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:

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$)?$