The Rasch model consistent with the Poisson model, when we consider the mean of the Poisson model as the ratio of an ability to a difficulty parameter with dichotomous data, is known as one popular model for measurement situations in educational fields. However, we investigate problems of the Poisson model from data with excess zeros. In such a situation, we concluded that the zero-inflated Poisson (ZIP) model could be an effective alternative to the Poisson model. In this study, we derive some specific guidelines on the use of the ZIP model by comparing the model fit between this and the Poisson model based on empirical as well as simulated data. Specifically, in the empirical study, the MLE method generally provided good estimates, although w and l estimates obtained from the MME and the MLE methods displayed patterns similar to their true values. In the simulation study, the null hypothesis, the Poisson model, was rejected using all test statistics from S1 to S5 if w was larger than or equal to 0.5 and l was larger than or equal to 1.0. In the aspect of the AIC and the BIC, the ZIP model fitted well comparing to the Poisson model for all w and l values except that w was 0.14 and l was 0.25. We also found that the critical proportions of zeros were 0.7. In most cases, the difference between the Poisson model and the ZIP model increased at this point, after which it decreased. To sum up, when the ZIP model is adapted, we have to consider the proportions of zeros and the mean of the Poisson model; however, there are situations where the ZIP model is not required.