$(20\%)$ Polychlorinated biphenyls $($PCBs$)$ have been found to be dangerously high in some game

birds found along the marshlands of the southeastern coast of the United States. The Federal Drug

Administration $($FDA$)$ considers a concentration of PCBs higher than $5$ parts per million $($ppm$)$

in these game birds to be dangerous for human consumption. A sample of $38$ game birds produced

an average of $7\mathrm{.}2ppm$ with a standard deviation of $6\mathrm{.}2ppm.$

$($a$)$ Is there sufficient evidence to indicate that the mean ppm of PCBs in the population of game

birds exceeds the FDA's recommended limit of $5$ ppm$?$ Use $=0\mathrm{.}05.$

$($b$)$ Calculate the type II error if the true mean ppm of PCBs is $6ppm.$

$($c$)$ Calculate the decision power when the true mean ppm of PCBs is $7ppm.$

$($d$)$ For what values of the true mean ppm of PCBs does the test in $($a$)$ have power greater than or

equal to $0\mathrm{.}90?$