Confidence Interval for the Ratio of Variances of Two Normally Distributed Populations


This file is part of a program based on the Bio 4835 Biostatistics class taught at Kean University in Union, New Jersey.  The course uses the following text:
Daniel, W. W. 1999.  Biostatistics: a foundation for analysis in the health sciences.  New York: John Wiley and Sons.  
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, sigma 1 squared /sigma2 squared .  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.

Sampling distribution

The sampling distribution of (s1-squared /sigma1-squared )/(s2-squared /sigma2-squared ) is used.  Since the population variances are usually not known, the sample variances are used.  The assumptions are that s1-squared and is2-squared are computed from independent samples of size n1 and n2 , respectively, drawn from two normally distributed populations.  If the assumptions are met,  (s1-squared /sigma1-squared )/(s2-squared /sigma2-squared ) 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  n1 -1 which is used in calculating s1-squared .  The denominator degrees of freedom is the value of  n2 -1 which is used in calculating s2-squared .

Reading F tables

F tables come in denominations based on F-1-(alpha/2) which are F.995F.99F.975F.95 and F.90 with one tail.  For two tail intervals, the lower boundary, F(alpha/2), must be calculated to give values of F.05F.025 and F.005 .

Confidence interval for sigma1-squared /sigma2-squared

The distribution (s1-squared /sigma1-squared )/(s2-squared /sigma2-squared ) is used to establish the 100(1-alpha ) percent confidence interval for sigma1-squared /sigma2-squared .  The staring point is

        Starting point for VR 

From this relation, it can be shown that the 100(1-alpha ) percent confidence interval for sigma1-squared /sigma2-squared is

        Calculation of VR


Example


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.

(1) Given

        n1 = 11        s1-squared5.8-squared = 33.64
        n2 = 4           s2-squared3.4-squared = 11.56

        alpha = .05

        F.975 10, 3 = 14.42

        F.025= 1/F.975 3, 10 = 1/4.83 = .20704

(2) Calculations

Calculation of the 95% confidence interval for sigma1-squared /sigma2-squared

        Calculation