Difficult# Establishing Normality Based on the $5$-Number Summary

APSTAT-BTFV4A

The $5$-number summary for a certain distribution is as follows:

$28$

$53$

$75$

$97$

$122$

The mean of the distribution is $75.10$ and the standard deviation is $31.27$.

Based upon this information, is the distribution approximately normal? Why or why not?