In order to understand the role of radio-quiet quasars (RQQs) in galaxy evolution, we must determine the relative levels of accretion and star-formation activity within these objects. Previous work at low radio flux-densities has shown that accretion makes a significant contribution to the total radio emission, in contrast with other quasar studies that suggest star formation dominates. To investigate, we use 70 RQQs from the Spitzer-Herschel Active Galaxy Survey. These quasars are all at $z$ ~ 1, thereby minimising evolutionary effects, and have been selected to span a factor of ~100 in optical luminosity, so that the luminosity dependence of their properties can be studied. We have imaged the sample using the Karl G. Jansky Very Large Array (JVLA), whose high sensitivity results in 35 RQQs being detected above 2 $sigma$. This radio dataset is combined with far-infrared luminosities derived from grey-body fitting to Herschel photometry. By exploiting the far-infrared--radio correlation observed for star-forming galaxies, and comparing two independent estimates of the star-formation rate, we show that star formation alone is not sufficient to explain the total radio emission. Considering RQQs above a 2-$sigma$ detection level in both the radio and the far-infrared, 92 per cent are accretion-dominated, and the accretion process accounts for 80 per cent of the radio luminosity when summed across the objects. The radio emission connected with accretion appears to be correlated with the optical luminosity of the RQQ, whilst a weaker luminosity-dependence is evident for the radio emission connected with star formation.
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