In the simple linear regression model Y = .alpha.$_{0}$ + .beta.$_{0}$Z + .epsilon. under the right censorship of the response variables, the estimation of the mean lifetime E(Y) is an interesting problem. In this paper we propose a method of estimating E(Y) based on the observations modified by the arguments of Buckley and James (1979). It is shown that the proposed estimator is consistent and our proposed procedure in the simple linear regression case can be naturally extended to the multiple linear regression. Finally, we perform simulation studies to compare the proposed estimator with the estimator introduced by Gill (1983).83).