The exponential better (worse) than used EBU (EWU) class of life distributions is considered. A moment inequality is derived for EBU (EWU) distributions which demonstrate that if the mean life is finite, then all moments exist. Based on this inequality, a new test statistic for testing exponentiality against EBU (EWU) is introduced. It is shown that the proposed test is simple, enjoys good power and has high relative efficiency for some commonly used alternatives. Critical values are tabulated for sample sizes n = 5(1)40. A set of real data is used as a practical application of the proposed test in the medical science.