In statistical applications involving linear models, many inferences can only be made based on statistics that are functions of positive linear combinations of mean squares. The Satterthwaite procedure is commonly used to approximate the distribution of such a linear combination to a chi-squared with appropriate degrees of freedom. This article presents a new procedure, based on recent work on confidence intervals, for determining this approximate degrees of freedom. Numerical examples are also given to compare the performance of the proposed method with other approximations