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Lecture8.chapt12

Lecture8.chapt12 - Lecture 8 Chapter 12 One-Way Analysis of...

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Lecture 8, Chapter 12 One-Way Analysis of Variance: It is always interesting to compare data (salaries of women and men, responses to different treatments in a clinical trial). We have learned to display comparisons with back to back stemplots and side-by-side boxplots. To answer the question “is the difference statistically significant?”, we have used two sample t procedures. 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 environmentalist 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. As with the t- test, the F statistic compares the variation among the means of several groups with the variation within the groups. In the ANOVA test, a 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. Lecture 8, Chapter 12 Page 1
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Assumptions of the ANOVA: 1. The data is normally distributed. 2. The population standard deviations are equal. The one-way ANOVA model is: x i ik ik μ ε = + for i = 1,….., I and k = 1,….., i n . The ik ε are assumed to be from a (0, ) N σ distribution. The parameters of the model are the population means 1 2 , , ...... , I μ μ μ and the common standard deviation σ. Note: I = the number of groups. N = the total sample size. i n = the sample size for group i. Example: The strength of concrete depends upon the formula used to prepare it. One study compared five different mixtures. Six batches of each mixture were prepared, and the strength of the concrete made from each batch was measured. What is the response variable? Give the values for I, the i n , and N.
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