Test of Two Variances Quiz

PDF page 783: Another use 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. 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. To perform a F test of two variances, it is important that the following are true:

PDF page 783: • The populations from which the two samples are drawn are normally distributed.

PDF page 783: • The two populations are independent of each other.

PDF page 783: Unlike most other tests in this book, the F test for equality of two variances is very sensitive to deviations from normality. If the two distributions are not normal, the test can give higher p-values than it should, or lower ones, in ways that are unpredictable. Many texts suggest that students not use this test at all, but in the interest of completeness we include it here. 2 2 Suppose we sample randomly from two independent normal populations. Let σ and σ be the population variances and 1 2

PDF page 784: F = . ⎡ 2⎤ (s ) ⎢ 2 ⎥ 2 ⎣(σ ) ⎦ 2 F has the distribution F ~ F(n1 – 1, n2 – 1), where n1 – 1 are the degrees of freedom for the numerator and n2 – 1 are the degrees of freedom for the denominator. ⎡ 2⎤ (s ) ⎢ 1 ⎥ 2 2 ⎣(σ ) ⎦ (s ) 2 2 1 1

PDF page 784: F = . ⎡ 2⎤ (s ) ⎢ 2 ⎥ 2 ⎣(σ ) ⎦ 2 F has the distribution F ~ F(n1 – 1, n2 – 1), where n1 – 1 are the degrees of freedom for the numerator and n2 – 1 are the degrees of freedom for the denominator. ⎡ 2⎤ (s ) ⎢ 1 ⎥ 2 2 ⎣(σ ) ⎦ (s ) 2 2 1 1

PDF page 784: 2 2 1 If the two populations have equal variances, then s and s are close in value and F = is close to 1. But if the 1 2 2

PDF page 784: 2 2 1 1 variance causes the ratio to be greater than 1. If s and s are far apart, then F = is a large number. 1 2 2 2

PDF page 784: 2 2 Therefore, if F is close to 1, the evidence favors the null hypothesis (the two population variances are equal). But if F is much larger than 1, then the evidence is against the null hypothesis. A test of two variances may be left-tailed, right-tailed, or two-tailed.

PDF page 784: 2 2 Therefore, if F is close to 1, the evidence favors the null hypothesis (the two population variances are equal). But if F is much larger than 1, then the evidence is against the null hypothesis. A test of two variances may be left-tailed, right-tailed, or two-tailed.

PDF page 784: 2 2 Therefore, if F is close to 1, the evidence favors the null hypothesis (the two population variances are equal). But if F is much larger than 1, then the evidence is against the null hypothesis. A test of two variances may be left-tailed, right-tailed, or two-tailed.

PDF page 784: Two college instructors are interested in whethe 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 percent.