  A judicious application of the Berry-Esseen theorem via suitable Augustin
information measures is demonstrated to be sufficient for deriving the sphere
packing bound with a prefactor that is
$\mathit{\Omega}\left(n^{-0.5(1-E_{sp}'(R))}\right)$ for all codes on certain
families of channels -- including the Gaussian channels and the non-stationary
Renyi symmetric channels -- and for the constant composition codes on
stationary memoryless channels. The resulting non-asymptotic bounds have
definite approximation error terms. As a preliminary result that might be of
interest on its own, the trade-off between type I and type II error
probabilities in the hypothesis testing problem with (possibly non-stationary)
independent samples is determined up to some multiplicative constants, assuming
that the probabilities of both types of error are decaying exponentially with
the number of samples, using the Berry-Esseen theorem.
