The Behrens-Fisher Distribution and Weighted Means

SUMMARY (1) The Behrens-Fisher and Welch solutions of the two-means problem are compared, and the former shown to be lacking in sensitivity. (2) Yates’s and the author’s tests for a weighted mean are compared, and the former shown to reject the true value of the mean, in error, more often than is indicated by the nominal significance level (in extreme cases nearly twice as often). (3) A new test for weighted means is put forward which does not neglect the information on the variances supplied by the difference between the means, and which is “exact”. (4) An incidental by-product is the Cornish-Fisher expansion for a symmetrical distribution, taken to some further terms.