Stat Papers (2010) 51:209–226 DOI 10.1007/s00362-008-0133-4 REGULAR ARTICLE Fuzzy p-value in testing fuzzy hypotheses with crisp data Abbas Parchami · S. Mahmoud Taheri · Mashaallah Mashinchi Received: 11 April 2007 / Accepted: 5 March 2008 / Published online: 22 April 2008 © Springer-Verlag 2008 Abstract In testing statistical hypotheses, as in other statistical problems, we may be confronted with fuzzy concepts. This paper deals with the problem of testing hypothe- ses, when the hypotheses are fuzzy and the data are crisp. We first introduce the notion of fuzzy p-value, by applying the extension principle and then present an approach for testing fuzzy hypotheses by comparing a fuzzy p-value and a fuzzy significance level, based on a comparison of two fuzzy sets. Numerical examples are also provided to illustrate the approach. Keywords Testing hypotheses · Monotone likelihood ratio · Fuzzy hypothesis · Fuzzy p-value · Fuzzy significance level 1 Introduction and background Statistical analysis, in traditional form, is based on the crispness of data, random vari- ables, hypotheses, decision rules and parameters. In classical testing hypotheses the hypotheses are crisp. For example, when we test the difference between two pop- ulation means, the ordinary null hypothesis stipulates the difference between these A. Parchami · M. Mashinchi Department of Statistics, Faculty of Mathematics and Computer Sciences, Shahid Bahonar University of Kerman, Kerman, Iran S. M. Taheri (B ) Department of Mathematical Sciences, Isfahan University of Technology (IUT), Isfahan 84156-83111, Iran e-mail: taheri@cc.iut.ac.ir S. M. Taheri Statistical Research and Training Center (SRTC), Tehran, Iran 123