<p>Cronbach’s coefficient alpha is one of the most commonly used measures for assessing the internal consistency or reliability of a set of items, ensuring that they measure the same research objectives. It is used as a measure of reliability in fields like education, psychology, sociology, medicine, accounting and economics. Cronbach’s alpha will be estimated for a general covariance matrix using a Bayesian approach and comparing these results to the asymptotic frequentist interval valid under a general covariance matrix framework. Most of the results used in the literature require the compound symmetry assumption for analyses of Cronbach’s alpha. Fiducial and posterior distributions will be derived for Cronbach’s alpha in the case of the bivariate normal distribution. Various objective priors will be considered for the variance components and the correlation coefficient. The posterior distribution of one of the priors considered corresponds to the fiducial distribution. The performance of these priors will be compared to an asymptotic frequentist interval often used in the literature. A simulation study will be considered to compare the performance of the priors and the asymptotic interval, where the coverage rates and average interval lengths will be computed. The simulation study showed that even though the asymptotic interval improved in the case of larger sample sizes, the Bayesian approach still consistently outperformed the asymptotic frequentist interval.</p>