Comparing the Means of Two Independent Samples

SUMMARY A practical procedure is given for assessing a P value for the hypothesis that two samples come from populations having the same mean, for the case when the two variances cannot be assumed equal. 1. The Problem, and a Procedure that sometimes Solves it Given two samples xl, x2,. . . , xm, and Yi, Y2, **. , Yn, supposed normal with means g, A + 8, and variances a2, p2 a2, respectively, standard texts are curiously silent on the question of testing whether 8 = 0 (or any other prescribed value) when it cannot be taken as known that the variances are equal, p2 = 1. A few texts give approximations which are valid for moderately large samples, and a few refer to the Tables for Welch's test given in Biometrika Tables, Volume I (Table 11, 3rd edition). Fewer still refer to Fisher and Yates' Statistical Tables (Table VI, 6th edition) which give critical values for the Behrens-Fisher test. And yet it is easy to see that if p can be taken to have any known value, the quantity t =x6 - -_) ( (I /m) + (p2/In) (m -1) s2 + (n - 1 s/p2 / (m + n- /2) ll2 ( (with the usual sample mean and variance notation) has Student's distribution with m + n -2 degrees of freedom. It is not difficult to program a hand calculator to take in p2, S2y m, n, and (j7- - 6) and to calculate this t. Then if a range of p2 values is considered plausible, p2 can be varied over this range to see what effect this variation has on the value of t. It will often happen (especially when m and n are nearly equal, and moderately large, and sX and sy do not differ by much) that t will remain nearly constant over a wide range of p2 values, and in this case the P value associated with the common t value can be taken as indicating the credibility or otherwise of the value of 6.