Estimating the confidence interval of the Weibull modulus is an important problem in the fracture strength modeling of ceramic and composite materials. It is particularly important in cases where the sample size is small due to high experimental costs. For this purpose, several classical methods, including the popular maximum likelihood method, and Bayesian methods have been developed in the literature. However, studies on Bayesian inference have remained very limited in the materials science literature. Recently a Bayesian Weibull model has been proposed for estimating confidence lower bounds for Weibull percentiles using the prior knowledge that the failure rates are increasing. This prior argument requires the Weibull modulus to be more than 1 due to wear-out failure. In this study, under the same prior information, two Bayesian Weibull models, one using the same prior argument and the other a relaxed version of it, have been developed for confidence interval estimation of the Weibull modulus. Their estimation performances have been compared against the maximum likelihood method with Monte Carlo simulations. The results show that the Bayesian Weibull models significantly outperform the maximum likelihood method for almost all Weibull modulus and sample size values.