The Pan-STARRS (PS1) Medium Deep Survey discovered over 5,000 likely supernovae (SNe) but obtained spectral classifications for just 10% of its SN candidates. We measured spectroscopic host galaxy redshifts for 3,147 of these likely SNe and estimate that $sim$1,000 are Type Ia SNe (SNe Ia) with light-curve quality sufficient for a cosmological analysis. We use these data with simulations to determine the impact of core-collapse SN (CC SN) contamination on measurements of the dark energy equation of state parameter, $w$. Using the method of Bayesian Estimation Applied to Multiple Species (BEAMS), distances to SNe Ia and the contaminating CC SN distribution are simultaneously determined. We test light-curve based SN classification priors for BEAMS as well as a new classification method that relies upon host galaxy spectra and the association of SN type with host type. By testing several SN classification methods and CC SN parameterizations on large SN simulations, we estimate that CC SN contamination gives a systematic error on $w$ ($sigma_w^{CC}$) of 0.014, 29% of the statistical uncertainty. Our best method gives $sigma_w^{CC} = 0.004$, just 8% of the statistical uncertainty, but could be affected by incomplete knowledge of the CC SN distribution. This method determines the SALT2 color and shape coefficients, $alpha$ and $beta$, with $sim$3% bias. However, we find that some variants require $alpha$ and $beta$ to be fixed to known values for BEAMS to yield accurate measurements of $w$. Finally, the inferred abundance of bright CC SNe in our sample is greater than expected based on measured CC SN rates and luminosity functions.
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