We exploit a set of high signal-to-noise (~70), low-resolution (R~800) quasar spectra to search for the signature of the so-called proximity effect in the HI Ly alpha forest. Our sample consists of 17 bright quasars in the redshift range 2.7<z<4.1. Analysing the spectra with the flux transmission technique, we detect the proximity effect in the sample at high significance. We use this to estimate the average intensity of the metagalactic UV background, assuming it to be constant over this redshift range. We obtain a value of J = (9+-4)x10^{-22}ergcm^{-2}s^{-1}Hz^{-1}sr^{-1}, in good agreement with previous measurements at similar z. We then apply the same procedure to individual lines of sight, finding a significant deficit in the effective optical depth close to the emission redshift in every single object except one (which by a different line of evidence does nevertheless show a noticeable proximity effect). Thus, we clearly see the proximity effect as a universal phenomenon associated with individual quasars. Using extensive Monte-Carlo simulations to quantify the error budget, we assess the expected statistical scatter in the strength of the proximity effect due to shot noise (cosmic variance). The observed scatter is larger than the predicted one, so that additional sources of scatter are required. We rule out a dispersion of spectral slopes as a significant contributor. Possible effects are long time-scale variability of the quasars and/or gravitational clustering of Ly alpha forest lines. We speculate on the possibility of using the proximity effect as a tool to constrain individual quasar ages, finding that ages between ~10^6 and ~10^8 yrs might produce a characteristic signature in the optical depth profile towards the QSO. We identify one possible candidate for this effect in our sample.
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