The statistical mechanics of polypeptide folding are studied by Monte Carlo simulations on a lattice protein model. Effective sampling of the protein conformations over the folding transition is achieved by an entropy sampling procedure, combined with a stochastically biased conformation-generating scheme and a jump-walking algorithm. Comparative analyses are carried out for optimally designed sequences and randomly selected sequences in terms of four properties : (a) the thermal folding/unfolding transitions of the different sequences; (b) the distributions of conformational populations at the corresponding transition temperatures of the sequences; (c) the fitting of the simulation results to an analytical theory; and (d) the distributions of local energy minima and the energy barriers between them in different conformational regions. It is found that all the polypeptides studied here exhibit rather sharp transitions at their folding temperatures. However, the designed/optimized sequences exhibit long-range cooperative behavior in the folding transition, while the random sequences reflect only short-range (i.e., nearest-neighbor) cooperative phenomena. Furthermore, it is shown that the optimized sequences fold to unique lowest-energy structures, but that the random sequences, when cooled, fold to compact but in general random structures. This study demonstrates that whether or not a polypeptide folds to the lowest-energy structure can be determined from its sequence by statistical mechanical characteristics. The implication of biological evolution in determining protein sequences is discussed.