Sometimes (often :-)) it is hard to say whether just the application or the whole application server is not working. Also debugging JNDI configuration may be challenging.
ping is a simplistic 5kb WAR which exposes JNDI, system properties and "echo" service through REST APIs.
Classic Java EE 7 Workshops: Bootstrap, Effective and Architectures, July, 13th-15th
Online workshop: Java EE 7 Bootstrap