Randomized Version of NP
The class you described is MA (interactive proofs consisting of one round where the prover, Merlin, sends one meassage to the verifier, Arthur, which then has to decide in probabilistic polynomial time whether to accept or reject).
Since $BPP\subseteq \Sigma_2\cap \Pi_2$  it immediately follows that $MA\subseteq \Sigma_2\cap \Pi_2$, which is believed to be a strict subset of PSPACE which contains the entire polynomial hierarchy.