Composites of direct-spun carbon nanotube (CNT) mats and epoxy are manufactured and tested in order to determine their mechanical and electrical properties. The mats are spun directly from a floating catalyst, chemical vapour deposition reactor. The volume fraction of epoxy is varied widely by suitable dilution of the epoxy resin with acetone. Subsequent evaporation of the acetone, followed by a cure cycle, leads to composites of varying volume fraction of CNT, epoxy and air. The modulus, strength, electrical conductivity and piezoresistivity of the composites are measured. The CNT mats and their composites exhibit an elastic-plastic stress-strain response under uniaxial tensile loading, and the degree of anisotropy is assessed by testing specimens in 0{\deg}, 45{\deg} and 90{\deg} directions with respect to the draw direction of mat manufacture. The electrical conductivity scales linearly with CNT volume fraction, irrespective of epoxy volume fraction. In contrast, the modulus and strength depend upon both CNT and epoxy volume fractions in a non-linear manner. The macroscopic moduli of the CNT mat-epoxy composites are far below the Voigt bound based on the modulus of CNT walls and epoxy. A micromechanical model is proposed to relate the macroscopic modulus and yield strength of a CNT mat-epoxy composite to the microstructure.