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 $\mathitΩ\left(n^{-0.5(1-E_{sp}'(R))}\right)$ 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.