$1.$ 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 when this service life is in control it 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$)$

$1495500505500$

$2525515505515$

$3470480460470$

$2.$ What is the mean of the sampling distribution of sample means when the service life is in control?

$3.515$ hours

$250$ hours

$500$ hours

$470$ hours

$495$ hours