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Based on the general formula, which depends on the length and difficulty of the test, the number of respondents and the number of ability levels, this study aims to provide a closed formula for the adaptive tests with medium difficulty (probability of solution is p = 1/2) to determine the accuracy of the parameters for each item and in the case of calibrated items, determine the required test length given number of respondents.

Empirical results have been obtained on computerized or multistage adaptive implementation. Simulation studies and classroom/experimental results show that adaptive tests can measure test subjects’ ability to the same quality over half the test length compared to linear versions. Due to the complexity of the problem, the authors discuss a closed mathematical formula: the relationship between the length of the tests, the difficulty of solving the items, the number of respondents and the levels of ability.

The authors present a closed formula that provides a lower bound for the minimum test length in the case of adaptive tests. The authors also present example calculations using the formula, based on the assessment framework of some student assessments to show the similarity between the theoretical calculations and the empirical results.

With this formula, we can form a connection between theoretical and simulation results.

For central banks who study the use of cash, acceptance of card payments is an important factor. Surveys to measure levels of card acceptance and the costs of payments can be complicated and expensive. In this paper, we exploit a novel data set from Hungary to see the effect of stratified random sampling on estimates of payment card acceptance and usage. Using the Online Cashier Registry, a database linking the universe of merchant cash registers in Hungary, we create merchant and transaction level data sets. We compare county (geographic), industry and store size stratifications to simulate the usual stratification criteria for merchant surveys and see the effect on estimates of card acceptance for different sample sizes. Further, we estimate logistic regression models of card acceptance/usage to see how stratification biases estimates of key determinants of card acceptance/usage.