A product is made of an assembly of

$1$

$0$

$$parts$,$ each produced at a different workcell. The

average defect rate of the parts before inspection in each cell is

$5$

$\%$

$.$

$$Before leaving the cell,

every part passes through a two

$-$

stage inspection process, each stage of which identifies

$9$

$5$

$\%$

$$of the remaining defects. No identified defects are allowed to proceed without being

corrected

$.$

a

$.$

$$What is the defect rate of parts following the inspections? What is this rate expressed as ppm

$?$

b

$.$

$$Suppose a defect in any of the

$1$

$0$

$$parts results in a defective assembled product. What is

the defect rate of assembled products in ppm

$?$

c

$.$

$$If a mistake occurs while assembling

$1$

$$in every

$1$

$0$

$0$

$0$

$$products

$($

assuming the assembly

uses the

$1$

$0$

$$parts described earlier

$)$

$,$

$$what then is the defect rate in ppm

$?$

d

$.$

$$Assume the maximum acceptable defect rate is

$2$

$5$

$0$

$$ppm

$.$

$$Assuming no change in the

quality of the

$1$

$0$

$$parts used in assembly, what must the mistake rate during assembly be

reduced to

$?$