According to a currently popular paradigm, nuclear activity in quasars is sustained via accretion of material onto super-massive black holes located at the quasar nuclei. A useful tracer of the gravitational field in the vicinity of such central black holes is available in the form of extremely dense gas clouds within the broad emission-line region (BLR) on the scale of ~ 1 parsec. Likewise, the radio sizes of the lobe-dominated radio sources are believed to provide a useful statistical indicator of their ages. Using two homogeneously observed (and processed) sets of lobe-dominated radio-loud quasars, taken from literature, we show that a positive correlation exists between the radio sizes of the quasars and the widths of their broad Hβ emission lines, and this correlation is found to be significantly stronger than the other well known correlations involving radio size. This statistical correlation is shown to be consistent with the largest (and, hence, very possibly the oldest) radio sources harboring typically an order-of-magnitude more massive central engines, as compared to the physically smaller and, hence, probably much younger radio sources. This inference is basically in accord with the "accreting central engine" picture for the radio-loud quasars.