lecture16 - ISYE 2028 A and B Spring 2009 Lecture 16 Dr...

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ISYE 2028 A and B Spring 2009 Lecture 16 Dr. Kobi Abayomi April 13, 2009 1 Introduction - The ”simplest” model - The ANOVA model In studying methods for the analysis of quantitative data, we first focused on problems involving a single sample of numbers and then turned to a comparative analysis of two different such samples. In one sample problems, the data consisted of observations of individuals randomly selected from a single population. In two sample problems, either the two samples were drawn from two different populations, or else two different treatments were applied to elements selected from a single population. The analysis of variance or ANOVA refers to a collection of procedures for the analysis of responses from experimental units. The simplest ANOVA problem is referred to as a single factor or one way ANOVA and involves analysis either of data sampled from two or more populations or data in which two or more treatments have been used. As such, the ANOVA setup is a generalization of the two sample t-test. The characteristic that differentiates the treatments or populations from one another is called the factor and the different treatments are referred to as the levels of the factor. Let’s begin with an example. .. 2 Single Factor or One Way ANOVA 2.1 Setup and Notation Briefly, say a farmer wants to investigate if flower production differs across gardens. There are three gardens: A , B , and C . We are given 10 weeks of data; the number of flowers grown per week per garden. As always we introduce some notation. Let: j An index for treatments or populations being compared. K The number of treatments in total. Here K = 3. 1
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i An index for the observations. n j The number of observations in each treatment. μ j The mean of population or treatment j. Here i = 1 is the garden A , i = 2 is garden B , i = 3 is the garden B . Of course x j is the sample mean of the jth treatment or strategy. We seek to test for a difference in gardens. Our null hypothesis is, then, that there is no difference in gardens vs. an alternative that there is at least one difference between gardens. In notation. ..: H o : μ 1 = μ 2 = μ 3 H a : At least two means differ. Now we need, of course, a test statistic - or a function of the observed values that we will link to some probability distribution. Let’s introduce some more notation. .. X i,j = the random variable that denotes the ith observation on the jth treatment. What is X 2 , 2 ? x i,j = the observed value of X i,j when the experiment is performed or the data is recorded. The individual treatment means, that is the mean across treatments for each observation are calculated. .
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lecture16 - ISYE 2028 A and B Spring 2009 Lecture 16 Dr...

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