We will investigate numerically a seesaw model with $A_4$ flavor symmetry to find allowed regions satisfying the current experimental neutrino oscillation data, then  use them to predict physical consequences.   Namely, the lightest active neutrino mass  is of the  order of $\mathcal{O}(10^{-2})$ eV.  The effective neutrino mass $|\langle m\rangle|$ associated with neutrinoless double beta decay is in the range $[0.002 \;\mathrm{eV},0.038\;\mathrm{eV}]$  and  $[0.048\;\mathrm{eV},0.058\;\mathrm{eV}]$, corresponding to the normal and the inverted hierarchy schemes, respectively. Other relations among relevant physical quantities are shown, so that they can be determined if some of them are confirmed experimentally.   The recent data of the baryon asymmetry of the Universe ($\eta_B$) can be explained via   leptogenesis caused by  the  effect of the renormalization group evolution on the Dirac Yukawa couplings, provided the right-handed neutrino mass scale $M_0$   ranges from $\mathcal{O}(10^8)$  GeV to $\mathcal{O}(10^{12})$ GeV for $\tan\beta =3$. This allowed $M_0$  range is different from the scale of  $\mathcal{O}(10^{13})$  GeV for other effects that also generate a consistent $\eta_B$  from leptogenesis.  The branching ratio of the decay $ \mu \rightarrow\,e\gamma$  may reach future experimental sensitivity for very light values of $M_0$. Hence, it will be  inconsistent with the $M_0$ range predicted from the $\eta_B$ data whenever  this decay is detected experimentally.