# Lesson 11.1 - Comparing Two Population Variances

 Unit Summary F-test to Compare Two Population Variances An Example to Compare Two Population Variances

An Introduction to Statistical Methods and Data Analysis, chapter 7.3.

### F-test to Compare Two Population Variances

To compare the variances of two quantitative variables, the hypotheses of interest are:

Ho: 12 - 22 = 0
Ha: 12 - 22 0

For one-tailed tests, you can follow section 7.3 of our textbook to set up the corresponding hypotheses and perform the test.

One of the major applications of a test to compare two population variances is for checking the equal variances assumption if you want to use the pooled variances t-test. In other situations, such as quality control problems, you may want to choose the process with smaller variances in the two processes of interest.

The Minitab command to use is homogeneity of variance test:

Stat > ANOVA > Homogeneity of Variance

### An Example to Compare Two Population Variances

Using the data in the package time given in Lesson 10, we want to check whether it is reasonable to assume that the two machines have equal population variances. Recall that the data are given as:

 New Machine Old Machine 42.1 41.3 42.4 43.2 41.8 42.7 43.8 42.5 43.1 44.0 41.0 41.8 42.8 42.3 42.7 43.6 43.3 43.5 41.7 44.1 = 42.14, s1 = 0.683 = 43.23, s2 = 0.750

Enter the package time for both machines into one column and name it "package time" with the type of machine indicated in another column named "machine."

The Minitab printout for the test for equal variances is as follows:

Homogeneity of Variance

 Response package time Factors machine ConfLv1 95.0000 Bonferroni confidence intervals for standard deviations

 Lower Sigma Upper N Factor Levels 0.447064 0.683455 1.37624 10 new 0.490520 0.749889 1.51002 10 old

F-Test (normal distribution)

 Test Statistic : 1.204 P-Value : 0.787

Levene's Test (any continuous distribution)

 Test Statistic : 0.024 P-Value : 0.879

How do we interpret the Minitab output?

Minitab gives us both the results for the F-test (assuming that the two populations follow a normal distribution) and the Levene's test. The hypotheses are:

Ho: 12 - 22 = 0
Ha: 12 - 22 0

In this example, the p-value for both the F-test and the Levene's test is very large (larger than 0.1). We don't need to check whether the two populations are normal and we can conclude that we cannot reject the null hypothesis. If there is a discrepancy between the F-test and the Levene's test, you need to check whether the normal assumption holds before you can use the F-test since the F-test is very sensitive to departure from the normal assumption.