Critical Values of Guessing on True-False and Multiple-Choice Tests
Wang, Jianjun, Education
Correction for guessing has been a persistent problem in the interpretation of true-false and multiple-choice test scores. Many authors have maintained that no solution to this problem is in sight. Thorndike (1971) pointed out: "Practice in United States testing organizations and among test publishers with respect to using the correction formula remains divided" (p. 59). Payne (1992) concurred: "Researchers have for more than 30 years been investigating the problem of whether or not to correct for guessing. There is till no definite answer or agreement among the experts" (p. 108).
One approach to the correction for guessing has been to investigate the conditions under which the influence of blind guessing on the scores of a test is negligible. Sax (1989) pointed out that teachers should include more items in tests to ignore the effect of guessing. Hopkins and Stanley (1981) asserted: "it should be evident that the greater the number of options per item, the less likely it is that one will select the correct option by chance and, hence, the less the magnitude of the weighting of an incorrect response" (p. 149). Most researchers agreed that the influence of blind guessing on the scores of a test diminishes as the length of the test and the number of options per item increase (e.g., Ebel and Frisbie, 1991; Brown, 1981; and Mehrens and Lehmann, 1984).
Nonetheless, when the correction for guessing is ignored, it becomes possible that a student may pass a test through guessing. In terms of statistics, an alternative hypothesis ([H.sub.a]) may be formulated that the effect of guessing is negligible. The mistake of ignoring the roles of guessing when the effect of guessing does exit is called Type I error. In social sciences, the acceptable risk of making Type I error is conventionally set at [Alpha] = .05.
Critical value is a statistic that marks the edge of the retaining region of [H.sub.a] at [Alpha] = .05 (Heiman, 1992). In a long true-false or multiple-choice test, the probability of obtaining a high scores through guessing is small (Sax, 1989). The passing scores of a test is the statistic that controls the risk of Type I error. The higher the passing scores, the less the risk of retaining the alternative hypothesis ([H.sub.a]). The retaining region of [H.sub.a] contains scores at which the probability of passing a test through guessing is less than 5%. The lowest passing score which guarantees a no larger than 5% risk is the critical value of a passing score for correction of guessing. By checking whether a passing score of a test is higher than the corresponding critical value, a decision can be made with 95% confidence as to whether the correction for guessing is necessary. Accordingly, although no solution to the correction [TABULAR DATA FOR TABLE 1 OMITTED] for guessing is in sight, it is possible to construct a table of critical values to evaluate the effect of guessing.
The critical value of a passing score is determined by the structure of a test and the stochastic model of guessing in which the probability of passing the test through guessing is delineated. Nevertheless, no such stochastic model has been stressed in educational and psychological measurements yet, needless to mention the construction of critical values to meet the structure of various tests (Brown, 1981; Mehrens & Lehmann, 1987). The purpose of this paper is to build a stochastic model describing the probability of passing a true-false or multiple-choice test through guessing, and to assemble a table of critical values for commonly used standardized tests.
The application of the table is straightforward. For a test with given total number of items (N) and the probability of guessing an item correctly (p), the table contains a critical value ([x.sub.o]) identified from the stochastic model. Based on the rationale of hypothesis testing, the correction for guessing can not be ignored unless the passing score (x) of the test has been set at an x [greater than] [x. …