In this paper, a family of statistical models, namely, a new exponential-X family is proposed. A sub-case of the introduced family, called the new exponential-Weibull (NE-Weibull) model, is studied. The NE-Weibull model is very competent and possessing heavy-tailed properties. The maximum likelihood estimators of its parameters are derived. The consistency and efficiency of these estimators are assessed in a brief simulation study. Finally, the effectiveness of the NE-Weibull distribution is illustrated by modeling real insurance claims data. The practical analysis shows that the NE-Weibull distribution outclassed other distributions and it can be a better choice for modeling data in the finance sector.