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Chapter12ANOVA

# Chapter12ANOVA - Chapter 12 ONE-WAY ANALYSIS OF VARIANCE We...

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Chapter 12 ONE-WAY ANALYSIS OF VARIANCE

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We wish to compare the mean responses for several populations where the levels of a single explanatory variable define the populations. Example: One-Way ANOVA Logic The two figures provide frequency plots for data obtained by taking independent random samples of size 10 from three populations. The three populations each had a normal distribution (normality is one of the assumptions of ANOVA) and the population means were 60, 65, and 70, respectively. So the population means are indeed not all equal. In Scenario I (Figure 11.2): the population standard deviations were all equal to 1.5. In Scenario II (Figure 11.3) the population standard deviations were all equal to 3. Another assumption for ANOVA is that the populations have equality standard deviations . a) Just looking at the frequency plots, which of the two scenarios do you think would provide more evidence that at least one of the population means is different from the others? b) The samples summary statistics, produced using MINTAB, for all six samples are provided next: Scenario I N MEAN MEDIAN STDEV MIN MAX Q1 Q3 Sample 1 10 60.6 60.7 1.6 58.6 62.8 59.1 62.2 Sample 2 10 64.5 64.7 1.1 62.3 66.1 64.0 65.1 Sample 3 10 70.2 70.3 2.2 66.3 73.3 69.0 71.6 Scenario II N MEAN MEDIAN STDEV MIN MAX Q1 Q3 Sample 1 10 60.9 60.3 3.5 55.7 67.3 58.2 63.6 Sample 2 10 66.9 66.9 2.9 60.6 70.3 65.8 69.1 Sample 3 10 69.4 71.1 3.7 61.2 73.2 66.5 72.2 How did the sample means compare to the mean of the population from which the sample was generated? How did the sample standard deviations in Scenario I compare to those in Scenario II?
Solution (a) It is a lot more obvious that the three different samples in Scenario I are from different populations. In Scenario II there is a lot more overlap between the populations. (b) Each sample mean is not exactly equal to the mean of the population from which the sample was generated. The two Sample 1 means of 60.6 and 60.9 were not equal to each other nor equal to the population mean of 60. The samples in Scenario I were generated from populations whose natural variation within each population was smaller compared to the natural variation within each population in Scenario II. For each scenario, even though the population standard deviations were equal, the sample standard deviations were not exactly equal, but they were comparable. So the sample means do vary, from about 60 to about 70, for each scenario. There is a good deal of variation between the sample means. c) What about Scenario III? In which we have frequency plots for three independent samples of size 10 each taken from a normal population with mean of 65 and a standard deviation of 1.5. So in this Scenario III, the population means are indeed all equal. Do the sample means vary? Is there variation within each sample? Does the data in Scenario III provide evidence that the population means are not all equal?

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Although the population means were all equal, there is still a small amount of variation between the sample means. The variation within each sample seems to mask any slight variation there is in the sample means. The data in Scenario III do not provide evidence that the
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Chapter12ANOVA - Chapter 12 ONE-WAY ANALYSIS OF VARIANCE We...

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