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}}.?

Question
Normal distributions
asked 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 \(\mu_{hat{p}}\) and \(\sigma_ {\hat{p}}\).?

Answers (1)

2021-02-14
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
\(\sigma_{hat{p}} = \sqrt{\frac{pq}{n}}\)
\(\sigma_{hat{p}} = \sqrt{\frac{0.21(1-0.21)}{33}}\)
\(\sigma_{hat{p}} = 0.071\)
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