A manufacturer of a polycrystalline silicon carbide $($ SiC $)$ mechanical component measured the

fracture strength of $21$ different samples. The values $($in MPa$)$ were: $334,289,232,294,252,$

$337,256,339,308,365,311,341,286,314,274,285,382,379,282,324,$ and $316.$ They would

like to sell this component for a few different applications, and they need to be able to give a

failure strength as an engineering specification.

What strength value could the company certify if they want to sell these components for an

application where replacement is just a minor inconvenience? What if they wanted to sell

them for an application where failure causes injury? What about an application where failure

could cause loss of life? Manufacturers typically look for failure probabilities in these three

cases of$,$ respectively, $1$ component in $100,1$ component in $1$ million, and $1$ component in $100$

million. Compare those values to each other and to the $21$ components measured. Next,

determine the Weibull modulus. Does the Weibull modulus value suggest that the

manufacturer has good, average, or poor quality control relative to typical ceramics? Explain

all of your results in a few sentences.