Analysis of Variance

# Analysis of Variance - Analysis of Variance Introduction We have just studied hypothesis tests about the differences of means of two independent

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Analysis of Variance -- Introduction We have just studied hypothesis tests about the differences of means of two independent populations. For example, 0 1 2 a 1 2 H : H : μ = μ μ μ Our decision about the null hypothesis is based on knowing about the probability distribution of a sample statistic -- a single number -- computed using 1 2 x x - , the difference of the means of random samples from the two populations. Recall that in the case of small sample size, it is important to be able to assume that the sampled populations are normal. nBut we may wish to compare the means of more than two populations. Clearly, now things are more complicated -- a single such difference cannot suffice. But we still do base the decision on the value of a test statistic -- just not simply z or t. Since ANOVA involves somewhat forbidding nomenclature, we'll first try to get the basic idea without all the jargon. For example, we might be interested in the following complementary hypotheses: 0 1 2 3 a H : H : These 3 means are not all the same. μ = μ = μ Notice that the null hypothesis is not one equation but three, and that the alternative hypothesis includes more than one possibility. How do we proceed? nFirst of all, we make some not very surprising assumptions. By now you are well aware that in this crapshoot called statistics, we usually have to do this to say anything significant at all. 1. All 3 populations are assumed to be normal. Not shocking -- what does "normal" mean? 2. All 3 populations have the same variance, 2 2 2 1 2 3 σ = σ = σ , so just call them all 2 σ . Recall that it is not shocking that different populations can have different means, but the same amount of variability. 3. All the observations of the variable involved are independent. And for the purposes of this introduction, we make the simplifying assumption that the 3 samples taken from the 3 populations are all the same size n. We will abandon this assumption when we develop the full ANOVA terminology. nNow, to start the analysis, suppose that the null hypothesis is true. But then 1 2 3 μ = μ = μ and 2 2 2 1 2 3 σ = σ = σ actually means that there is really just one population with mean μ and variance 2 σ . The 3 populations we are interested in can just be thought of as manifestations of this one population. In this case, the 3 sample means, 1 2 3 x ,x ,x , should be close together, i.e., have small variability. In fact we could say that small variability of these means favors the null hypothesis, while large variability contradicts it. Of course, we need more precision than just this.

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## This note was uploaded on 05/07/2008 for the course QMBE 2786 taught by Professor Easly during the Spring '08 term at University of New Orleans.

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Analysis of Variance - Analysis of Variance Introduction We have just studied hypothesis tests about the differences of means of two independent

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