$1.(20$ points$)$ A semiconductor fabrication process has four process steps with data as follows:

The process has five failure mechanisms: Failures to meet process specifications for each of the four steps above, and particle contamination.

$($a$)(8$ points$)$ Estimate the probability that specification limits for all four steps are met by a random unit of production.

$($b$)(4$ points$)$ The overall yield of the process is $\backslash (84\backslash \%\backslash ).$ Estimate the contaminationlimited yield.

$($c$)(2$ points$)$ Statistical process control is applied to monitor and control the presence of particles on the wafers. When contamination is in statistical control, the mean number of particles deposited per wafer is $450.$ Measurements of five wafers were as follows:

Was particle contamination in statistical control on these wafers?

$($d$)$ The contamination engineers estimate that $25\backslash \%$ of deposited particles are fatal. Assuming that the baseline level of contamination corresponds to the mean number of particles deposited when in statistical control, estimate the baseline contamination$-$limited yield for a chip with an area of $0\mathrm{.}25$ sq cm $.$ Assume the wafer diameter is $200$ mm $.$

$($e$)$ Estimate the yield loss from particle excursions.