Page 178 - Statistics II for Dummies

162
Part III: Analyzing Variance with ANOVA
For the seed-spitting data, the variances for each age group are listed in Figure 9-2.
These variances are close enough to say the equal variance condition is met.

Setting Up the Hypotheses

Step two of ANOVA is setting up the hypotheses to be tested. You’re testing
to see if all the population means can be deemed equal to each other. The
null hypothesis for ANOVA is that all the population means are equal. That
is, Ho: μ = μ = . . . = μ , where μ is the mean of the first population, μ is the
1 2 k 1 2
mean of the second population, and so on until you reach μ (the mean of the
k
th
k population).
What appears in the alternative hypothesis (Ha) must be the opposite of
what’s in the null hypothesis (Ho). What’s the opposite of having all k of the
population’s means equal to each other? You may think the opposite is that
they’re all different. But that’s not the case. In order to blow Ho wide open, all
you need is for at least two of those means to not be equal. So, the alternative
hypothesis, Ha, is that at least two of the population means are different from
each other. That is, Ha: At least two of μ , μ , . . . μ are different.
1 2 k
Note that Ho and Ha for ANOVA are an extension of the hypotheses for a two-
sample t-test (which only compares two independent populations). And even
though the alternative hypothesis in a t-test may be that one mean is greater
than, less than, or not equal to the other, you don’t consider any alterna-
tive other than ≠ in ANOVA. (Statisticians use more in-depth models for the
others. Aren’t you glad someone else is doing it?)

You only want to know whether or not the means are equal — at this stage of
the game anyway. After you reach the conclusion that Ho is rejected in
ANOVA, you can proceed to figure out how the means are different, which
ones are bigger than others, and so on, using multiple comparisons. Those
details appear in Chapter 10.

Doing the F-Test

Step three of ANOVA is collecting the data, and it includes taking k random
samples, one from each population. Step four of ANOVA is doing the F-test on
this data, which is the heart of the ANOVA procedure. This test is the actual
hypothesis test of Ho: μ = μ = . . . = μ versus Ha: At least two of μ , μ , . . . μ
1 2 k 1 2 k
are different.

7/23/09 9:31:29 PM
15_466469-ch09.indd 162 7/23/09 9:31:29 PM
15_466469-ch09.indd 162