Standard Normal Random Variable $(30$ pts$)$

$2.(30$ pts $)$ The pitch diameter of the thread on a fitting is normally distributed with a mean of $0\mathrm{.}4008$ centimeters $(cm)$ and a standard deviation of $0\mathrm{.}0004$ cm $.$ the design specification is $0\mathrm{.}4000+-0\mathrm{.}0010cm.$

$($a$)$ Notice that the operation follows the mean, not the specifications. Determine the probability that a thread falls into the design specification. $(15$ pts $)$

$5$

$($b$)$ The engineers have studied the results of such calculations. Now they decide to replace a worn cutting tool and adjust the machine producing the fittings so that the new mean falls directly at the center of the design specification$-$i$.$e$.,0\mathrm{.}4000.$ Determine the probability that a thread falls into the design specification. $(15$ pts$)$