Yes. Suppose N is a subgroup of S_n of index 2. Then we have a surjective homomorphism f:S_n->Z_2 (the cyclic group of order 2) such that ker f= H.

Observe now that the image of any transposition is 1 (why?)

This implies that the image of every even permutation is 0 hence Ker f=A_n. So we must have H=A_n