The hypotheses of interest in an ANOVA are as follows:

where µ = group mean and k = number of groups. If, however, the one-way ANOVA returns a statistically significant result, we accept the alternative hypothesis (HA), which is that there are at least two group means that are statistically significantly different from each other.

The test statistic for testing H0: μ1 = μ2 = ... =   μk is:

You should decide whether to use the one-tailed or two-tailed probability before you collect your data, of course. A one-tailed probability is more powerful, in the sense of having a lower chance of false negatives, but you should only use a one-tailed probability if you really, truly have a firm prediction about which direction of deviation you would consider interesting. In the chicken example, you might be tempted to use a one-tailed probability, because you're only looking for treatments that decrease the proportion of worthless male chickens. But if you accidentally found a treatment that produced 87% male chickens, would you really publish the result as "The treatment did not cause a significant decrease in the proportion of male chickens"? I hope not. You'd realize that this unexpected result, even though it wasn't what you and your farmer friends wanted, would be very interesting to other people; by leading to discoveries about the fundamental biology of sex-determination in chickens, in might even help you produce more female chickens someday. Any time a deviation in either direction would be interesting, you should use the two-tailed probability. In addition, people are skeptical of one-tailed probabilities, especially if a one-tailed probability is significant and a two-tailed probability would not be significant (as in our chocolate-eating chicken example). Unless you provide a very convincing explanation, people may think you decided to use the one-tailed probability after you saw that the two-tailed probability wasn't quite significant, which would be cheating. It may be easier to always use two-tailed probabilities. For this handbook, I will always use two-tailed probabilities, unless I make it very clear that only one direction of deviation from the null hypothesis would be interesting.

We will be interested in testing the null hypothesis

The probability that was calculated above, 0.030, is the probability of getting 17 or fewer males out of 48. It would be significant, using the conventional PP=0.03 value found by adding the probabilities of getting 17 or fewer males. This is called a one-tailed probability, because you are adding the probabilities in only one tail of the distribution shown in the figure. However, if your null hypothesis is "The proportion of males is 0.5", then your alternative hypothesis is "The proportion of males is different from 0.5." In that case, you should add the probability of getting 17 or fewer females to the probability of getting 17 or fewer males. This is called a two-tailed probability. If you do that with the chicken result, you get P=0.06, which is not quite significant.

The null hypothesis and the alternate hypothesis are types of ..

The significance level you choose should also depend on how likely you think it is that your alternative hypothesis will be true, a prediction that you make before you do the experiment. This is the foundation of Bayesian statistics, as explained below.

Null hypotheses of homogeneity are used to verify that ..

The most familiar one-way anovas are "fixed effect" or "model I" anovas. The different groups are interesting, and you want to know which are different from each other. As an example, you might compare the AAM length of the mussel species Mytilus edulis, Mytilus galloprovincialis, Mytilus trossulus and Mytilus californianus; you'd want to know which had the longest AAM, which was shortest, whether M. edulis was significantly different from M. trossulus, etc.

There are three sets of hypothesis with the two-way ANOVA

The other kind of one-way anova is a "random effect" or "model II" anova. The different groups are random samples from a larger set of groups, and you're not interested in which groups are different from each other. An example would be taking offspring from five random families of M. trossulus and comparing the AAM lengths among the families. You wouldn't care which family had the longest AAM, and whether family A was significantly different from family B; they're just random families sampled from a much larger possible number of families. Instead, you'd be interested in how the variation among families compared to the variation within families; in other words, you'd want to partition the variance.

Null hypothesis for a one-way anova - SlideShare

There are also tests to compare different sets of groups; for example, you could compare the two Oregon samples (Newport and Tillamook) to the two samples from further north in the Pacific (Magadan and Petersburg). The Scheffé test is probably the most common. The problem with these tests is that with a moderate number of groups, the number of possible comparisons becomes so large that the P values required for significance become ridiculously small.