# Basic Computation:hat{p} Distribution Suppose we have a binomial experiment in which success is defined to be a particular quality or attribute that i

Basic Computation:$\stackrel{^}{p}$ Distribution Suppose we have a binomial experiment in which success is defined to be a particular quality or attribute that interests us.
(c) Suppose $n=48$ and $p=0.15$. Can we approximate the $\stackrel{^}{p}$ distribution by a normal distribution? Why? What are the values of ${\mu }_{hatp}$ and ${\sigma }_{p}$.?
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Layton

We have binomial experiment with $n=48$ and $p=0.15$
$np=48\left(0.15\right)$
$np=7.2$
$nq=48\left(1—0.15\right)$
$nq=40.8$
Since both the values np and nq are greater than 5, hence, we can approximate the hat p distribution by a normal distribution.
The formula for the mean of the $\stackrel{^}{p}$ distribution is ${\mu }_{hatp}=\stackrel{^}{p}$.
${\mu }_{hatp}=0.15$
The formula for the standard error of the normal approximation to the $\stackrel{^}{p}$ distribution is
${\sigma }_{\stackrel{^}{p}}=\sqrt{\frac{pq}{n}}$
${\sigma }_{\stackrel{^}{p}}=\sqrt{\frac{0.15\left(1-0.15\right)}{48}}$
${\sigma }_{\stackrel{^}{p}}=0.052$