In this paper we investigate trade-offs between speed and accuracy that are produced by humans when confronted with a sequence of choices between two alternatives. We assume that the choice process is described by the drift diffusion model, in which the speed-accuracy trade-off is primarily controlled by the value of the decision threshold. We test the hypothesis that participants choose the decision threshold that maximizes reward rate, defined as an average number of rewards per unit of time. In particular, we test four predictions derived on the basis of this hypothesis in two behavioural experiments. The data from all participants of our experiments provide support only for some of the predictions, and on average the participants are slower and more accurate than predicted by reward rate maximization. However, when we limit our analysis to subgroups of 30-50% of participants who earned the highest overall rewards, all the predictions are satisfied by the data. This suggests that a substantial subset of participants do select decision thresholds that maximize reward rate. We also discuss possible reasons why the remaining participants select thresholds higher than optimal, including the possibility that participants optimize a combination of reward rate and accuracy or that they compensate for the influence of timing uncertainty, or both.