Lack of sleep costs companies about $$18$ billion a year in lost productivity, according to the National

Sleep Foundation. Companies are waking up to the problem, however. Some even have quiet rooms

available for study or sleep. A major airline recently began encouraging reservation agents to nap during

their breaks. The number of complaints received about each of a sample of $10$ reservation agents during

the $6$ months before naps were encouraged and during the $6$ months after the policy change is saved in

NAPS.txt$.$

a$)$ Do the data present sufficient evidence to conclude that the new napping policy reduced the mean

number of customer complaints about reservation agents? Test using $=0\mathrm{.}05.$

b$)$ What assumptions must hold to ensure the validity of the test?

c$)$ What variables, not controlled in the study, could lead to an invalid conclusion?