Adaptive designs are commonly used in clinical and drug development studies for optimum utilization of available resources. In this article, we consider the problem of estimating the effect of the selected (better) treatment using a two-stage adaptive design. Consider two treatments with their effectiveness characterized by two normal distributions having different unknown means and a common unknown variance. The treatment associated with the larger mean effect is labeled as the better treatment. In the first stage of the design, each of the two treatments is independently administered to different sets of $n_1$ subjects, and the treatment with the larger sample mean is chosen as the better treatment. In the second stage, the selected treatment is further administered to $n_2$ additional subjects. In this article, we deal with the problem of estimating the mean of the selected treatment using the above adaptive design. We extend the result of \cite{cohen1989two} by obtaining the uniformly minimum variance conditionally unbiased estimator (UMVCUE) of the mean effect of the selected treatment when multiple observations are available in the second stage. We show that the maximum likelihood estimator (a weighted sample average based on the first and the second stage data) is minimax and admissible for estimating the mean effect of the selected treatment. We also propose some plug-in estimators obtained by plugging in the pooled sample variance in place of the common variance $σ^2$, in some of the estimators proposed by \cite{misra2022estimation} for the situations where $σ^2$ is known. The performances of various estimators of the mean effect of the selected treatment are compared via a simulation study. For the illustration purpose, we also provide a real-data application.