fail to reject H 0 when H 0 is false, then we say that we have committed an error of type II and β ( θ 1 ) = P (type II error) = P ( fail to reject H 0 when θ = θ 1 H 1 ) 4

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