  By studying the Paschos-Wolfenstein (PW) ratio of deep inelastic $\nu{\rm
Fe}$ and $\bar \nu{\rm Fe}$ scattering cross sections, we show that it should
be possible to explain the NuTeV $\sin ^2 \theta_{\rm W}$ anomaly with quite
conventional physics, by introducing mutually different nuclear modifications
for the valence-$u$ and valence-$d$ quark distributions of the protons in iron.
Keeping the EKS98 nuclear modifications for $u_V+d_V$ as a baseline, we find
that some 20-30 % nuclear modifications to the $u_V$ and $d_V$ distributions
account for the change induced in the PW ratio by the NuTeV-suggested increase
$\Delta \sin ^2 \theta_{\rm W}=0.005$. We show that introduction of such
nuclear modifications in $u_V$ and $d_V$ individually, does not lead into
contradiction with the present global DGLAP analyses of the nuclear parton
distributions, where deep inelastic $lA$ scattering data and Drell-Yan dilepton
data from $pA$ collisions are used as constraints. We thus suggest that the
NuTeV result serves as an important further constraint in pinning down the
nuclear effects of the bound nucleon PDFs. We also predict that if the NuTeV
anomaly is explained by this mechanism, the NOMAD experiment should see an
increase in the weak mixing angle quite close to the NuTeV result.
