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The conventional approach for testing the equality of two normal mean vectors is to test first the equality of covariance matrices, and if the equality assumption is tenable, then use the two-sample Hotelling T (2) test. Otherwise one can use one of the approximate tests for the multivariate Behrens-Fisher problem. In this article, we study the properties of the Hotelling T (2) test, the conventional approach, and one of the best approximate invariant tests (Krishnamoorthy & Yu, 2004) for the Behrens-Fisher problem. Our simulation studies indicated that the conventional approach often leads to inflated Type I error rates. The approximate test not only controls Type I error rates very satisfactorily when covariance matrices were arbitrary but was also comparable with the T (2) test when covariance matrices were equal.

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The problems of hypothesis testing and interval estimation of the squared multiple correlation coefficient of a multivariate normal distribution are considered. It is shown that available one-sided tests are uniformly most powerful, and the one-sided confidence intervals are uniformly most accurate. An exact method of calculating sample size to carry out one-sided tests (null hypothesis may involve a nonzero value for the multiple correlation coefficient) to attain a specified power is given. Sample size calculation for computing confidence intervals for the squared multiple correlation coefficient with a specified expected width is also provided. Sample sizes for powers and confidence intervals are tabulated for a wide range of parameter configurations and dimensions. The results are illustrated using the empirical data from Timm (1975) that related scores from the Peabody Picture Vocabulary Test to four proficiency measures.

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