This paper addresses the modeling and the design of an interval state observer for a linear time-invariant plant in presence of sporadically available measurements corrupted by unknown-but-bounded errors and noise. The interval observer is modelled as an impulsive system where an impulsive correction is made whenever a measurement is available. The nonnegativity of the observation error between two successive measurements is preserved by applying the internal positivity based on the Müller's existence theorem, while at measurement times a linear programming constraint is added. A new methodology for designing the discrete-time observer gain is proposed that guarantees both nonnegativity and stability of the estimation error. The synthesis is performed by solving a set of Bilinear Matrix Inequalities (BMIs). The theoretical result is supported by numerical simulation. Index Terms Interval observers, LTI systems, sparse output measurements, hybrid system.