A study of the annual growth of certain cactus showed that, $64$ of them, selected at random in a desert region,

grew on the average $52\mathrm{.}80mm$ with a standart deviation of $4\mathrm{.}5mm.$

$($a$)$ Construct a $95\%$ confidence interval for the true average annual growth of the cactus.

$($b$)$ With what degree of confidence could we say that the mean of annual growth of all $64$ cactus is $(51\mathrm{.}5,54\mathrm{.}1)?$