We examine the effect of an embedding medium refractive index on the plasmon resonant properties of silica core-gold shell nanoshells. The plasmon response is shifted to longer wavelengths with increasing refractive index of the dielectric host matrix, increasing in overall amplitude for nanoparticles in the dipole limit. For nanoshells of constant core-shell ratio, this plasmon shift increases with absolute particle size. We also observe that the plasmon shift is the same for nanoparticles of the same size, independent of core-shell ratio. For larger nanoshells we observe an increase in amplitude of the quadrupole plasmon resonance relative to the dipole plasmon.