Measuring the relative efficiency of a finite fixed set of service-producing units (hospitals, state services, libraries, banks,...) is an important purpose of Data Envelopment Analysis (DEA). We illustrate an innovative way to measure this efficiency using stochastic indexes of the quality from these services. The indexes obtained from the opinion-satisfaction of the customers are estimators, from the statistical view point, of the quality of the service received (outputs); while, the quality of the offered service is estimated with opinion-satisfaction indexes of service providers (inputs). The estimation of these indicators is only possible by asking a customer and provider sample, in each service, through surveys. The technical efficiency score, obtained using the classic DEA models and estimated quality indicators, is an estimator of the unknown population efficiency that would be obtained if in each one of the services, interviews from all their customers and all their providers were available. With the object of achieving the best precision in the estimate, we propose results to determine the sample size of customers and providers needed so that with their answers can achieve a fixed accuracy in the estimation of the population efficiency of these service-producing units through the use of a novel one bootstrap confidence interval. Using this bootstrap methodology and quality opinion indexes obtained from two surveys, one of doctors and another of patients, we analyze the efficiency in the health care system of Spain.
