Hypothesis Test for Comparison of Multivariate Means

Often it is useful to compare a multivariate measurement for an object with the mean of a multivariate population [5], To perform the test, we will determine if a 1 x p vector xT is an acceptable value for the mean of a multivariate normal distribution, /jlt, according to the null hypothesis and alternative hypothesis shown in Equation 3.34. 

The true values of X and jx are usually estimated from a small sample of size n. When n is very large, the estimates x and S are very good however, n is usually small, and thus the estimates x and S have a lot of uncertainty. In this case it is necessary to make an adjustment for the confidence interval, 100%(1 - a), of the sample mean and scatter matrix by use of Hotelling s T2 statistic. 

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