  We use 32 age measurements of passively evolving galaxies as a function of
redshift to test and compare the standard model ($\Lambda$CDM) with the $R_{\rm
h}=ct$ Universe. We show that the latter fits the data with a reduced
$\chi^2_{\rm dof}=0.435$ for a Hubble constant $H_{0}= 67.2_{-4.0}^{+4.5}$ km
$\rm s^{-1}$ $\rm Mpc^{-1}$. By comparison, the optimal flat $\Lambda$CDM
model, with two free parameters (including $\Omega_{\rm
m}=0.12_{-0.11}^{+0.54}$ 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 $\chi^2_{\rm dof}=0.428$.
Based solely on their $\chi^2_{\rm dof}$ values, both models appear to account
for the data very well, though the optimized $\Lambda$CDM parameters are only
marginally consistent with those of the concordance model ($\Omega_{\rm
m}=0.27$ and $H_{0}= 70$ km $\rm s^{-1}$ $\rm Mpc^{-1}$). Fitting the age-$z$
data with the latter results in a reduced $\chi^2_{\rm dof}=0.523$. 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 $\Lambda$CDM with a likelihood of $\sim 66.5\%-80.5\%$
versus $\sim 19.5\%-33.5\%$. 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 $\sim
99.7\%$ confidence level. We find that if the real cosmology is $\Lambda$CDM, a
sample of $\sim 45$ galaxy ages would be sufficient to rule out $R_{\rm h}=ct$
at this level of accuracy, while $\sim 350$ galaxy ages would be required to
rule out $\Lambda$CDM if the real Universe were instead $R_{\rm h}=ct$.
