fortdefruitI

2021-02-13

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.
(a) Suppose $n=33$ and $p=0.21$. Can we approximate the $\stackrel{^}{p}$
distribution by a normal distribution? Why? What are the values of ${\mu }_{\stackrel{^}{p}}$ and ${\sigma }_{\stackrel{^}{p}}$.?

We have binomial experiment with $n=33$ and $p=0.21$
$np=33\left(0.21\right)$
$np=6.93$
$nq=33\left(1—0.21\right)$
$nq=26.07$
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 hat p distribution is ${\mu }_{\stackrel{^}{p}}=\stackrel{^}{p}$.
${\mu }_{\stackrel{^}{p}}=0.21$
The formula for the standard error of the normal approximation to the $\stackrel{^}{p}$ distribution is
${\sigma }_{\stackrel{^}{p}}=\sqrt{\frac{pq}{n}}$
${\sigma }_{\stackrel{^}{p}}=\sqrt{\frac{0.21\left(1-0.21\right)}{33}}$
${\sigma }_{\stackrel{^}{p}}=0.071$

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