17.3.6.2 Algorithms (Two Sample Test for Variance)

The F-test calculates the ratio of two sample variance to test whether or not the two data samples come from populations with equal variances. And the hypotheses take the form:

\(H_0:\frac{\sigma_1^2}{\sigma_2^2}=1\) vs \(H_1:\frac{\sigma_1^2}{\sigma_2^2}\ne 1\) Two Tailed

\(H_0:\frac{\sigma_1^2}{\sigma_2^2} \le 1\) vs \(H_1:\frac{\sigma_1^2}{\sigma_2^2} > 1\) Upper tailed

\(H_0:\frac{\sigma_1^2}{\sigma_2^2} \ge 1\) vs \(H_1:\frac{\sigma_1^2}{\sigma_2^2} < 1\) Lower tailed

Test Statistics

And the F-test statistic is calculated as: \(F=\frac{s_1^2}{s_2^2}\)

where \(s_1^2\,\!\) and \(s_2^2\,\!\) are observed sample variances. A ratio of 1 implies equal sample variances, while ratios that deviate from 1 indicate unequal population variances. The hypothesis that the variances of the two samples are equal is rejected if \(p < \sigma\,\!\), where p is the calculated probability and \(\sigma\,\!\) is the chosen significance level.

Confidence Intervals

The upper and lower confidence limit values for F-test statistic is:

Null Hypothesis	Confidence Interval
\[H_0:\frac{\sigma_1^2}{\sigma_2^2}=1\]	\[\left[\frac{F}{F_{1-\alpha/2}},\frac{F}{F_{\alpha/2}}\right]\]
\[H_0:\frac{\sigma_1^2}{\sigma_2^2} \le 1\]	\[\left[\frac{F}{F_{1-\alpha}},\infty\right]\]
\[H_0:\frac{\sigma_1^2}{\sigma_2^2} \ge 1\]	\[\left[0,\frac{F}{F_{\alpha}}\right]\]

where \(F_{1-\sigma/2}\,\!\) and \(F_{\sigma/2}\,\!\) represents the lower and upper critical values for an F-distribution with \(n_1-1\,\!\) and \(n_2-1\,\!\) degrees of freedom, and \(\sigma\,\!\) level of significance.