  The present paper concludes our investigations on the QCD cross-over
transition temperatures with 2+1 staggered flavours and one-link stout
improvement. We extend our previous two studies [Phys. Lett. B643 (2006) 46,
JHEP 0906:088 (2009)] by choosing even finer lattices ($N_t$=16) and we work
again with physical quark masses. The new results on this broad cross-over are
in complete agreement with our earlier ones. We compare our findings with the
published results of the hotQCD collaboration. All these results are confronted
with the predictions of the Hadron Resonance Gas model and Chiral Perturbation
Theory for temperatures below the transition region. Our results can be
reproduced by using the physical spectrum in these analytic calculations. The
findings of the hotQCD collaboration can be recovered by using a distorted
spectrum which takes into account lattice discretization artifacts and heavier
than physical quark masses. This analysis provides a simple explanation for the
observed discrepancy in the transition temperatures between our and the hotQCD
collaborations.
