Problem $1$: $5$ points

A quality characteristic has a specification $($in inches$)$ of $0\mathrm{.}200+-0\mathrm{.}020.$ If the value of the quality characteristic exceeds $0\mathrm{.}200$ by the tolerance of $0\mathrm{.}020$ on either side, the product will require a repair of $$150.$ Develop the appropriate Taguchi loss function. $($Hint: find the constant $k$ in the Tagulchi function$)$

Problem $2$: $5$ points

Assume the specifications for a part $($in inches$)$ are $5\mathrm{.}00+-0\mathrm{.}37,$ and the Taguchi loss function is estimated to be $L(x)=9,300(x-T{)}^2.$ Estimate the loss for a part if the quality characteristic under study takes on a value of $5\mathrm{.}50$ inches.

Problem $3$: $10$ points

Suppose that the design specifications for a hydraulic cylinder are $10\mathrm{.}00+-0\mathrm{.}10$ centimeters and that the Taguchi loss function is estimated to be $L(x)=1,400(x-T{)}^2.$ Answer the following questions

a$)$ Estimate the loss for a production if the characteristic under study takes on a value of $10\mathrm{.}04$ and $100$ parts are produces

b$)$ Assume the production process is recalibrated weekly and a new sample of cylinders after recalibrations reveals an $x-$bar of $9\mathrm{.}789.$ What action, if any, is needed in this situation? Explain.

Problem $4$: $5$ points

The service center for a brokerage company provides three functions to callers: account status, order confirmations, and stock quotes in series system. The reliability was measured for each of these services over $1$ month with these results: $90\%,80\%,$ and $96\%,$ respectively. What is the overall reliability of the call center?