Braxton Pugh

2020-10-21

Distribution Suppose we have a binomial experiment in which success is defined to be a particular quality or attribute that interests us.
(a) Suppose $n=100$ and $p=0.23$. Can we safely approximate the $\stackrel{^}{p}$ distribution by a normal distribution? Why?
Compute ${\mu }_{\stackrel{^}{p}}$ and ${\sigma }_{\stackrel{^}{p}}$.

berggansS

We have binomial experiment with $n=100$ and $p=0.23$
$np=100\left(0.23\right)$
$np=23$
$nq=100\left(1—0.23\right)$
$nq=77$
Since both the values np and ng are greater than 5, hence, we can approximate the $\stackrel{^}{p}$ distribution by a normal distribution.
The formula for the mean of the $\stackrel{^}{p}$ distribution is ${\mu }_{\stackrel{^}{p}}=\stackrel{^}{p}$.
${\mu }_{\stackrel{^}{p}}=0.23$
The formula for the standard error of the normal approximation to the $\stackrel{^}{p}$ distribution is
${\sigma }_{\stackrel{^}{p}}=\sqrt{\frac{pq}{n}}$
${\sigma }_{\stackrel{^}{p}}=\sqrt{0.23\frac{1-0.23}{100}}$
${\sigma }_{\stackrel{^}{p}}=0.042$

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