Test of Two Variances Quiz

PDF page 764: 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:

PDF page 764: 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 2 2 s and s be the sample variances. Let the sample sizes be n1 and n2. Since we are interested in comparing the two sample 1 2 variances, we use the F ratio:

PDF page 764: 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 2 2 s and s be the sample variances. Let the sample sizes be n1 and n2. Since we are interested in comparing the two sample 1 2 variances, we use the F ratio:

PDF page 764: 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 2 2 s and s be the sample variances. Let the sample sizes be n1 and n2. Since we are interested in comparing the two sample 1 2 variances, we use the F ratio:

PDF page 765: 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 765: 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 765: 2 2 1 is close to one. But if the If the two populations have equal variances, then s and s are close in value and F = 1 2 2

PDF page 765: 2 2 2 2 two population variances are very different, s and s tend to be very different, too. Choosing s as the larger sample 1 2 1 2 2

PDF page 765: 2 2 1 1 variance causes the ratio to be greater than one. If s and s are far apart, then F = is a large number. 1 2 2 2

PDF page 765: 2 2 Therefore, if F is close to one, the evidence favors the null hypothesis (the two population variances are equal). But if F is much larger than one, then the evidence is against the null hypothesis. A test of two variances may be left, right, or two-tailed.

PDF page 765: 2 2 Therefore, if F is close to one, the evidence favors the null hypothesis (the two population variances are equal). But if F is much larger than one, then the evidence is against the null hypothesis. A test of two variances may be left, right, or two-tailed.

PDF page 765: 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%.

PDF page 765: 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%.