Given samples from two non-negative random variables, we propose a family of tests for the null hypothesis that one random variable stochastically dominates the other at the second order. Test statistics are obtained as functionals of the difference between the identity and the Lorenz P-P plot, defined as the composition between the inverse unscaled Lorenz curve of one distribution and the unscaled Lorenz curve of the other. We determine upper bounds for such test statistics under the null hypothesis and derive their limit distribution, to be approximated via bootstrap procedures. We then establish the asymptotic validity of the tests under relatively mild conditions and investigate finite sample properties through simulations. The results show that our testing approach can be a valid alternative to classic methods based on the difference of the integrals of the cumulative distribution functions, which require bounded support and struggle to detect departures from the null in some cases.