THE CONFIDENCE INTERVALS: A DIFFICULT MATTER, EVEN FOR EXPERTS

We present the results of a research whose purpose was to find out what a sample of experts (statisticians and statistics university professors) and university students understood exactly by confidence intervals. To this end, a questionnaire was answered by 41 experts and 297 students. The results show that both, students and professors, possess misconceptions regarding confidence intervals. The conception that these intervals contain sample means or single values of the population instead of possible parameter values, and the interpretation of significance levels as a measure of certainty, without any frequency referent, were found to be the most generalized misconceptions. INTRODUCTION The confidence intervals comprise of the statistical inference and are very useful to estimate population parameters from a sample of values of the population. The CIs contain a set of possible values of the searched parameter and have a confidence measurement associated that is responsible for the probability of finding the searched value among them. Although it is certain that the CI in principle can help to surpass some of the errors that can be committed when the hypothesis tests and p-values are used to decide on certain hypothesis around the searched parameter (Fidler, 2005), it is also certain the fact that the students have bad conceptions around them. (Bower, 2003; Fidler, 2005). A very usual one is to interpret CIs as a statistical descriptive object, in the sense of containing possible values of the sample mean or of being a range of individual values ("95% of the population values are in the interval"), and not as a statistical inferential object that provides information about the value of a population parameter (Fidler, 2005). But not only students misconstrue CIs. In an e-mail survey carried out by Cumming, Williams and Fidler (2004), a group of researchers and authors of scientific papers were asked to estimate 10 future sample means from a given sample mean and a 95% CI. 78% of the subjects (105 of 134) placed 9 of the means inside the interval. They made it clear that they had tried to adjust the number of future means within the interval to the confidence level (95%). That is, the confidence level is understood as the probability that the CI contain the values of the sample means. Concerning the relations between interval size with confidence level and the sample size, it has been reported that students are not clear on the direct or inverse relations between the interval width, the sample size and the confidence level. In fact, Fidler (2005) informs that only 16% of 180 examined students answered that increasing the sample size decreases the interval width, and 73% answered that the 90% CI is wider than the 95% CI. With this outlook of difficulties we ask ourselves, what about the professors? More precisely, what meaning do professors and experts have regarding confidence intervals? To provide an answer, we conducted a research that allows us to know the exact nature of the conceptions that statistics professors have about confidence intervals. Part of the results of this research is presented in this paper. In order to interpret the obtained results we use conceptual analysis (Glasersfeld, 1995) which attempts to build a model or conceptual scheme that is compatible with the observed outcomes. Therefore, our explanations are but hypotheses about the motives that drive people to answer like they did, put forward for corroboration or disproof by further studies who would look deeper into the logic of our results. METHODOLOGY