Equivalent widths (EWs) observed in high-redshift Lyman alpha galaxies could be stronger than the EW intrinsic to the stellar population if dust is present residing in clumps in the inter-stellar medium (ISM). In this scenario, continuum photons could be extinguished while the Lyman alpha photons would be resonantly scattered by the clumps, eventually escaping the galaxy. We investigate this radiative transfer scenario with a new sample of six Lyman alpha galaxy candidates in the GOODS CDF-S, selected at z = 4.4 with ground-based narrow-band imaging obtained at CTIO. Grism spectra from the HST PEARS survey confirm that three objects are at z = 4.4, and that another object contains an active galactic nuclei (AGN). If we assume the other five (non-AGN) objects are at z = 4.4, they have rest-frame EWs from 47 -- 190 A. We present results of stellar population studies of these objects, constraining their rest-frame UV with HST and their rest-frame optical with Spitzer. Out of the four objects which we analyzed, three objects were best-fit to contain stellar populations with ages on the order of 1 Myr and stellar masses from 3 - 10 x 10^8 solar masses, with dust in the amount of A_1200 = 0.9 - 1.8 residing in a quasi-homogeneous distribution. However, one object (with a rest EW ~ 150 A) was best fit by an 800 Myr, 6.6 x 10^9 solar mass stellar population with a smaller amount of dust (A_1200 = 0.4) attenuating the continuum only. In this object, the EW was enhanced ~ 50% due to this dust. This suggests that large EW Lyman alpha galaxies are a diverse population. Preferential extinction of the continuum in a clumpy ISM deserves further investigation as a possible cause of the overabundance of large-EW objects that have been seen in narrow-band surveys in recent years.