We study tilting objects of the stable category CM_ZA of graded Cohen-Macaulay modules over a finite dimensional graded Iwanaga-Gorenstein algebra A. We first show that if there exists a tilting object in CM_ZA, then its endomorphism algebra always has finite global dimension. Next, to study the existence of a tilting object, we introduce a numerical invariant g(A). In the case where A is 1-Iwanaga-Gorenstein, we give a sufficient condition on g(A) for the existence of a tilting object. As an application, for a truncated preprojective algebra Π(Q)w of a tree quiver Q, we prove that CM_ZΠ(Q)w always admits a tilting object.