  The scenario consistent with a wealth of observations for the missing mass
problem is that of weakly interacting dark matter particles. However, arguments
or proposals for a Newtonian or relativistic modified gravity scenario continue
to be made. A distinguishing characteristic between the two scenarios is that
dark matter particles can produce a gravitational effect, in principle, without
the need of baryons while this is not the case for the modified gravity
scenario where such an effect must be correlated with the amount of baryonic
matter. We consider here ultra-faint dwarf (UFD) galaxies as a promising arena
to test the two scenarios based on the above assertion. We compare the
correlation of the luminosity with the velocity dispersion between samples of
UFD and non-UFD galaxies, finding a significant loss of correlation for UFD
galaxies. For example, we find for 28 non-UFD galaxies a strong correlation
coefficient of -0.688 which drops to -0.077 for the 23 UFD galaxies. Incoming
and future data will determine whether the observed stochasticity for UFD
galaxies is physical or due to systematics in the data. Such a loss of
correlation (if it is to persist) is possible and consistent with the dark
matter scenario for UFD galaxies but would constitute a new challenge for the
modified gravity scenario.
