This file is part of a program based on the Bio 4835 Biostatistics class taught at
Daniel, W. W. 1999. Biostatistics: a foundation for analysis in the health sciences.
The file follows this text very closely and readers are encouraged to consult the text for further information.
F) Confidence interval for the ratio of variances of two normally distributed populations
A way to compare the variances of two normally distributed populations is to use the variance ratio, / . The variance ratio is used, among other things, as the test statistic for analysis of variance (ANOVA). If the two variances are equal, then V. R. = 1.
The sampling distribution of ( / )/( / ) is used. Since the population variances are usually not known, the sample variances are used. The assumptions are that and i are computed from independent samples of size and , respectively, drawn from two normally distributed populations. If the assumptions are met, ( / )/( / ) follows a distribution known as the F distribution with two values used for degrees of freedom.
Degrees of freedom
The F distribution uses two values for degrees of freedom. The numerator degrees of freedom is the value of -1 which is used in calculating . The denominator degrees of freedom is the value of -1 which is used in calculating .
Reading F tables
F tables come in denominations based on which are , , , and with one tail. For two tail intervals, the lower boundary, , must be calculated to give values of , and .
Confidence interval for /
The distribution ( / )/( / ) is used to establish the 100(1- ) percent confidence interval for / . The staring point is
From this relation, it can be shown that the 100(1- ) percent confidence interval for / is
Among 11 patients in a certain study, the standard deviation of the property of interest was 5.8. In another group of 4 patients, the standard deviation was 3.4. We wish to construct a 95 percent confidence interval for the ratio of the variances of these two populations.
= 11 = = 33.64
= 4 = = 11.56
10, 3 = 14.42
= 1/ 3, 10 = 1/4.83 = .20704
Calculation of the 95% confidence interval for /