In this paper, a new class of life distribution namely exponential better than used in convex average and denoted by (EBUCA), or its dual (EWUCA) is introduced. Testing exponentiality against ( EBUCA) based on U-statistics is investigated. The percentiles of these tests are tabulated for samples sizes n=5 (1) 40. The power of the test is simulated for some commonly used distributions in reliability. Pitman's asymptotic efficiency of the test is calculated and compared. Data for 40 patients suffering from blood cancer disease (Leukemia) is considered as a practical application of the proposed test on medical data.