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This work describes an empirical investigation of the cost effectiveness of well-known state-based testing techniques for classes or clusters of classes that exhibit a state-dependent behavior. This is practically relevant as many object-oriented methodologies recommend modeling such components with statecharts which can then be used as a basis for testing. Our results, based on a series of three experiments, show that in most cases state-based techniques are not likely to be sufficient by themselves to catch most of the faults present in the code. Though useful, they need to be complemented with black-box, functional testing. We focus here on a particular technique, Category Partition, as this is the most commonly used and referenced black-box, functional testing technique. Two different oracle strategies have been applied for checking the success of test cases. One is a very precise oracle checking the concrete state of objects whereas the other one is based on the notion of state invariant (abstract states). Results show that there is a significant difference between them, both in terms of fault detection and cost. This is therefore an important choice to make that should be driven by the characteristics of the component to be tested, such as its criticality, complexity, and test budget.