The new better than used failure rate (NBUFR), Abouammoh and Ahmed (1988), and new better than average failure rate (NBAFR) Loh (1984) classes of life distributions, have been considered in the literature as natural weakenings of NBU (NWU) property. The paper considers testing exponentiality against strictly NBUFR (NBAFR) alternatives, or their duals, based on goodness of fit approach that is possible in life testing problems and that it results in simpler procedures that are asymptotically equivalent or better than standard ones. They may also have superior finite sample behavior. The asymptotic normality are proved. Powers, Pitman asymptotic efficiency and critical points are computed. Dealing with censored data case also studied. Practical applications of our tests in the medical sciences are present.