$\backslash$table$[[$Spec$.$ Limit,Percent Inside Specs,Dpm Defective$],[+-1$ Sigma,$68\mathrm{.}27,317300],[+-2$ Sigma,$95\mathrm{.}45,45500],[+-3$ Sigma,$99\mathrm{.}73,2700],[+-4$ Sigma,$99\mathrm{.}9937,63],[+-5$ Sigma,$99\mathrm{.}999943,0\mathrm{.}57],[+-6$ Sigma,$99\mathrm{.}9999998,0\mathrm{.}002]]$

$($a$)$ Normal distribution centered at the target $(T)$

$\backslash$table$[[$Spec$.$ Limit,Percent inside specs,ppm Defective$],[+-1$ Sigma,$30\mathrm{.}23,697700],[+-2$ Sigma,$69\mathrm{.}13,608700],[+-3$ Sigma,$93\mathrm{.}32,66810],[+-4$ Sigma,$99\mathrm{.}3790,6210],[+-5$ Sigma,$99\mathrm{.}97670,233],[+-6$ Sigma,$99\mathrm{.}999660,3\mathrm{.}4]]$

$($b$)$ Normal distribution with the mean shifted by $+-1\mathrm{.}5$ from the target

$(2$ points$)$ Show that the ppm for $+-3$ sigma is $2700$ ppm in figure $($a$)?$

$(2$ points$)$ Show that the ppm for $+-3$ sigma is $66810$ ppm in figure $($b$)?$

$(2$ points$)$ A process was working on $+-3$ sigma in figure $($a$)$ before being improved. After improvement, the process is working on $+-5$ sigma in figure $($b$).$ What is the percentage of improvement does the process have after improvement?

$(4$ points$)$ Suppose that your business is operating at the five$-$sigma quality level, assuming the $1\mathrm{.}5$ sigma shift in the mean that is customary with six$-$sigma applications$).$ If projects have an average improvement rate of $5\%$ monthly, how many years will it take to achieve six$-$sigma quality assuming $1\mathrm{.}5$ sigma shift in the mean?