  Entanglement is often verified by a violation of an inequality like a Bell
inequality or an entanglement witness. Considerable effort has been devoted to
the optimization of such inequalities in order to obtain a high violation. We
demonstrate theoretically and experimentally that such an optimization does not
necessarily lead to a better entanglement test, if the statistical error is
taken into account. Theoretically, we show for different error models that
reducing the violation of an inequality can improve the significance.
Experimentally, we observe this phenomenon in a four-photon experiment, testing
the Mermin and Ardehali inequality for different levels of noise. Furthermore,
we provide a way to develop entanglement tests with high statistical
significance.
