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accept the hypothesis Accepting when H alternative hypothesis Beta functions CATHOLIC UNIVERSITY characteristic function critical region degrees of freedom dissertation distributed according drawn independently errors of Type expressed four events FUNCTIONS AND DISTRIBUTION Gamma function given sample gives the distribution H is false H is true hypothesis H hypothesis under consideration independently distributed integral joint distribution Let H likelihood ratio test normal populations null hypothesis number of rejections obtain the distribution population means population variances power function proba probability distribution probability law probability of accepting PROBABILITY OF OCCURRENCE probability of rejecting quantity random variable reject the hypothesis rejecting H rejecting when H respect sample point sample space samples from normal statistical populations statistical test Student Suppose the variances t-test test the hypothesis theoretical probabilities given three samples UNIVERSITY OF AMERICA unknown variances variances are equal variances are known variances are unequal variances o-i2 y1 and y2 z-test