Problem: SlimCase

A company making smartphone cases is concerned about the exact length of their cases. The company has a lower specification limit of $12\mathrm{.}75$ cm and an upper specification limit of $13\mathrm{.}25$ cm$.$ The standard deviation is $0\mathrm{.}125$ cm and the mean is $13$ cm$.$

$1.$

Question $1$

What is the process capability index for the process?

$($Your answer should be formatted as a decimal $-0\mathrm{.}00)$

$1$ point

Enter answer here

$2.$

Question $2$

What is the probability that a smartphone case falls outside the specification limits$?$

$1$ point

Enter answer here

$3.$

Question $3$

If the company aims to produce $80,000$ non$-$defective cases every month, how many cases should it produce in total? Assume that rework is not possible. Round up to the next largest integer number.

$($Your answer should be formatted $00,000$ or $00000)$

$1$ point

Enter answer here

$4.$

Question $4$

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

$($Write answer as a decimal $0\mathrm{.}0000$ or $0\mathrm{.}00000)$