Drell-Yan lepton pairs are produced in the process $pbar{p} rightarrow mu^+mu^- + X$ through an intermediate $gamma^*/Z$ boson. The forward-backward asymmetry in the polar-angle distribution of the $mu^-$ as a function of the invariant mass of the $mu^+mu^-$ pair is used to obtain the effective leptonic determination $sin^2 theta^{lept}_{eff}$ of the electroweak-mixing parameter $sin^2 theta_W$, from which the value of $sin^2 theta_W$ is derived assuming the standard model. The measurement sample, recorded by the Collider Detector at Fermilab (CDF), corresponds to 9.2 fb-1 of integrated luminosity from $pbar{p}$ collisions at a center-of-momentum energy of 1.96 TeV, and is the full CDF Run II data set. The value of $sin^2 theta^{lept}_{eff}$ is found to be 0.2315 +- 0.0010, where statistical and systematic uncertainties are combined in quadrature. When interpreted within the context of the standard model using the on-shell renormalization scheme, where $sin^2 theta_W = 1 - M_W^2/M_Z^2$, the measurement yields $sin^2 theta_W$ = 0.2233 +- 0.0009, or equivalently a W-boson mass of 80.365 +- 0.047 GeV/c^2. The value of the W-boson mass is in agreement with previous determinations in electron-positron collisions and at the Tevatron collider.
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