Problem $3$:

Deep Sea Diggers estimate that the time until failure for the battery used on its blowout preventers should be at least $36$ months and at most $96$ months. The manufacturing process is such that the time until failure is $($approximately$)$ normally distributed with a mean of $57$ months and standard deviation of $8$ months. The company aims for $3$ performance for this process.

a$.$ Calculate $C_p.$ Is the process capable of $3$ performance based on $C_p?$ Explain your answer.

b$.$ Calculate $C_{pk}.$ Does your answer from part $($a$)$ regarding process capability change when you also use $C_{pk}?$ Explain your answer.