Where S is the pooled sample covariance matrix of X and Y, namely the null hypothesis H 0: μ X = μ Y.ĭefinition 1: The Two sample Hotelling’s T-square test statistic is We now look at a multivariate version of the problem, namely to test whether the population means of the k × 1 random vectors and Y are equal, i.e. Also note that by Property 1 of F Distribution, an equivalent test can be made using the test statistic t 2 and noting that The null hypothesis is rejected if | t| > t crit. It turns out that the t-test is pretty robust for violations of the normality assumption provided each population is relatively symmetric about its mean. Regarding the normality assumption, if n x and n y are sufficiently large, the Central Limit Theorem holds, and we can proceed as if the populations were normal. The sample for x and y are random with each element in the sample taken independently.The variances of the two populations are equal (homogeneity of variances).The populations of x and y have a normal distribution.
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