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

\(\sigma_{\hat{p}} = \sqrt{\frac{pq}{n}}\)

\(\sigma_{\hat{p}} = \sqrt{\frac{0.21(1-0.21)}{33}}\)

\(\sigma_{\hat{p}} = 0.071\)