Page 193 - Intermediate Statistics for Dummies

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Part III: Comparing Many Means with ANOVA
From the ANOVA table for the seed spitting data in Figure 9-3, you can see
that column two has the heading DF, which stands for degrees of freedom.
You can find the degrees of freedom for SST in the Factor row (row two); this
value is equal to k – 1 = 4 – 1 = 3. The degrees of freedom for SSE is found to
be n – k = 20 – 4 = 16. (Remember you have four age groups and five children
in each group for a total of n = 20 data values.) The degrees of freedom for
SSTO is n – 1 = 20 – 1 = 19 (found in the Total row under DF.) You can verify
that the degrees of freedom for SSTO = degrees of freedom for SST + degrees
of freedom for SSE.
The values of MST and MSE are shown in column four of Figure 9-3, with the
heading MS. You can see the MST in the Factor row, which is 29.92. This value
was calculated by taking SST = 89.75, and dividing it by degrees of freedom, 3.
You can see MSE in the Error row, equal to 3.55. MSE is found by taking SSE =
56.80 and dividing that value by its degrees of freedom, 16.
By finding the mean sums of squares, you’ve completed step two of the F-test,
but don’t stop here! You need to continue to the next section if you want to
complete the process.
Figuring the F-statistic
The test statistic for the test of the equality of the k population means is
MST
F = . The result of this formula is called the F-statistic. The F-statistic
MSE
has an F-distribution, which is equivalent to the square of a t-test (when the
numerator degrees of freedom is 1). All F-distributions start at zero and are
skewed to the right. The degree of curvature and the height of the curvature
of each F-distribution is reflected in two degrees of freedom, represented by
k – 1 and n – k. (These come from the denominators of MST and MSE, respec-
tively, where n is the total sample size and k is the total number of treatments
or populations.) A shorthand way of denoting the F-distribution for this test
is F (k – 1,n – k).
In the watermelon seed spitting example, you’re comparing four means and
have a sample of size five from each population. Figure 9-4 shows the corre-
sponding F-distribution, which has degrees of freedom 4 – 1 = 3 and 20 – 4 =
16; in other words F (3, 16) .
You can see the F-statistic on the Minitab ANOVA output (see Figure 9-3) in
the Factor row, under the column indicated by F. For the seed spitting exam-
ple, the value of the F-statistic is 8.43. This number was found by taking MST =
29.92 divided by MSE = 3.55. You can then locate 8.43 on the F-distribution in
Figure 9-4 to see where it stands. (More on that in the next section.)