Testing ‘Two-Sided’ Hypothesis about the Mean of an Interval-Valued Random Set Manuel Montenegro, Mar´ ıa Rosa Casals, Ana Colubi, and Mar´ ıa ´ Angeles Gil Departamento de Estad´ ıstica e I.O. y D.M., Universidad de Oviedo, Oviedo, Spain Abstract. Interval-valued observations arise in several real-life situations, and it is convenient to develop statistical methods to deal with them. In the literature on Statistical Inference with single-valued observations one can find different studies on drawing conclusions about the population mean on the basis of the information supplied by the available observations. In this paper we present a bootstrap method of testing a ‘two-sided’ hypothesis about the (interval-valued) mean value of an interval-valued random set based on an extension of the t statistic for single-valued data. The method is illustrated by means of a real-life example. Keywords: Random interval, Interval mean, Hypothesis testing. 1 Introduction In previous papers it has been pointed out that in many real-life situations observations are essentially (or customary) interval-valued rather than single-valued. For instance, some observations correspond to ranges or fluctuations (like price fluctuations, blood pressure fluctuations, income ranges, and so on), or they are engineering/physical data (as descriptions of amount, bounds, and limits, speed, mass, etc.), or interval-censoring times, or simply incomplete data which are treated as grouped ones. In the last decade the interest for the statistical analysis of interval-valued data has increased, especially in which concerns descriptive aspects. In 2000 Billard and Diday [3] and Gil et al. [7] (see also [8], [9], [16], for a more detailed study) have considered different approaches for the regression (and also the correlation in the second one) anal- ysis of interval-valued data: the symbolic data analysis and the random sets approach. The last approach has been also considered to deal with other descriptive problems (see, for instance, [13]). An approach, which has been shown to be certainly valuable for the statistical man- agement of interval-valued data, is the one based on the mid-spread (or centre-radium) approach and the use of interval arithmetic (see, for instance, Gil et al. [8], [9], and Marino and Palumbo [14]). In some recent papers (cf. Montenegro et al. [17], Gil et al. [6], Gonz´ alez-Rodr´ ıguez et al. [10]) we have developed some inferential procedures on the problems of least- squares regression and correlation between interval-valued random elements. The sta- tistical analysis of these random elements has been developed by modelling them as particular random sets, using the set-valued arithmetic and a suitable metric between D. Dubois et al. (Eds.): Soft Methods for Hand. Var. and Imprecision, ASC 48, pp. 133–139, 2008. springerlink.com c  Springer-Verlag Berlin Heidelberg 2008