## Quick Reference

A problem concerned with the comparison of the means of two populations having normal distributions with different variances. The problem was first discussed by B. V. Behrens in 1929. Although Behrens's method of solution was unclear, his conclusions were confirmed by Sir Ronald Fisher in 1935.

The null hypothesis is that the populations have the same mean, and the suggested solution is an application of the Welch statistic. In this case, the test statistic *t* is given by where *x̄*_{1} and *x̄*_{2} are the sample means, *n*_{1} and *n*_{2} are the sample sizes, and *s*_{1}^{2} and *s*_{2}^{2} are the sample variances (using the divisors *n*_{1}−1 and *n*_{2}−1). If the populations do have the same mean then *t* is an observation from an approximate *t*-distribution with ν degrees of freedom, where ν is taken as the nearest integer to , where *a*=s^{2}_{1}/ *n*_{1} and *b*=s^{2}_{2} / *n*_{2}. If *n*_{1}≤*n*_{2}, then the above formula ensures that (*n*_{1}−1)≤*ν*≤(*n*_{1}+*n*_{2}−2).

If the populations can be assumed to have the same variance then the problem is simple and the standard *t*-test is appropriate.

*Subjects:*
Probability and Statistics.