Dichotomous decisions based on dichotomously scored items: a case study

In a course in elementary statistics for psychology students using criterion-referenced achievement tests, the total test score, based on dichotomously scored items, was used for classifying students into those who passed and those who failed. The score on a test is considered as depending on a latent variable; it is assumed that the students can be dichotomized into the categories "mastery" (with scores on the latent variable above a cutting score), and "no mastery" (with scores below the cutting score on the latent variable). Two problems are considered: (a) How many students are classified incorrectly? Using the binomial error model a procedure is described for computing the classification proportions: p(mastery, passed), p(mastery, failed), p(no mastery, passed), and p(no mastery, failed), (b) What is the optimal cutting score on a test? Using a loss function a procedure for computing the optimal curring score is described.