THIS PAPER DEALS WITH SOME TESTS OF HYPOTHESIS FREQUENTLY ENCOUNTERED IN THE ANALYSIS OF MULTIVARIATE DATA. THE TYPE OF HYPOTHESIS CONSIDERED IS THAT WHICH THE STATISTICIAN CAN ANSWER IN THE NEGATIVE OR AFFIRMATIVE. THE DOOLITTLE METHOD MAKES IT POSSIBLE TO EVALUATE THE DETERMINANT OF A MATRIX OF HIGH ORDER, TO SOLVE A MATRIX EQUATION, OR TO INVERT A MATRIX OF HIGH ORDER. HYPOTHESES WHICH, IN THE UNIVARIATE CASE, WOULD LEAD TO A USE OF THE STUDENT T-DISTRIBUTION ARE ALSO PRESENTED. THE ANALYSIS OF DATA COLLECTED FROM A "DESIGNED EXPERIMENT" IS DISCUSSED, INCLUDING THE LATIN SQUARE, RANDOMIZED BLOCK, CROSS-CLASSIFICATION, AND FACTORIAL DESIGNS. AN ANALYSIS OF REGRESSION IS PRESENTED ALONG WITH A TEST OF HYPOTHESIS ON THE DISPERSION MATRIX. (HW)