10. Comparing Two Population Means

# 10.3 Difference between Two Variances – the F Distributions

Here we have to assume that the two populations (as opposed to sample mean distributions) have a distribution that is almost normal as shown in Figure 10.2.

Figure 10.2: Two normal populations lead to two distributions that represent distributions of sample variances. The distribution results when you build up a distribution of the ratio of the two sample values.

The ratio follows an -distribution if . That distribution has two degrees of freedom: one for the numerator (d.f.N. or ) and one for the denominator (d.f.D. or ). So we denote the distribution more specifically as . For the case of Figure 10.2, and . The ratio, in general is the result of the following stochastic process. Let be random variable produced by a stochastic process with a distribution and let be random variable produced by a stochastic process with a distribution. Then the random variable will, by definition, have a distribution.

The exact shape of the distribution depends on the choice of and , But it roughly looks like a distribution as shown in Figure 10.3.

and are related :

so the statistic can be viewed as a special case of the statistic.

For comparing variances, we are interested in the follow hypotheses pairs :

 Right-tailed Left-tailed Two-tailed

We’ll always compare variances () and not standard deviations () to keep life simple.

The test statistic is

where (for finding the critical statistic), and .

Note that when , a fact you can use to get a feel for the meaning of this test statistic.

Values for the various critical values are given in the F Distribution Table in the Appendix. We will denote a critical value of with the notation :

Where:

= Type I error rate
= d.f.N.
= d.f.D.

The F Distribution Table gives critical values for small right tail areas only. This means that they are useless for a left-tailed test. But that does not mean we cannot do a left-tail test. A left-tail test is easily converted into a right tail test by switching the assignments of populations 1 and 2. To get the assignments correct in the first place then, always define populations 1 and 2 so that . Assign population 1 so that it has the largest sample variance. Do this even for a two-tail test because we will have no idea what on the left side of the distribution is.

Example 10.3 : Given the following data for smokers and non-smokers (maybe its about some sort of disease occurrence, who cares, let’s focus on dealing with the numbers), test if the population variances are equal or not at .

 Smokers Nonsmokers

Note that so we’re good to go.

Solution :

1. Hypothesis.

2. Critical statistic.

Use the F Distribution Table; it is a bunch of tables labeled by “” that we will designate at , the table values that signify right tail areas. Since this is a two-tail test, we need . Next we need the degrees of freedom:

So the critical statistic is

3. Test statistic.

With this test statistic, we can estimate the -value using the F Distribution Table. To find , look up all the numbers with d.f.N = 25 and d.f.N = 17 (24 17 are the closest in the tables so use those) in all the the F Distribution Table and form your own table. For each column in your table record and the value corresponding to the degrees of freedom of interest. Again, corresponds to for a two-tailed test. So make a row above the row with . (For a one-tailed test, we would put .)

 0.20      0.10      0.05      0.02      0.01 0.10       0.05     0.025    0.01      0.005 1.84       2.19      2.56       3.08      3.51         3.6 is over here somewhere so

Notice how we put an upper limit on because was larger than all the values in our little table.

Let’s take a graphical look at why we use in the little table and for finding for two tailed tests :

But in a two-tailed test we want split on both sides:

4. Decision.

Reject . The -value estimate supports this :

5. Interpretation.

There is enough evidence to conclude, at with an -test, that the variance of the smoker population is different from the non-smoker population.