TABLE OF CONTENTS In this appendix we describe the rationale for the MLS and GCI approaches. We do this in the context of a balanced random model with the ANOVA shown in Table B.1. Given the standard normality and independence assumptions made in this book, 
are jointly independent chi-squared random variables with n q degrees of freedom.
| Source of variation | Degrees of freedom | Mean square | Expected mean square |
|---|---|---|---|
| Factor 1 | n1 | | ? 1 |
| Factor 2 | n 2 | | ? 2 |
| | | | |
| Replicates | n Q | | ? Q |
The term modified large-sample was introduced by Graybill and Wang [25] in their paper proposing a confidence interval for nonnegative linear combinations of variance components. To apply this method, one starts with an approximate large-sample confidence interval. The interval is then modified to make it exact under certain parameter conditions. To demonstrate, consider the problem of constructing a confidence interval for ? = ? 1 + ? 2, where ? 1 and ? 2 are defined in Table B.1. The uniformly minimum variance unbiased estimator for ? is 
and this estimator has variance 
. Thus, the 100 (1 ? ?)% large-sample confidence interval for ? is
and
where Z 1? ?/ 2 is the percentile from a standard normal distribution with area 1 ? ?/2 to the left. We now replace
