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Chapter 12

# Chapter 12 - CHAPTER 12 One-Way Analysis of Variance(ANOVA...

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1 CHAPTER 12 One-Way Analysis of Variance (ANOVA) One-way analysis of variance is used when you want to compare more than two means. It is a technique that generalizes the two-sample t procedure which compares two means. Like the two-sample t test, it is robust and useful. Examples: 1. The presence of harmful insects in farm fields is detected by erecting boards covered with a sticky material and then examining the insects trapped on the boards. To investigate which colors are most attractive to cereal leaf beetles, researchers placed six boards of each of four colors in a field of oats in July. 2. An ecologist is interested in comparing the concentration of the pollutant cadmium in five streams. She collects 50 water specimens for each stream and measures the concentration of cadmium in each specimen. Note: The first example is an experiment with four treatments (the colors) and the second example is an observational study where the concentration of cadmium is compared between the five streams. In both cases we can use ANOVA to compare the mean responses. We will use the F statistic to compare the variation among the means of several groups with the variation within the groups. In the ANOVA test, an SRS from each population is drawn and the data is used to test the null hypothesis that the populations are all equal against the alternative that not all are equal. If we reject the null, we need to perform some further analysis to draw conclusions about which population means differ.

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2 Assumptions of the ANOVA: 1. The data is normally distributed. 2. The population standard deviations are equal. Estimating the population parameters: The unknown parameters in the statistical model for ANOVA are the I population means i μ and the common standard deviation . To estimate i μ we use the sample mean for the i th group: 1 1 i n i ij i j x x n = = To estimate σ the common standard deviation: Our second assumption in the ANOVA model was that our population standard deviations are all equal. An official test is not recommended, so we use the following rule of thumb: If the largest standard deviation is less than twice the smallest standard deviation, we can use methods based on the assumption of equal standard deviations and our results will still be approximately correct. Pooled Estimator of : If we assume all the population standard deviations are equal, each s is an estimate of . We combine these into a Pooled Estimator of . 2 2 2 1 1 2 2 1 2 ( 1) ( 1) ...... ( 1) ( 1) ( 1) ..... ( 1) I I p I s n s n s n s n n n = - + - + + - - + - + + - In the SPSS ANOVA output, p s MSE =
3 The best way to approach a problem which involves comparing more than two groups is as follows: 1. Find an estimate for the mean and standard deviation for each group and plot the means on a graph.

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Chapter 12 - CHAPTER 12 One-Way Analysis of Variance(ANOVA...

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