  After correcting for their light-curve shape and color, Type Ia supernovae
(SNe Ia) are precise cosmological distance indicators. However, there remains a
non-zero intrinsic scatter in the differences between measured distance and
that inferred from a cosmological model (i.e., Hubble residuals or HRs),
indicating that SN Ia distances can potentially be further improved. We use the
open-source relational database kaepora to generate composite spectra with
desired average properties of phase, light-curve shape, and HR. At many phases,
the composite spectra from two subsamples with positive and negative average
HRs are significantly different. In particular, in all spectra from 9 days
before to 15 days after peak brightness, we find that SNe with negative HRs
have, on average, higher ejecta velocities (as seen in nearly every optical
spectral feature) than SNe with positive HRs. At +4 days relative to B-band
maximum, using a sample of 62 SNe Ia, we measure a 0.091 +/- 0.035 mag HR step
between SNe with Si II 6355 line velocities higher/lower than -11,000 km/s (the
median velocity). After light-curve shape and color correction, SNe with higher
velocities tend to have underestimated distance moduli relative to a
cosmological model. The intrinsic scatter in our sample reduces from 0.094 mag
to 0.082 mag after making this correction. Using the Si II 6355 velocity
evolution of 115 SNe Ia, we estimate that a velocity difference > 500 km/s
exists at each epoch between the positive-HR and negative-HR samples with 99.4%
confidence. Finally at epochs later than +37 days, we observe that negative-HR
composite spectra tend to have weaker spectral features in comparison to
positive-HR composite spectra.
