I have simulation results of 2 methods, say A and B. As a means of comparison, I have plotted the sample means and corresponding 95% confidence intervals.
Now I would like to get a confidence level for the following statement: "Method A performs better than B", based on the simulation results. The rule of thumb is to state that A performs better than B if their confidence intervals do not overlap, or to be inconclusive when they do overlap. Yet I would like to be able to express more accurate whether A performs better than B.
So in short, I would like to make a statement like: with a confidence of X%, A performs better than B. But how to obtain X?

Does anyone know how to do this? Please note that my knowledge about probability theory is limited, so please provide an example or a reference.