ABSTRACT: Linear regression analysis is a statistical phenomenon in order to evaluate the association between the variables. Multiple linear regression models are the one in which there is one dependent variable and more than one independent variables. Regression analysis is an important tool to identify and characterize the relationshipsof multiple factors. The goal of this article is to introduce some methods and applications of linear regression models. The central concepts in linear regression analysis namely estimation theory, maximum likelihood, and linear hypothesis are comprehensively discussed. Moreover an innovative proof of Gauss –Markov theorem in full rank case has been proposed here.