Answer
The ball went less than a mile.
Work Step by Step
We know that a mile is 1.6 kilometers. Thus, we compare the ratio of gravity on earth to that on the moon:
$ = 257 \times\frac{9.81}{1.625}=1.55\ km$
This is less than 1.62 km, so the astronaut is wrong: the ball went less than a mile.
