Abstract: Let $X = (X_1,X_2, \cdots, X_m)$ and $Y = (Y_1, Y_2, \cdots , Y_n)$ be independent random observations from two normal populations with a common unknown mean $\mu$ and possibly different variances $\sigma_1^2$ and $\sigma_2^2$ respectively. Estimation of $\mu$ and quantile of the first population with respect to a quadratic loss function is considered. In the case of estimation of $\mu$, sufficient conditions for the inadmissibility of estimators equivariant under location, scale and affine groups of transformations have been derived. A massive simulation study with a high precision is done to numerically compare the risk performance of some well known estimators for the common mean. In the case of quantile estimation, some new estimators for the quantile are proposed using some previously known estimators of a common mean. Inadmissibility results are proved for estimators which are equivariant under affine and location groups of transformations. Risk values of various estimators of a quantile are compared numerically using a detailed simulation study.