  Testing differences between a treatment and control group is common practice
in biomedical research like randomized controlled trials (RCT). The standard
two-sample t-test relies on null hypothesis significance testing (NHST) via
p-values, which has several drawbacks. Bayesian alternatives were recently
introduced using the Bayes factor, which has its own limitations. This paper
introduces an alternative to current Bayesian two-sample t-tests by
interpreting the underlying model as a two-component Gaussian mixture in which
the effect size is the quantity of interest, which is most relevant in clinical
research. Unlike p-values or the Bayes factor, the proposed method focusses on
estimation under uncertainty instead of explicit hypothesis testing. Therefore,
via a Gibbs sampler the posterior of the effect size is produced, which is used
subsequently for either estimation under uncertainty or explicit hypothesis
testing based on the region of practical equivalence (ROPE). An illustrative
example, theoretical results and a simulation study show the usefulness of the
proposed method, and the test is made available in the R package bayest.
