Interval for a ratio of variances
Section of Confidence intervals, part of Statistical inference
Interval for a ratio of variances
| Type | Section |
| Taught in | Confidence intervals |
| Part | Statistical inference |
The pooled interval is valid only if the variances are equal. That can be checked with an interval for \sigma_1^2/\sigma_2^2, and seeing whether it contains 1.
Theorem 1 (Variance-ratio F pivot) For two independent normal samples,
\frac{s_1^2/\sigma_1^2}{s_2^2/\sigma_2^2}
has the F distribution with (n_1-1,\,n_2-1) degrees of freedom.
Write U=(n_1-1)s_1^2/\sigma_1^2 and V=(n_2-1)s_2^2/\sigma_2^2. These are independent \chi^2_{n_1-1} and \chi^2_{n_2-1} variables, and the ratio is \{U/(n_1-1)\}/\{V/(n_2-1)\}, which is the definition of F_{n_1-1,n_2-1}. Placing the pivot between its quantiles and using 1/F_{m_1,m_2;\,\alpha/2}=F_{m_2,m_1;\,1-\alpha/2} gives the interval
\left(\frac{1}{F_{n_1-1,n_2-1;\,1-\alpha/2}}\,\frac{s_1^2}{s_2^2},\ \ F_{n_2-1,n_1-1;\,1-\alpha/2}\,\frac{s_1^2}{s_2^2}\right).
For the safety measures, s_1^2/s_2^2=14.1^2/14.3^2=0.972 and F_{49,49;0.975}=1.762, so the 95% interval for \sigma_1^2/\sigma_2^2 is (0.552,\ 1.713). It contains 1, so there is no reason at the 5% level to doubt the equal-variance assumption behind the pooled interval.
Used in
See also
Confidence interval · Pivot · One-sample t pivot · Pooled variance · Two-sample t pivot · Variance-ratio F pivot · Wald interval · Coverage