Using molecular dynamics simulation of a standard coarse-grained polymer glass model, we investigate by means of the stress-fluctuation formalism the shear modulus μ as a function of temperature T and sampling time Δt. While the ensemble-averaged modulus μ(T) is found to decrease continuously for all Δt sampled, its standard deviation δμ(T) is nonmonotonic, with a striking peak at the glass transition. Confirming the effective time-translational invariance of our systems, μ(Δt) can be understood using a weighted integral over the shear-stress relaxation modulus G(t). While the crossover of μ(T) gets sharper with an increasing Δt, the peak of δμ(T) becomes more singular. It is thus elusive to predict the modulus of a single configuration at the glass transition.