To test the cooling flow model of early-type galaxies, we obtained a complete magnitude-limited sample of 34 early-type galaxies, observed with the PSPC and HRI on ROSAT. The X-ray to optical distribution of galaxies implies a lower envelope that is consistent with the stellar emission inferred from Cen A. When this stellar component is removed, the gaseous emission is related to the optical luminosity by Lx proportional to L_B^m, where m = 3.0-3.5, significantly steeper than the standard theory (m = 1.7). The dispersion about the correlation is large, with a full range of 30-100 in Lx for a fixed L_B. The X-ray temperature is related to the velocity dispersion temperature as Tx proportional to Tsigma^n, where n = 1.43 +/- 0.21, although for several galaxies, Tx is about twice Tsigma. The excessively hot galaxies are generally the most luminous and are associated with the richest environments. We suggest a model whereby environment influences the X-ray behavior of these galaxies: early-type galaxies attempt to drive partial or total galactic winds, which can be stifled by the pressure of their environment. Stifled winds should lead to hotter and higher luminosity systems, which would occur most commonly in the richest environments, as observed.