Summary: This module provides the assumptions to be considered in order to calculate a Test of Two Variances and how to execute the Test of Two Variances. An example is provided to help clarify the concept.

Another of the uses of the F distribution is testing two variances. It is often desirable to compare two variances rather than two averages. For instance, college administrators would like two college professors grading exams to have the same variation in their grading. In order for a lid to fit a container, the variation in the lid and the container should be the same. A supermarket might be interested in the variability of check-out times for two checkers.

In order to perform a F test of two variances, it is important that the following are true:

- The populations from which the two samples are drawn are normally distributed.
- The two populations are independent of each other.

Suppose we sample randomly from two independent normal populations. Let

where

If the null hypothesis is

The F F ratio could also be
(
s 2
) 2
(
s 1
)
2
(
s 2
) 2
(
s 1
)
2
.
It depends on H a H a and on which sample variance is larger.

If the two populations have equal variances, then

Therefore, if

*A test of two variances may be left, right, or two-tailed.*

Two college instructors are interested in whether or not there is any variation in the way they grade math exams. They each grade the same set of 30 exams. The first instructor's grades have a variance of 52.3. The second instructor's grades have a variance of 89.9.

Test the claim that the first instructor's variance is smaller. (In most colleges, it is desirable for the variances of exam grades to be nearly the same among instructors.) The level of significance is 10%.

Let 1 and 2 be the subscripts that indicate the first and second instructor, respectively.

*Calculate the test statistic:* By the null hypothesis

*Distribution for the test:*

*Graph:**This test is left tailed.*

Draw the graph labeling and shading appropriately.

*Probability statement:*

*Compare *

*Make a decision:* Since

*Conclusion:* With a 10% level of significance, from the data, there is sufficient
evidence to conclude that the variance in grades for the first instructor is smaller.

*TI-83+ and TI-84:* Press `STAT`

and arrow over to `TESTS`

. Arrow down to
`D:2-SampFTest`

. Press `ENTER`

. Arrow to `Stats`

and press `ENTER`

. For
`Sx1`

, `n1`

, `Sx2`

, and `n2`

, enter

, `30`

,

, and `30`

. Press `ENTER`

after
each. Arrow to `σ1:`

and

. Press `ENTER`

. Arrow down to `Calculate`

and
press `ENTER`

. `Draw`

instead of `Calculate`

.

Comments:"Reviewer's Comments: 'I recommend this book. Overall, the chapters are very readable and the material presented is consistent and appropriate for the course. A wide range of exercises introduces […]"