  We report correlation measurements on two $^9$Be$^+$ ions that violate a
chained Bell inequality obeyed by any local-realistic theory. The correlations
can be modeled as derived from a mixture of a local-realistic probabilistic
distribution and a distribution that violates the inequality. A statistical
framework is formulated to quantify the local-realistic fraction allowable in
the observed distribution without the fair-sampling or
independent-and-identical-distributions assumptions. We exclude models of our
experiment whose local-realistic fraction is above 0.327 at the 95 \%
confidence level. This bound is significantly lower than 0.586, the minimum
fraction derived from a perfect Clauser-Horne-Shimony-Holt inequality
experiment. Furthermore, our data provides a device-independent certification
of the deterministically created Bell states.
