A design engineer wants to construct a sample mean chart for controlling the service life of a halogen headlamp his company produces. He knows from numerous previous samples that this service life is normally distributed with a mean of $500$ hours and a standard deviation of $20$ hours. On three recent production batches, he tested service life on random samples of four headlamps, with these results

Sample Service Life $($hours$)$

$1495500505500$

$2525515505515$

$3470480460470$

If he uses upper and lower control limits of $520$ and $480$ hours, what is his risk $($alpha$)$ of concluding that service life is out of control when it is actually under control $($Type I error$)?$

Group of answer choices

$0\mathrm{.}0026$

$0\mathrm{.}0456$

$0\mathrm{.}3174$

$0\mathrm{.}6826$

$0\mathrm{.}9544$