1. AP Statistics: ANOVA Section 2

2. In the previous section, we saw how to use ANOVA to test for a difference in means among several groups. However, that test only tells us when differences exist, not which specific groups differ. The goal of this section is to adapt the inference procedures of section 13.1 to use the results of the ANOVA analysis. This will allow us to find a confidence interval for the mean of any group, find a confidence interval for a difference in means between two groups and test when that difference is significant.

3. In earlier inference work, we used formulas such as those below for doing inference about a single mean or a difference in means.

4. If we have found an ANOVA table based on samples from several groups, we make a couple of small adjustments to these formulas. * Estimate any standard deviation with from the ANOVA table Since one of the conditions for the ANOVA is that the standard deviation is the same for each group, using gives an estimate that is based on all of the samples and not just one. * Use the error degrees of freedom, n – k, for any t- distributions We often call the pooled standard deviation

5. Inference for Means After ANOVA After doing an ANOVA for a difference in means among k groups based on sample of size   Confidence interval for :       Confidence interval for :

6. If the ANOVA indicates that there are differences among the means:     Pairwise test of vs :     where MSE is the mean square error from the ANOVA table and the t-distribution use n – k degrees of freedom

7. Example: Let’s go back to the ants and sandwich fillings example from section 1.  

8.   Find and interpret a 95% confidence interval for the mean number of ants attracted to a peanut butter sandwich.

9. Find and interpret a 95% confidence interval for the difference in average ant counts between vegemite and ham & pickles sandwiches.

10. Note: Since our confidence interval contains only negative numbers, this implies that our two population means differ – this is not surprising since ANOVA done earlier indicated that at least two of the groups have different means.

11. Test at the 5% level for a difference in mean number of ants between vegemite and peanut butter sandwiches.