We use 32 age measurements of passively evolving galaxies as a function of redshift to test and compare the standard model ($Λ$CDM) with the $R_{\rm h}=ct$ Universe. We show that the latter fits the data with a reduced $χ^2_{\rm dof}=0.435$ for a Hubble constant $H_{0}= 67.2_{-4.0}^{+4.5}$ km $\rm s^{-1}$ $\rm Mpc^{-1}$. By comparison, the optimal flat $Λ$CDM model, with two free parameters (including $Ω_{\rm m}=0.12_{-0.11}^{+0.54}$ and $H_{0}=94.3_{-35.8}^{+32.7}$ km $\rm s^{-1}$ $\rm Mpc^{-1}$), fits the age-\emph{z} data with a reduced $χ^2_{\rm dof}=0.428$. Based solely on their $χ^2_{\rm dof}$ values, both models appear to account for the data very well, though the optimized $Λ$CDM parameters are only marginally consistent with those of the concordance model ($Ω_{\rm m}=0.27$ and $H_{0}= 70$ km $\rm s^{-1}$ $\rm Mpc^{-1}$). Fitting the age-$z$ data with the latter results in a reduced $χ^2_{\rm dof}=0.523$. However, because of the different number of free parameters in these models, selection tools, such as the Akaike, Kullback and Bayes Information Criteria, favour $R_{\rm h}=ct$ over $Λ$CDM with a likelihood of $∼ 66.5\%-80.5\%$ versus $∼ 19.5\%-33.5\%$. These results are suggestive, though not yet compelling, given the current limited galaxy age-$z$ sample. We carry out Monte Carlo simulations based on these current age measurements to estimate how large the sample would have to be in order to rule out either model at a $∼ 99.7\%$ confidence level. We find that if the real cosmology is $Λ$CDM, a sample of $∼ 45$ galaxy ages would be sufficient to rule out $R_{\rm h}=ct$ at this level of accuracy, while $∼ 350$ galaxy ages would be required to rule out $Λ$CDM if the real Universe were instead $R_{\rm h}=ct$. This difference in required sample size reflects the greater number of free parameters available to fit the data with $Λ$CDM.