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Count data are commonly found in a wide range of disciplines from medical to social sciences. Regarding an interval estimation for a mean, researchers tend to assume that count data have a Poisson distribution, in which the mean and variance are equal and the interarrival time has an Exponential distribution. A plausible explanation is that the confidence interval for the Poisson mean is simpler to compute and readily found in virtually all statistical programs. However, when count data are generated from a renewal process, the interarrival time of count data can possess any distribution. This paper, therefore, aims to study the performance of seven confidence intervals constructed for the Poisson mean, provided interarrival time is set to be Gamma or Weibull distribution. Because of the given distributions, variance of count data shows to be either greater or smaller than the average. Count data generated from a negative Binomial model is also investigated. The findings show that the Index of Dispersion (ID) can be used to predict the performance of the confidence intervals. If ID is less than or equal to one, all confidence intervals produce the coverage probabilities greater than or equal to the desired confidence level (0.95). On the contrary, if ID is greater than one, the resulting coverage probabilities are lower than the desired confidence level. Moreover, an increase in the sample size is found to have a marginal effect on the coverage probability.
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