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For MAX SAT, which is a well-known NP-hard problem, many approximation algorithms have been proposed. Two types of best approximation algorithms for MAX SAT were proposed by Asano and Williamson: one with best proven performance guarantee 0.7846 and the other with performance guarantee 0.8331 if the performance guarantee of the Zwick's algorithm for MAX SAT is, as conjectured. Both algorithms are based on their sharpened analysis of Goemans and Williamson's LP-relaxation for MAX SAT. In this paper, we present an improved analysis which is simpler than the previous analysis. Furthermore, algorithms based on this, analysis will play a role as a better building block in designing an improved approximation algorithm for MAX SAT. Actually we, show an example that algorithms based on this analysis lead to approximation algorithms with performance guarantee 0.7877 and, conjectured performance guarantee 0.8353 which are slightly better than the best known corresponding performance guarantees, 0.7846 and 0.8331, respectively.