We introduce a new two-parameter model related to the inverted Topp–Leone distribution called the power inverted Topp–Leone (PITL) distribution. Major properties of the PITL distribution are stated; including; quantile measures, moments, moment generating function, probability weighted moments, Bonferroni and Lorenz curve, stochastic ordering, incomplete moments, residual life function, and entropy measure. Acceptance sampling plans are developed for the PITL distribution, when the life test is truncated at a pre-specified time. The truncation time is assumed to be the median lifetime of the PITL distribution with pre-specified factors. The minimum sample size necessary to ensure the specified life test is obtained under a given consumer’s risk. Numerical results for given consumer’s risk, parameters of the PITL distribution and the truncation time are obtained. The estimation of the model parameters is argued using maximum likelihood, least squares, weighted least squares, maximum product of spacing and Bayesian methods. A simulation study is confirmed to evaluate and compare the behavior of different estimates. Two real data applications are afforded in order to examine the flexibility of the proposed model compared with some others distributions. The results show that the power inverted Topp–Leone distribution is the best according to the model selection criteria than other competitive models.