In the previous unit we learned how to compare means, medians and variances from two populations. Although it may be hard to believe, for many, this knowledge is sufficient for most research needs. But only being able to compare statistics from samples of two populations severely restricts what hypotheses can be addressed and puts limitations on the efficiency and effectiveness of the study designs we can attempt.

In this Unit we will extend the two-population design to the multiple population design. This allows us to address questions of differences among a group of means rather than between two means. We will have the same issues as with the t-test of what kind of pooled variance estimate should be used. We no longer have a simple test statistic, such as the difference between two means, to use but will define a new test statistic that measures among mean variation. Finally we combine all computations related to this among means test and place them in a table we refer to as the "Analysis of Variance" table.

If you conjecture specific relationships among means prior to computing the analysis of variance table and associated overall test of means, the related hypotheses can be tested with linear contrasts. We will not see this come into its full importance until we get to the Units on experimental design. If, instead, our interest in comparison of specific means is the result of a significant analysis of variance test, post hoc or multiple comparison procedures are employed.

We end this section by exploring hypotheses and statistical tests for parameters from populations that yield quantitative random variables. The random variable provides us with a classification of the individual sample unit and the resulting data are summarized in a table of counts for the set of possible categories. Our statistical interest in these data typically lie in comparisons of the probability that an individual will belong to a specific category. These test of proportions are unlike tests for population means and different test statistics are used. Related to this important class of hypotheses are those that explore whether two categorical random variables are independent. Unfortunately in this course we do not spend near enough time exploring some of the modern tools for analyzing categorical data. We will revisit this topic briefly again when we talk about response transformations.

1. To expand testing of population means and variances from two populations to more than two populations.
2. To develop testing of linear contrasts of means.
3. To explore our options for post hoc analysis of means.
4. To develop hypotheses testing for population proportions in categorical data.