A simple random sample of size $n=200$ is obtained from a population whose size is $N=15,000$ and whose population proportion with a specified characteristic is $p=0\mathrm{.}6.$ Complete parts $($a$)$ through $($c$).$

Determine the mean of the sampling distribution of hat$(p).$

${}_{\text{h}\text{a}\text{t}(p)}=($Round to one decimal place as needed.$)$

Determine the standard deviation of the sampling distribution of hat$(p).$

${}_{\text{h}\text{a}\text{t}(p)}=,($Round tofy six decimal places as needed.$)$

$($b$)$ What is the probability of obtaining $x=126$ or more individuals with the characteristic? That is$,$ what is $P(hat(p)0\mathrm{.}63)?$

$P(hat(p)0\mathrm{.}63)=($Round to four decimal places as needed.$)$

$($c$)$ What is the probability of obtaining $x=108$ or fewer individuals with the characteristic? That is$,$ what is $P(hat(p)0\mathrm{.}54)?$

$P(hat(p)0\mathrm{.}54)=($Round to four decimal places as needed.$)$