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Several approximations to the binomial distribution were compared in 1956 by Raft [9], including the normal, arcsine, normal Gram-Charlier, Camp-Paulson, Poisson, and Poisson Gram-Charlier approximations. The closeness of these approximations is judged by the maximal error which can arise in evaluating any sum of consecutive binomial terms. Borges [3] investigated a general class of approximations (which includes some of the above) and showed that a certain beta-transformation of a binomially distributed variable is the unique member of this class which is approximately normal with error 0(l/n). In 1963, Bol'shev [2] gave two new Poisson-type approximations. In this paper, some of Raft's tables are extended and tables for the error of these newer approximations are added. As a consequence of the numerical results obtained, the Poisson Gram-Charlier approximation is recommended for small values of p while the Camp-Paulson and Borges' approximations should generally be preferred for large p. [ABSTRACT FROM AUTHOR]

In this paper the first two moments of the Statistic (treatment sum of squares)/(treatment +error sums of squares) over all possible random assignments of treatments to the experimental plots are obtained for two associate PBIB designs. They are compared with the corresponding moments of a central beta distribution to study the extent to which the normal theory test may serve as an approximation to the randomization test. It is found that the approximation is reasonable for some classes of PBIB designs. [ABSTRACT FROM AUTHOR]