Bayesian methods are utilized to analyse the one-way random effect modelyij= μ +ai+eijin whicheijare assumed normal,N(0, σ2e), andaiare assumed to have a mixture of two normals, .95N(−.05ø(k– 1)σ, σ2) + .05N(.95ø(k– l)σ,k2ø2). It is shown by a moderately sized sample that inferences regarding σ2e= var (eij) are insensitive but those of σ2e= var (ai) are very sensitive to changes of the two non-normality parameters (k, ø). The data are also used to illustrate what inferences can be made about (k, ø). Further, for σ = 0 it is found in general that extreme observations have less importance in comparison to others in the posterior expectation of σ2e2. Finally, the ways of assessing the contributions fromaiandeiito the variation ofyijare considered.