  A key objective in conducting a Bell test is to quantify the statistical
evidence against a local-hidden variable model (LHVM) given that we can collect
only a finite number of trials in any experiment. The notion of statistical
evidence is thereby formulated in the framework of hypothesis testing, where
the null hypothesis is that the experiment can be described by an LHVM. The
statistical confidence with which the null hypothesis of an LHVM is rejected is
quantified by the so-called P-value, where a smaller P-value implies higher
confidence. Establishing good statistical evidence is especially challenging if
the number of trials is small, or the Bell violation very low. Here, we derive
the optimal P-value for a large class of Bell inequalities. What's more, we
obtain very sharp upper bounds on the P-value for all Bell inequalities. These
values are easily computed from experimental data, and are valid even if we
allow arbitrary memory in the devices. Our analysis is able to deal with
imperfect random number generators, and event-ready schemes, even if such a
scheme can create different kinds of entangled states. Finally, we review
requirements for sound data collection, and a method for combining P-values of
independent Bell experiments.
