We considerer a sequential, response-adaptive design for clinical trials which is characterized by the fact that it assigns patients to the best treatment with a probability converging to one. This property is optimal from an ethical point of view; in this paper we analyze some inferential problems related to the design. In particular, we want to establish, by means of a test of hypothesis, which treatment is superior, in the sense that it has greater mean response. Together with the natural generalization of the classical t-statistic, we introduce a statistic based on the probability of assigning patients to a treatment conditional on past observations. Theoretical properties of the tests are studied, together with numerical evaluations of the power for dichotomous responses.