We consider a (k+1)-variate distribution in which k variates are continuous and I variate are categorical. The k variates are assumed to have a conditional multivariate normal distribution with respect to the I categorical variates which are assumed to have multinomial distribution. Appropriate hypotheses are framed in this situation, analogous to the customary hypotheses on a single multivariate normal distribution, large sample tests of such hypotheses are developed and some of their properties studied. Next, instead of assuming a single multinomial distribution on the 1 categorical variate, a product of multinomial distributions is assumed (in case some of the 1-categorical are ways of classification) and hypotheses are framed in this situation analogous to the customary ones for several multivariate normal distributions, and large sample tests of such hypotheses and some of their properties are studied.