The mass-dependent symmetry energy coefficients asym（A） has been extracted by analysing the heavy nuclear mass differences reducing the uncertainties as far as possible in our previous work. Taking advantage of the obtained symmetry energy coefficient asym（A） and the density profiles obtained by switching off the Coulomb interaction in ^208Pb, we calculated the slope parameter L0.11 of the symmetry energy at the density of 0.11 fm^-3. The calculated L0.11 ranges from 40.5 MeV to 60.3 MeV. The slope parameter L0.11 of the symmetry energy at the density of 0.11 fm^-3 is also calculated directly with Skyrme interactions for nuclear matter and is found to have a fine linear relation with the neutron skin thickness of ^208spb, which is the difference of the neutron and proton rms radii of the nucleus. With the linear relation the neutron skin thickness ARnp of ^208spb is predicted to be 0.15-0.21 fm.
