A new four-parameter lifetime distribution called the beta binomial exponential 2 (BBE2) distribution is proposed. Some mathematical features, including quantile function, moments, generating function and characteristic function, of the BBE2 distribution, are computed. When the life test is truncated at a predetermined time, acceptance sampling plans (ASP) are constructed for the BBE2 distribution. The truncation time is supposed to represent the median lifetime of the BBE2 distribution with predetermined factors for the smallest sample size required to guarantee that the prescribed life test is achieved at a given consumer’s risk. Some numerical results for a given consumer’s risk, BBE2 distribution parameters and truncation time are derived. Classical (maximum likelihood and maximum product of spacing estimation methods) and Bayesian estimation approaches are utilized to estimate the model parameters. The performance of the model parameters is examined through the simulation study by using the three different approaches of estimation. Subsequently, we examine real-world data applications to demonstrate the versatility and potential of the BBE2 model. A real-world application demonstrates that the new distribution can offer a better fit than other competitive lifetime models.