In this paper, we present the U-Statistic test for testing exponentiality against new renewal better than used (NRBU) based on a goodness of fit approach. Selected critical values are tabulated for sample sizes n=5(1)30(10)50. The asymptotic Pitman relative efficiency relative to (NRBU) test given in the work of Mahmoud et all (2003) is studied. The power estimates of this test for some commonly used life distributions in reliability are also calculated. Some of real examples are given to elucidate the use of the proposed test statistic in the reliability analysis. The problem in case of right censored data is also handled.