Design specifications require that a key dimension on a product measure $100$ $ $10$ units. A process being considered for producing this product has a standard deviation of $4$ units.

$($a$)$ What can you say $($quantitatively$)$ regarding the process capability?

$($b$)$ What are the minimum design specifications that would make this process capable?

$($c$)$ Suppose the process average shifts to $92.$ Calculate and comment on the new process capability.

$($d$)$ What can you say about the process after the shift? Approximately what percentage of the items produced will be defective?

$($e$)$ With a process average of $92,$ what are the design specification limits that correspond to containing $90\%$ of the products within the limits$?$