The authors address the interannual signal in the tropical mean temperature of the free troposphere. This signal was shown by Soden to be well simulated by a wide range of general circulation models, given the observed sea surface temperature (SST) as a boundary condition. This suggests that the mechanism by which SST anomalies are communicated to the atmosphere might be understood simply as moist convective adjustment. This in turn suggests the hypothesis that the free tropospheric temperature might be more sensitive to SST anomalies in regions of high mean SST and frequent deep convection than anomalies in regions of low mean SST and rare deep convection. However, it is shown that a time series of interannual SST anomalies averaged over convective regions only is very similar to the time series of total tropical mean SST anomalies, and both are very similar to the time series of tropospheric temperature anomalies (all for the period 1982-1998), so that this hypothesis is not easily testable using observations. Several atmospheric models are driven by observed SST fields for the period 1982-1998: the Quasi-equilibrium Tropical Circulation Model (QTCM) developed by Neelin and Zeng, a one-dimensional reduced model derived from that model but making a number of additional strong simplifying assumptions, and a zero-dimensional model which simply assumes moist convective adjustment to the {\em tropical mean} SST. All three models simulate the tropospheric temperature signal fairly well. These and the observational results show that the interannual tropospheric temperature signal is consistent with moist convective adjustment, and that for purposes of explaining this observed signal at a gross level one need not consider subtleties associated with changes in the distribution of SST (as opposed to its mean value) because the histogram of SST shifts fairly uniformly with interannual variability. There is, however, a slight suggestion in both the observational and model results that changes in the histogram shape may be significant for the tropospheric temperature anomalies associated with the largest El Nino events, and that in those events it is indeed the SST anomalies in the convective regions which are most important in controlling the tropospheric temperature.