In this paper, we use the observed mass splittings among low-lying baryons (i.e., the flavor-SU(3) octet and decuplet) to perform an extensive search for chiral-bag-model parameters including the strong coupling constant ..cap alpha../sub s/ (equivalent(4..pi..)/sup -1/g/sup 2/), the effective pion-quark coupling ..cap alpha../sub ..pi../ (equivalent(4..pi..)/sup -1/g/sub piQ/ /sup 2/), the quark mass differences m/sub d/-m/sub u/ and m/sub s/-m/sub d/, and the nucleon bag radius R/sub N/. It is found that a remarkably good simultaneous fit exists at a nucleon bag radius of around 0.45 fm for the baryon mass splittings including n-p, ..sigma../sup -/-..sigma../sup +/, ..xi../sup -/-..xi../sup 0/, ..sigma../sup 0/-..lambda.., ..sigma../sup -asterisk/-..sigma../sup +asterisk/, ..xi../sup -asterisk/-..xi../sup 0asterisk/, ..delta../sup + +/-p, ..xi../sup 0asterisk/-p, and ..sigma../sup +asterisk/-p. In addition, the quality of the fit, as measured in terms of a chi/sup 2/ defined in the text, is worsened by almost two orders of magnitude if the nucleon bag radius is chosen to be greater than approx.0.8 fm or less than approx.0.3 fm or if the effective pion-quark coupling is set to zero.