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28_anova - DEVEMC28_0321500458.qxd 12:26 PM Page 1 CHAPTER...

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28-1 CHAPTER 28 Analysis of Variance Where are we going? In Chapter 24 we compared the mean lifetimes of generic and brand-name batteries. But our supermarket carries four different “name” brands of batteries and two cheaper generic brands. Are all these brands equally good? How can we compare them all? We could run a t -test for each of the 15 head-to-head comparisons, but we’ll learn a better way to compare more than two groups in this chapter. D id you wash your hands with soap before eating? You’ve undoubtedly been asked that question a few times in your life. Mom knows that washing with soap eliminates most of the germs you’ve managed to collect on your hands. Or does it? A student decided to investigate just how effective washing with soap is in eliminating bacteria. To do this she tested four different methods—washing with water only, washing with regular soap, washing with antibacterial soap (ABS), and spraying hands with antibacterial spray (AS) (containing 65% ethanol as an active ingredient). Her experiment con- sisted of one experimental factor, the washing Method, at four levels. She suspected that the number of bacteria on her hands before washing might vary considerably from day to day. To help even out the effects of those changes, she generated random numbers to determine the order of the four treatments. Each morning she washed her hands according to the treatment randomly chosen. Then she placed her right hand on a sterile media plate designed to encourage bacteria growth. She incubated each plate for 2 days at 36°C, after which she counted the bacteria colonies. She replicated this procedure 8 times for each of the four treatments. A side-by-side boxplot of the numbers of colonies seems to show some dif- ferences among the treatments: WHO Hand washings by four different methods, assigned randomly and replicated 8 times each WHAT Number of bacteria colonies HOW Sterile media plates incubated at 36°C for 2 days
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28-2 CHAPTER 28 Analysis of Variance 1 The alternative hypothesis is that “the means are not all equal.” Be careful not to confuse that with “all the means are different.” With 11 groups we could have 10 means equal to each other and 1 different. The null hypothesis would still be false. 2 You might think of testing all pairs, but that method generates too many Type I errors. We’ll see more about this later in the chapter. When we first looked at a quantitative variable measured for each of several groups in Chapter 5, we displayed the data this way with side-byside boxplots. And when we compared the boxes, we asked whether the centers seemed to dif- fer, using the spreads of the boxes to judge the size of the differences. Now we want to make this more formal by testing a hypothesis. We’ll make the same kind of comparison, comparing the variability among the means with the spreads of the boxes. It looks like the alcohol spray has lower bacteria counts, but as always, we’re skeptical. Could it be that the four methods really have the same mean
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