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# tests of hypotheses:

The ANOVA procedure tests these hypotheses:

H0: μ1 = μ2 = … = μr, all the means are the same

H1: two or more means are different from the others

Let’s test these hypotheses at the α = 0.05 significance level.

You might wonder why you do analysis of variance to test means, but this actually makes sense. The question, remember, is whether the observed difference in means is too large to be the result of random selection. How do you decide whether the difference is too large? You look at the absolute difference of means between treatments (samples), but you also consider the variability within each treatment. Intuitively, if the difference betweentreatments is a lot bigger than the difference within treatments, you conclude that it’s not due to random chance and there is a real effect.

And this is just how ANOVA works: comparing the variation between groups to the variation within groups. Hence, analysis of variance.

### Requirements for ANOVA

1. You need r simple random samples for the r treatments, and they need to be independent samples. The sample sizes need not be the same, though it’s best if they’re not very different.
2. The underlying populations should be normally distributed. However, the ANOVA test is robust and moderate departures from normality aren’t a problem, especially if sample sizes are large and equal or nearly equal (Kuzma & Bohnenblust 2005 page 180).
3. The samples should all have the same standard deviation, theoretically. Because the ANOVA test is robust, Sullivan 2011 page C–21 (on CD) says it’s good enough if the largest standard deviation is less than double the smallest standard deviation.Miller 1986 (pages 90–91) is more cautious. When sample sizes are equal but standard deviations are not, the actual p-value will be slightly larger than what you find in the tables. But when sample sizes are unequal, and the smaller samples have the larger standard deviations, the actual p-value “can increase dramatically above” what the tables say, even “without too much disparity” in the standard deviations. “Falsely reporting significant results when the small samples have the larger variances is a serious worry. The lesson to be learned is to balance the experiment [equal sample sizes] if at all possible.