  We investigate the transverse momentum weighted Sivers asymmetries in the
processes with transversely polarized proton target, including the
$\frac{P_{hT}}{zM_p}\sin (\phi_h-\phi_S)$ weighted asymmetry in charged hadron
production in semi-inclusive deeply inelastic scattering (SIDIS) and the
$\frac{q_T}{M_p}\sin (\phi_S)$ weighted asymmetry in $\pi^- p$ Drell-Yan
process. Due to the integration over the transverse momentum, the weighted
asymmetries can be expressed as the product of the transverse-moments of the
transverse momentum dependent (TMD) parton distribution functions (PDFs) and
fragmentation functions (FFs). Using the parametrization for the Sivers
function of the proton, the unpolarized PDFs of proton and pion, and the
unpolarized FF of charged hadron, we present the numerical calculation for
weighted Sivers asymmetries in SIDIS and Drell-Yan processes at the kinematics
of COMPASS, and compare them with experimental data. We find that our
prediction on the weighted Sivers asymmetry in SIDIS process is in agreement
with the recent COMPASS measurement. Due to the relatively large uncertainties
of the preliminary data from COMPASS Drell-Yan program, high precision
experimental data are needed to test the sign change property of the Sivers
function between SIDIS and Drell-Yan processes and to constrain the sea quark
Sivers function.
