A universal combination of amplitudes for the end-to-end distance of N-step self-avoiding walks and for the number of N step self-avoiding polygons is estimated numerically for various lattices. In d=2, universality of this amplitude combination is confirmed to good accuracy, while in d=3, the data are consistent with universality but the error limits are rather large. For finite-width, square-lattice strips the authors calculate universal critical point correlation amplitudes by fixed-fugacity Monte Carlo simulations. In addition, these amplitudes are calculated analytically either exactly or approximately by Cardy's conformal mapping approach. Numerical values agree well with the theoretical predictions.