http://www.ck12.org Chapter 11. Analysis of Variance and the F-Distribution
Establishing the critical values in anF-test is a bit more complicated than when doing so in other hypothesis tests.
Most tables contain multipleF-distributions, one for each of the following: 1 percent, 5 percent, 10 percent and
25 percent of the area are in the right-hand tail (please see the supplemental links for an example of the table). We
also need to use the degrees of freedom fromeachof the samples to determine the critical values.
Say, for example, that we are trying to determine the critical values for the scenario above and we set the level of
significance at. 02 (α=. 02 ). Because we have a two-tailed test, we assign .01 to the area of the right of the critical
value. Using theF-table forα=.01 (for example, see http://www.statsoft.com/textbook/sttable.html#f01)) , we find
the critical value at 2.20 (d f=30 and 40 for the numerator and denominator with aα=.01 to the area to the right
of the tail).
Once we set our critical values and calculate our test statistic, we perform the hypothesis test the same way we do
with the hypothesis tests using the normal and the Student’st-distributions.
Example:
Using our example above, suppose a teacher administered two different reading programs to two different groups
of students and was interested if one program produced a greater variance in scores. Perform a hypothesis test to
answer her question.
Solution:
In the example above, we calculated anFratio of 2.909 and found a critical value of 2.20.
Since the observed test statistic exceeds the critical value, we reject the null hypothesis. Therefore, we can con-
clude that the observed ratio of the variances from the independent samples would have occurred by chance if the
population variances were equal less than 2%(. 02 )of the time. We can conclude that the variance of the student
achievement scores for the second sample is less than the variance for the students in the first sample. We can also
see that the achievement test means are practically equal so the variance in student achievement scores may help the
teacher in her selection of a program.
The Limits of Using the F-Distribution to Test Variance
The test of the null hypothesisH 0 :σ 12 =σ 22 using theF-distribution is only appropriate when it can be safely
assumed that the population is normally distributed. If we are testing the equality of standard deviations between
two samples, it is important to remember that theF-test is extremely sensitive. Therefore, if the data displays even
small departures from the normal distribution including non-linearity or outliers, the test is unreliable and should not
be used. In the next lesson, we will introduce several tests that we can use when the data are not normally distributed.
Lesson Summary
- We use theF-Max test and theF-distribution when testing if two variances from independent samples are
equal. - TheF-distribution differs from the Student’st-distribution. Unlike the normal and thet-distributions, the
F-distributions are not symmetrical and go from zero to infinity(∞)not from−∞to∞as the others do. - When testing the variances from independent samples, we calculate theF-ratio, which is the ratio of the
variances of the independent samples. - When we reject the null hypothesisH 0 :σ 12 =σ 22 we conclude that the variances of the two populations are
not equal. - The test of the null hypothesisH 0 :σ 12 =σ 22 using theF-distribution is only appropriate when it can be safely
assumed that the population is normally distributed.