Page 195 - Intermediate Statistics for Dummies

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Part III: Comparing Many Means with ANOVA
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 9-3 is 0.001.
That means the results are highly statistically significant. You reject Ho and
conclude that at least one pair of age groups differ 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
going on in their lives than 17-year-olds do.)
Using Figure 9-4, 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 it’s way off to the
right, out of sight. It makes sense that the p-value, which measures the proba-
bility of being beyond that F-statistic, is 0.001.
If you’ve gotta use 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. However, statistical
software packages automatically calculate all p-values exactly (so on any
computer output you can see them as such).
To approximate the p-value from your F-statistic (in the event you don’t have
a computer or computer output available), you find a cutoff value on the
F-distribution with (k – 1, n – k) degrees of freedom that draws a line in the
sand between rejecting Ho and not rejecting Ho. This cutoff (also known as
the critical value) is determined by your prespecified α (typically 0.05). You
choose the critical value so that the area to its right on the F-distribution is
equal to α.
Table A-5 in the Appendix shows the critical values of the F-distribution with
various degrees of freedom, all using α = 0.05. Other F-distribution tables
are available in various statistics textbooks and Internet links for other
values of α; however, α = 0.05 is by far the most common α level used for
the F-distribution and is sufficient for your purposes.
This table of values for the F-distribution is called the F-table (students are
typically given these with their exams). For the seed spitting example, the
F-statistic has an F-distribution with degrees of freedom (3, 16), which I calcu-
late in a previous section. To find the critical value, go to Table A-5 in the
Appendix. Because the degrees of freedom are (3, 16), go to column 3 and
row 16 on the F-table. The critical value is 3.2389 (or 3.24). Your F-statistic for
the seed spitting example is 8.43, which is well beyond this critical value (you
can see how 8.43 compares to 3.24 by looking at Figure 9-4). Your conclusion
is to reject Ho at level α. At least two of the age groups differ on mean seed
spitting distances.