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# Lecture7_print - Chapter 12: One-way Analysis of Variance...

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Chapter 12: One-way Analysis of Variance (ANOVA) Readings: Chapter 12 1 Introduction The One-Way ANOVA is used when you have one categorical and one quantitative variable and you want to compare the means of the quantitative variable across diﬀerent categories. The lifetime of D batteries from two brands are compared for their lifetime in a ﬂashlight. 50 ﬂashlights are loaded with Brand 1 batteries and 45 with Brand 2 batteries. The lifetime (in hours) of each battery is then recorded. In order to examine whether the drug Paxil aﬀect serotonin levels in healthy young men, a random sample of 15 men are split into 3 groups of 5. They receive 0, 20 and 40mg of the drug per day for a week and then their serotonin levels are measured. If the categorical variable has two groups (Brand 1 and Brand 2 batteries), use the two- sample t procedures in Chapter 7. If the categorical variable has more than 2 groups (Paxil doses=0, 20, and 40mg), then use the one-way ANOVA . ANOVA (ANalysis Of VAriance): used for comparing several means. It is a technique that generalizes the two-sample t procedure which compares two means to a situation with more than two sample means. Assumptions of the ANOVA: Each distribution have a normal distribution Samples are independent and random The population standard deviations are equal 2 One-way ANOVA Goal of ANOVA: Is there at least one mean that is statistically signiﬁcantly diﬀerent from the others? Notation: I = number of groups n i = then number of observations in the i th group N = n 1 + n 2 + ... + n I = total number of observations μ i = the population mean of the i th group ¯ x i = the sample mean of the i th group σ = the common population standard deviation 1

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Standard Deviations We assume that in the ANOVA models, the population standard deviations are all equal. The oﬃcial test is quite complicated, so we use the following rule of thumb: If the largest standard deviation is less than twice the smallest standard devia- tion, we can use methods based on the assumption of equal standard deviations If we assume all the standard deviations are equal, each s is an estimate of σ . We combine these into a Pooled Estimator of σ : s p = s ( n 1 - 1) s 2 1 + ( n 2 - 1) s 2 2 + ... + ( n I - 1) s 2 I ( n 1 - 1) + ( n 2 - 1) + ... + ( n I - 1) Example 1 : An experiment was run to compare four groups. The sample sizes were 20, 220, 18, and 15, and the corresponding sample standard deviations were 62, 40, 52, and 48. a. Is it reasonable to use the assumption of equal standard deviations when we analyze
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## Lecture7_print - Chapter 12: One-way Analysis of Variance...

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