Case Part B $10-1($Algo$)$ c$.$ What is the $C_{pk}=$min$\{\frac{\text{U}\text{T}\text{L}-x}{3},\frac{x-LTL}{3}\}=$min$\{\frac{0\mathrm{.}5350}{0\mathrm{.}6606},\frac{0\mathrm{.}4650}{0\mathrm{.}6606}\}=$min$\{0\mathrm{.}8099,0\mathrm{.}7039\}=0\mathrm{.}7039$ for the process?

Note: Round your answer to $4$ decimal places.

Capability index

d$.$ What would be the $C_{pk}=$min$\{\frac{\text{U}\text{T}\text{L}-x}{3},\frac{x-LTL}{3}\}=$min$\{\frac{0\mathrm{.}50}{0\mathrm{.}6606},\frac{0\mathrm{.}50}{0\mathrm{.}6606}\}=$min$\{0\mathrm{.}7569,0\mathrm{.}7569\}=0\mathrm{.}7569$ for the process if it

were centered $($assume the process standard deviation is the same$)?$

Note: Round your answer to $4$ decimal places.

Capability index

e$.$ What percentage of output would be expected to be out of tolerance if the process were centered?

Note: Round your answer to $2$ decimal places.

Percentage of output f$.$ If the process could be improved so that the standard deviation was only about $0\mathrm{.}14$ millimeters, what would be the best that could

be expected with the process relative to fraction defective?

Note: Round your answer to the nearest whole number.

The process to be centered at

defects per billion washers.

g$.$ Setup X$-$bar and Range control charts for the current process. Assume the operators will take samples of $10$ washers at a time.

Note: Round your answer to $3$ decimal places.

As part of the process for improving the quality of their cars, engineers have identified a potential improvement to the process that

makes a washer that is used in the accelerator assembly. The tolerances on the thickness of the washer are fairly large since the fit

can be loose, but if it does happen to get too large, it can cause the accelerator to bind and create a potential problem for the driver.

$($Note: This part of the case has been fabricated for teaching purposes and none of these data were obtained from Toyota.$)$

Let's assume that as a first step to improving the process, a sample of $40$ washers coming from the machine that produces the

washers was taken and the thickness measured in millimeters. The following table has the measurements from the sample:

Use the approprlate Excel function to compute normal distribution probabilities in Parts $a,b,$ e and f $.$

a$.$ If the specification is such that no washer should be greater than $2\mathrm{.}4$ millimeters, assuming that the thicknesses are distributed

normally, what percentage of output is expected to be greater than this thickness?

Note: Round your answer to $2$ decimal places.

b$.$ What if there is an upper and lower specification, where the upper thickness limit is $2\mathrm{.}4$ and the lower thickness limit $1\mathrm{.}4,$ what

percentage of output is expected to be out of tolerance?

Note: Round your answer to $2$ decimal places.