On the Distribution of a Linear Combination of Independent Chi Squares

In several models of the analysis of variance, inferences can be made using the statistics of the analysis of variance table only by taking linear combinations of mean squares. A form for the exact distribution function of such a linear combination is given. While not new, this form does not seem to have been sufficiently exploited. Its use is illustrated in finding, by numerical integration, exact probabilities associated with two methods of assigning approximate confidence limits to a positive linear combination of variance components.