Question: A local contractor manufactures the speakers used in telephones. The phone company requires the speakers to ring at a specified noise level (in decibels). An X‾ ­ chart is being designed to monitor this variable. The process of sampling speakers from the line requires hitting the speakers with a fixed-force clapper and measuring the decibel level on a meter designed for that purpose. The cost of sampling is $1.25 per speaker. When the process goes out of control, the thickness of the speakers is incorrect. The cost of searching for an assignable cause is estimated to be $50. The cost of operating the process in an out-of-control state is estimated to be $180 per hour. Out of control corresponds to a shift of 2σ in the decibel level, and the probability that the process shifts out of control in any hour is .03.

a. The company uses an X‾ chart based on 3σ limits and subgroups of size 4. What is its hourly cost?

b. What are the optimal values of n and k for this process and the associated optimal cost?