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2021-02-13

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.
(a) Suppose $n = 33$ and $p = 0.21$ . Can we approximate the $\hat{p}$
distribution by a normal distribution? Why? What are the values of $μ_{\hat{p}}$ and $σ_{\hat{p}}$ .?

Answer & Explanation

SkladanH

Skilled2021-02-14Added 80 answers

We have binomial experiment with $n = 33$ and $p = 0.21$
$n p = 33 (0.21)$
$n p = 6.93$
$n q = 33 (1 — 0.21)$
$n q = 26.07$
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 $μ_{\hat{p}} = \hat{p}$ .
$μ_{\hat{p}} = 0.21$
The formula for the standard error of the normal approximation to the $\hat{p}$ distribution is
$σ_{\hat{p}} = \sqrt{\frac{p q}{n}}$
$σ_{\hat{p}} = \sqrt{\frac{0.21 (1 - 0.21)}{33}}$
$σ_{\hat{p}} = 0.071$

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