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Part III: Analyzing Variance with ANOVA
Making conclusions from ANOVA

If you’ve completed the F-test and found your F-statistic (step four in the
ANOVA process), you’re ready for step five of ANOVA: making conclusions
for your hypothesis test of the k population means. If you haven’t already
done so, you can compare the F-statistic to the corresponding F-distribution
with (k – 1, n – k) degrees of freedom to see where it stands and make a
conclusion. You can make the conclusion in one of two ways: the p-value
approach or the critical-value approach. The approach you use depends pri-
marily on whether you have access to a computer, especially during exams. I
describe these two approaches in the following sections.

Using the p-value approach
On Minitab ANOVA output (see Figure 9-4), the value of the F-statistic is
located in the Factor row, under the column noted by F. The associated
p-value for the F-test is located in the Factor row under the column headed by
P. The p-value tells you whether or not you can reject Ho.

✓ If the p-value is less than your predetermined α (typically 0.05), reject
Ho. Conclude that the k population means aren’t all equal and that at
least two of them are different.
✓ If the p-value is greater than α, then you can’t reject Ho. You don’t
have enough evidence in your data to say the k population means have
any differences.
The F-statistic for comparing the mean watermelon seed-spitting distances
for the four age groups is 8.43. The p-value as indicated in Figure 9-4 is 0.001.
That means the results are highly statistically significant. You reject Ho and
conclude that at least one pair of age groups differs in its mean watermelon
seed-spitting distances. (You would hope that a 17-year-old could do a lot
better than a 6-year-old, but maybe those 6-year-olds have a lot more spitting
practice than 17-year-olds do.)

Using Figure 9-5, you see how the F-statistic of 8.43 stands on the F-distribution
with (4 – 1, 20 – 4) = (3, 16) degrees of freedom. You can see that it’s way off
to the right, out of sight. It makes sense that the p-value, which measures the
probability of being beyond that F-statistic, is 0.001.

Using critical values
If you’re in a situation where you don’t have access to a computer (as is still
the case in many statistics courses today when it comes to taking exams),
finding the exact p-value for the F-statistic isn’t possible using a table. You

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