Mediating/intervening variable effects can be examined and demonstrated in a variety of ways in psychological research. One approach is the so-called difference in coefficients method, where the significance of the difference between the direct path from X to Y with no mediated path via M and the direct path from X to Y when a mediated path (X->M->Y) exists is tested. The attached simple Excel worksheet provides an easy way to do this using the Freedman-Schatzkin approach to estimating the standard error of the difference. The sources used are detailed in the worksheet.
As mentioned, there are many other ways to examine mediator effects, so only use this if you know what you’re doing. This method appears to be superior to, e.g., the old Baron & Kenny steps approach, but personally, I’d recommend boostrapping for most uses.