Studies have shown that people who work with benzene longer than five years have $20$ times the incidence of

leukemia than the general population. As a result, OSHA

$($Occupational Safety and Health Administration$)$ has set

a time$-$weighted average permissible exposure limit of

$1$ ppm$.$ A steel manufacturing plant, which exposes its

workers to benzene daily, is under investigation for possible violations of the permissible exposure limit$.$ Thirty$-$six

measurements, taken over a period of $1\mathrm{.}5$ months, yielded

X $=2\mathrm{.}1$ ppm$,$ S $=4\mathrm{.}1$ ppm$.$ Complete the following parts

to determine if there is sufficient evidence to conclude

that the steel manufacturing plant is in violation of the

OSHA exposure limit$.$