The Range of confidence Intervals for ${\sigma}^{2}_{A}/{\sigma}^{2}_{B}$ in Two-Factor Nested Variance Component Model

The two-factor nested variance component model with equal numbers in the cells are given by $y_{ijk}\;=\;{\mu}\;+\;A_i\;+\;B_{ij}\;+\;C_{ijk}$ and the confidence intervals for the ratio of variance components, ${\sigma}^{2}_{A}/{\sigma}^{2}_{B}$ are obtained in various forms by many authors. This article shows the probability ranges of these confidence intervals on ${\sigma}^{2}_{A}/{\sigma}^{2}_{B}$ proved by the mathematical computation.