There is a Fisher's criterion whose statistic seems like what you put down here. It's used to reject 0-hypothesis of equal variances. The statistic is (s_1)^2/(s_2)^2, where (s_1)^2 and (s_2)^2 are group variances estimates. It has F-distribution provided sigma_1=sigma_2 equation is true. So your CI left and right bound estimates look credible.