A designer estimates that she can $($a$)$ increase the average time between failures of a part by $5$ percent at a cost of $$450,$ or $($b$)$ reduce the average repair time by $10$ percent at a cost of $$200.$ Which option would be more cost$-$effective? Currently, the average time between failures is $100$ hours and the average repair time is $4$ hours.

Auto batteries have an average life of $2\mathrm{.}7$ years. Battery life is normally distributed with a mean