In this paper, a new class of life distribution is named new better than renewal used in moment generating function (NBRUmgf). A new test for exponentiality versus NBRUmgf based on moment inequalities is established. Pitman's asymptotic efficiencies, powers and critical values of the new test are calculated to assess the performance of the test. The right censored data is handled also. Finally some applications are applied to the new test.