The overhead reach distances of adult females are normally distributed with a mean of $195cm$ and a standard deviation of $8\mathrm{.}9cm.$

a$.$ Find the probability that an individual distance is greater than $205\mathrm{.}00cm.$

b$.$ Find the probability that the mean for $15$ randomly selected distances is greater than $192\mathrm{.}80cm.$

c$.$ Why can the normal distribution be used in part $($b$),$ even though the sample size does not exceed $30?$

a$.$ The probability is

$($Round to four decimal places as needed.$)$

b$.$ The probability is

$($Round to four decimal places as needed.$)$

c$.$ Choose the correct answer below.

A$.$ The normal distribution can be used because the original population has a normal distribution.

B$.$ The normal distribution can be used because the mean is large

C$.$ The normal distribution can be used because the finite population correction factor is small.

D$.$ The normal distribution can be used because the probability is less than $0\mathrm{.}5$