 # 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 mu_{hat{p}} and sigma_{hat{p}}. Braxton Pugh 2020-10-21 Answered

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 $$\mu_{\hat{p}}$$ and $$\sigma_{\hat{p}}$$.

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We have binomial experiment with $$n = 100$$ and $$p = 0.23$$
$$np = 100(0.23)$$
$$np = 23$$
$$nq = 100(1 — 0.23)$$
$$nq = 77$$
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 $$\mu_{\hat{p}} = \hat{p}$$.
$$\mu_{\hat{p}} = 0.23$$
The formula for the standard error of the normal approximation to the $$\hat{p}$$ distribution is
$$\sigma_{\hat{p}}=\sqrt{\frac{pq}{n}}$$
$$\sigma_{\hat{p}}=\sqrt{0.23\frac{1-0.23}{100}}$$
$$\sigma_{\hat{p}}=0.042$$