We report on a conceptually new test of the equivalence principle performed by measuring the acceleration in Earth's gravity field of two isotopes of strontium atoms, namely, the bosonic (88)Sr isotope which has no spin versus the fermionic (87)Sr isotope which has a half-integer spin. The effect of gravity on the two atomic species has been probed by means of a precision differential measurement of the Bloch frequency for the two atomic matter waves in a vertical optical lattice. We obtain the values η=(0.2±1.6)×10(-7) for the Eötvös parameter and k=(0.5±1.1)×10(-7) for the coupling between nuclear spin and gravity. This is the first reported experimental test of the equivalence principle for bosonic and fermionic particles and opens a new way to the search for the predicted spin-gravity coupling effects.