In an experiment, a scientific constant $C$ is determined to be $2.43865$ with an error of at most $\pm 0.00312$. 
The experimenter wishes to announce a value for $C$ in which every digit is significant. 
That is, whatever $C$ is, the announced value must be the correct result when $C$ is rounded to that number of digits. 
The most accurate value the experimenter can announce for $C$ is
$\textbf{(A)}\ 2\qquad \textbf{(B)}\ 2.4\qquad \textbf{(C)}\ 2.43\qquad \textbf{(D)}\ 2.44\qquad \textbf{(E)}\ 2.439$
If added together, we have:
\[2.43865+0.00312=2.44177\]
This rounds to $2.44$. If they subtracted, we have:
\[2.43865-0.00312=2.43553.\]
This rounds to $2.44$. Therefore, we have the answer to be $\fbox{\textbf{(D)} 2.44}$.