F Test

Test for Equality of Variances or Homogeneity of Variances

The F test is used to test whether two population variances are equal (H0: σ21 = σ22) versus either a onetail alternative, upper or lower, Ha: σ21 > σ22 or Ha: σ21 < σ22, respectively, or a two-tail alternative, Ha: σ21 61/4 s22. The test statistic is the ratio of F = s21= s22, where s21 is the sample variance from Sample 1 with df = n1 − 1and s22 is the sample variance from Sample 2 with df = n2 − 1. If H0 is true, then the ratio would be close to 1. The farther away the ratio is from 1, the stronger the evidence for Ha. (Note that 0 < f < ∞ since s2 > 0.) One rejects the null hypothesis in the following cases, where Fa,m,n is the critical value of the F distribution with numerator df = m, denominator df = n, and a significance value α:

• F > Fα,n1 − 1, n2 − 1 for an upper one-tailed test.
• F < f1 − α, n1 − 1, n2 − 1 for a lower one-tailed test.
• F < f1 − α/2,n1 − 1, n2 − 1 or F > Fa=2,n1 − 1, n2 − 1 for a two-tailed test, although some texts may recommend that the F test statistic be found where the largest sample variance is in the numerator and reject H0 if F > Fα/2,n1 − 1, n2 − 1

ANOVA and the F Test

In many experimental design situations, the data are analyzed using an ANOVA table. The plethora of problems makes it too difficult to explain each of them here. Instead, the general idea is presented using the one-way ANOVA table. In this situation, one wishes to test if the means are equal versus at least one population mean is different. A sample ANOVA table is given in Table 1. The details of where the values were derived from are not given since the interest is in the F test itself. In general, df gives the degrees of freedom and SS gives the sum of squares. The MS stands for mean squares and is calculated by SS/df. Then, the F ratio is calculated using the ratio of the MS. For example, F = 48.8/13.6 = 3.58. The null hypothesis is rejected when F > Fa, dfnum, dfdenom: In this case, Fa = 0:05,3,12 = 4.49, so in this situation, one would reject H0.

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