Question

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}}.

Normal distributions
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asked 2020-10-21

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 \(\sigma_{\hat{p}}\).

Answers (1)

2020-10-22

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
\(\sigma_{\hat{p}}=\sqrt{\frac{pq}{n}}\)
\(\sigma_{\hat{p}}=\sqrt{0.23\frac{1-0.23}{100}}\)
\(\sigma_{\hat{p}}=0.042\)

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