In preparation for a public hearing, an engineer needs to estimate the travel time savings

for a proposed highway bypass of a small city in a rural area. The existing highway goes

through the downtown area for $5\mathrm{.}5$ kilometers between the points where the bypass will

connect.

$($a$)$ If the engineer anticipates a standard deviation of $8$ km$/$h and would like a $95\%$

confidence of estimating the mean speed within $\{4\mathrm{.}8$ km$/$h$,$ how many runs should

a travel time study include?

$($b$)$ If the engineer measures an average speed of $55\mathrm{.}8$ km$/$h through the town and the

bypass is expected to operate at speeds of $105$ km$/$h over a distance of $9\mathrm{.}0$ kilometers,

what is the expected travel time saving per vehicle?