Median survival times and their associated confidence intervals are often used to summarize the survival outcome of a group of patients in clinical trials with failure-time endpoints. Although there is an extensive literature on this topic for the case in which the patients come from a homogeneous population, few papers have dealt with the case in which covariates are present as in the proportional hazards model. In this talk we propose a new test-based approach to this problem and demonstrate its advantages over existing methods, not only for the proportional hazards model but also for the widely studied cases where covariates are absent and where there is no censoring. Asymptotic theory and simulation studies show that the proposed method indeed yields confidence intervals with accurate coverage errors. The test-based approach is extended to handle the more difficult problem of constructing confidence intervals for the regression parameter of the Cox model following a time-sequential clinical trial with censored survival data. Asymptotic theory and simulation studies show that the confidence intervals thus constructed have coverage probabilities close to the nominal values.