Problem $2$

ACME Thermostats are subjected to rigorous testing before they are shipped to air conditioning

technicians around the world.

a$)(15$ points$)$ Construct $3-$sigma $\stackrel{}{x}-$ and $R-$charts for the length in centimeters of a particular part based on

the data from the following table. Be sure to clearly specify the upper and lower control limits for each

chart.

b$)(5$ points$)$ Is the process under control? Explain.

c$)(5$ points$)$ Suppose the engineering design specifications state that this particular part must be $0\mathrm{.}500$ cm

long with a tolerance of $\frac{+}{-}0\mathrm{.}050cm.$ The current data indicate that the standard deviation for the part

length is $0\mathrm{.}014$ cm $.$ Is the process capable at the four$-$sigma level? Explain using the process capability

index and$/$or the process capability ratio.