$1$i$.$ A company daims that the mean deflection of their $10-$foot steel beams is $0\mathrm{.}012$ inches. A construction

contractor who purchases large quantities of steel beams suspects that the manufacturer misleads its

customers and that the true deflection is$,$ in fact, larger than the one claimed. To see whether this

suspicion is justified, the contractor selects $n=10$ beams at random from his inventory, determines their

deflection as follows:

Conduct a full hypothesis testing to determine if the contractor's suspicion is valid at $a=0\mathrm{.}05.$

$1$ii In a survey poll of $500$ adults, $205$ prefer public transit in urban centers, $112$ use their own vehicles,

and $197$ adults prefer other means of transportation $($walking$,$ biking, etc.$).$

a$)$ Find the probability that an adult uses only the public transit OR their own vehicle in urban centers.

$($Draw a Venn diagram$)$

b$)$ Find the probability that an adult uses both public transit AND their own vehicle in urban centers. $($Draw

a Venn diagram$)$

c$)$ Are the two events mutually exclusive? Why or why not?