Scattering cross sections were precisely measured for natural carbon and polyethylene samples at neutron energies of 6.23, 7.21, 8.16, 9.16, 11.05, 11.90, 12.84 and 13.75 MeV. A proton recoil telescope served as a neutron fluence reference detector. Measured neutron tim-of-flight spectra were compared with Monte-Carlo simulations performed on the basis of evaluated cross sections (ENDF/B-V). For the elastic and inelastic scattering on carbon the ratios of masured to expected yields ranged from 0.6 to 1.8, while the neutron-proton scattering was consistently confirmed within the uncertainties of 2.5 to 6% estimated for these experiments.