Odd square roots and the sum of element orders on Sn and An

For a finite group $G$ let $ψ(G)$ be the sum of the orders of its elements. The quotient $S_n/A_n$ has order two. Hence $ψ(S_n/A_n)=3$. The quantity to be compared with $ψ(S_n)$ is therefore $3ψ(A_n)$. Computation indicates that $3ψ(A_n)>ψ(S_n)$ for every $n\ge3$. We reduce this inequality to a ques