Solved: "Distribution Suppose we have a binomial experiment in..."

Braxton Pugh

Answered question

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 $\hat{p}$ distribution by a normal distribution? Why?
Compute $μ_{\hat{p}}$ and $σ_{\hat{p}}$ .

Answer & Explanation

berggansS

Skilled2020-10-22Added 91 answers

We have binomial experiment with $n = 100$ and $p = 0.23$
$n p = 100 (0.23)$
$n p = 23$
$n q = 100 (1 — 0.23)$
$n q = 77$
Since both the values np and ng 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 $μ_{\hat{p}} = \hat{p}$ .
$μ_{\hat{p}} = 0.23$
The formula for the standard error of the normal approximation to the $\hat{p}$ distribution is
$σ_{\hat{p}} = \sqrt{\frac{p q}{n}}$
$σ_{\hat{p}} = \sqrt{0.23 \frac{1 - 0.23}{100}}$
$σ_{\hat{p}} = 0.042$

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