The average cost for teeth straightening with metal braces is approximately $$5400.$ A nationwide franchise thinks that its cost is below that figure. A random sample of $28$ patients across the country had an average cost of $$5250$ and the standard deviation for this sample was $$629.$ At $=0\mathrm{.}025,$ can it be concluded that the mean is less than $$5400?(22$ points$)$

$($a$)$ State both hypotheses.

$($b$)$ Find the critical value$($s$).$

$($c$)$ Compute the test value

$($d$)$ Make the decision to reject or not to reject the null hypothesis by stating "Reject" or "Not Reject".