Problem: Tour Eiffel

Philippe is a French artisan specialized in building high$-$detail replicas of the Tour Eiffel. His replicas weigh on average $180$ grams, with a standard deviation of $5$ grams. The retailer that buys his creations demands upper and lower specification limits of $185$ and $175$ grams respectively.

$1.$

Question $1$

What is Philippe$$s current Capability Score?

$($Your answer should be formatted as $0\mathrm{.}000)$

$1$ point

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$2.$

Question $2$

How many replicas should Philippe produce to meet an order for $100$ good$-$quality units? Assume that rework is not possible. Round up to the next largest integer number.

$1$ point

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$3.$

Question $3$

Suppose Philippe wanted to improve its quality and bring the number of defective units to $27$ every $10,000$ units produced. What would be new standard deviation he needs to achieve to meet this target?

$1$ point

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$4.$

Question $4$

Phillipe now wants to reduce the defect probability and run a $$Six Sigma Process$.$ To what level would he have to reduce the standard deviation in the process to meet this target?

$($Your answer should be formatted $0\mathrm{.}000)$