This paper asks the question of how much should society spend on diagnosis as opposed to medical treatments and compensation when implementing its goals of distributive justice in health policy through publicly-funded medical entitlement programs in a world of limited information? A model is presented in which social planners seek to maximize expected social welfare by allocating medical goods, including diagnostic tests among medically dissimilar individuals when there is imperfect information about the medical condition of an individual, subject to the resource constraints of a medical entitlement fund. The goals of distributive justice underlying the social welfare function is governed by a parameter representing society's aversion to inequality. It is argued that society, given its aversion to inequality, need not always entitle the medically more fortunate individual to less treatment and compensation. Moreover, in most cases, it is socially desirable to spend a finite and equal amount on diagnostic tests for each individual, even though there is some probability of misdiagnosis. In some unusual cases, zero expenditures on diagnostic tests will be socially optimal.