This article presents a novel methodology for fault detection based on the design of an Unknown Input Interval Observer (UIIO). The approach begins with constructing an unknown input observer for systems affected by disturbances and noise, then reformulates it into a positive representation. From this, an interval observer is designed that enhances robustness to uncertainties and noise, while maintaining sensitivity to faults. A key contribution is the development of an interval residue generator, derived solely from interval state estimates, eliminating the need for noise filtering. The methodology, grounded in Interval Arithmetic theory, shows that UIIO effectively balances robustness and sensitivity in fault detection and offers new observer design techniques to improve fault diagnosis reliability.