This study reports on the development and application of a four-tier multiplechoice (4TMC) diagnostic instrument, which has not been reported in the literature. It is an enhanced version of the two-tier multiple-choice (2TMC) test. As in 2TMC tests, its answer and reason tiers measure students' content knowledge and explanatory knowledge, respectively. The two additional tiers measure the level of confidence of students in the correctness of their chosen options for the answer and reason tiers respectively. The 4TMC diagnostic test focused on the properties and propagation of mechanical waves. It was administered to 598 upper secondary students after they were formally instructed on the foregoing topics. The vast majority of the respondents were found to have an inadequate grasp of the topics tested. Mean scores and mean confidence associated with the answer tier was higher than those associated with the reason tier. The students tended to be poorly discriminating between what they know and what they do not know. Familiarity with the topic tested was associated with greater percentage of students giving correct answers, higher confidence, and better discrimination quotient. Nine genuine alternative conceptions (which were expressed with moderate levels of confidence by students) were identified.In the past two decades, a plethora of studies have focused on the identification and modification of conceptions that differ from established scientific knowledge, which are commonly labelled as alternative conceptions (ACs). In investigating students' ACs, multiple-choice questions (MCQs) tend to be a popular choice. Typical MCQs require students to choose the best answer to a given question from a given set of alternatives. MCQs

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P robabilistic selling-the sale of synthetic products consisting of a lottery between two distinct goods-has been extensively analyzed in horizontal markets. In this research, we investigate probabilistic selling in quality-differentiated markets. This is an important new dimension of inquiry because of the widespread prevalence of quality-differentiated markets as well as significant differences in the preference structure across these markets. In fact, this latter consideration casts doubt as to whether probabilistic selling will even emerge in quality-differentiated markets. We find that probabilistic selling emerges in quality-differentiated markets as a way to profitably dispose excess capacity; moreover, probabilistic selling remains viable even under endogenous quality choice. In addition, in markets where sellers employ "strong" quality differentiation, the introduction of an intermediate probabilistic good actually causes closer quality levels in a product line and enhances consumer welfare. In contrast, in markets where sellers employ "weak" quality differentiation, the introduction of an intermediate probabilistic good increases quality separation and degrades consumer welfare. Overall, we view our contribution as one of characterizing the optimality, implementation, and policy implications of probabilistic selling in quality-differentiated markets.