"Basic Computation:p^ Distribution Suppose we have a binomial..." was answered by our experts

UkusakazaL

Answered question

2020-11-05

Basic Computation:

\hat{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

\hat{p}

distribution by a normal distribution? Why? What are the values of

μ_{h a t p}

and

σ_{p}

Answer & Explanation

Layton

Skilled2020-11-06Added 89 answers

We have binomial experiment with $n = 48$ and $p = 0.15$
$n p = 48 (0.15)$
$n p = 7.2$
$n q = 48 (1 — 0.15)$
$n q = 40.8$
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 $\hat{p}$ distribution is $μ_{h a t p} = \hat{p}$ .
$μ_{h a t p} = 0.15$
The formula for the standard error of the normal approximation to the $\hat{p}$ distribution is
$σ_{\hat{p}} = \sqrt{\frac{p q}{n}}$
$σ_{\hat{p}} = \sqrt{\frac{0.15 (1 - 0.15)}{48}}$
$σ_{\hat{p}} = 0.052$

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