The distance (in $km$) of $40$ engineers from their residence to their place of work were found as follows.

$5$	$3$	$10$	$20$	$25$	$11$	$13$	$7$	$12$	$31$
$19$	$10$	$12$	$17$	$18$	$11$	$32$	$17$	$16$	$2$
$7$	$9$	$7$	$8$	$3$	$5$	$12$	$15$	$18$	$3$
$12$	$14$	$2$	$9$	$6$	$15$	$15$	$7$	$6$	$12$

What is the empirical probability that an engineer lives:
$(i)$ less than $7 \, km$ from her place of work?
$(ii)$ more than or equal to $7 \, km$ from her place of work?
$(iii)$ within $\frac{1}{2} \, km$ from her place of work?

(A) $(i)$ Total number of engineers $= 40$.
Number of engineers living less than $7 \, km$ from their place of work are: $5, 3, 3, 2, 7, 9, 7, 3, 2, 6, 6$ (Wait,let us re-count: $5, 3, 3, 2, 7$ is not less than $7$,so $5, 3, 3, 2, 3, 5, 2, 6, 6$ are less than $7$).
Counting values less than $7$: $5, 3, 2, 3, 5, 2, 6, 6$ (Total $8$ values). Let's re-verify: $5, 3, 2, 3, 5, 2, 6, 6$ are $8$ values. Actually,looking at the table: $5, 3, 3, 2, 7, 9, 7, 3, 2, 6, 6$. The values less than $7$ are $5, 3, 2, 3, 5, 2, 6, 6$. Total count is $8$.
Wait,let's list all values $< 7$: $5, 3, 2, 3, 5, 2, 6, 6$. Total $= 8$.
Probability $P = \frac{8}{40} = \frac{1}{5} = 0.2$.
$(ii)$ Number of engineers living more than or equal to $7 \, km = 40 - 8 = 32$.
Probability $P = \frac{32}{40} = \frac{4}{5} = 0.8$.
$(iii)$ Number of engineers living within $\frac{1}{2} \, km = 0$.
Probability $P = 0$.