Errors in observing and reporting sample data often complicate the problem of estimating parameters of the distribution being sampled. If neglected, such errors may lead to seriously biased estimates. There exits a large general class of such estimation problems involving numerous different distributions, different types and varying degrees of observational errors. This paper is limited, however, to maximum likelihood estimation in a Poisson distribution which has been modified to the extent that a proportion θ of the ones are reported as being zeros. An inspector who sometimes fails to see or at least fails to report items containing only a single Poisson distributed defect, while correctly observing and reporting results of inspecting items containing two or more defects, produces sample data of the type under consideration. Estimators are derived both for the Poisson parameter and for θ. Asymptotic variances and covariances are derived and an illustrative example is included.