Count data models are based on definite counts of events as dependent variables. But there are practical situations in which these counts may fail to be specific and are seen as imprecise. In this paper, an assumption that heaped data points are fuzzy is used as a way of identifying counts that are not definite since heaping can result from imprecisely reported counts. Because it is practically unlikely to report all counts in an entire dataset as imprecise, this paper proposes a likelihood function that not only considers both precise and imprecisely reported counts but also incorporates alpha - cuts of fuzzy numbers with the aim of varying impreciseness of fuzzy reported counts. The proposed model is then illustrated through a smoking cessation study data that attempts to identify factors associated with the number of cigarettes smoked in a month. Through the real data illustration and a simulation study, it is shown that the proposed model performs better in predicting the outcome counts especially when the imprecision of the fuzzy points in a dataset are increased. The results also show that inclusion of alpha - cuts makes it possible to identify better models, a feature that was not previously possible.