# Basic Computation:hat{p} Distribution Suppose we have a binomial experiment in which success is defined to be a particular quality or attribute that i

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

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