Raising the bar

Chapter 8. Estimation: One or Two Populations

Unit 8.3.3. Confidence interval for population proportion

From Eq. (5.14) we know that $latex E\hat p = p,\,\, V(\hat p) = \frac{p(1-p)}{n}.$ By the central limit theorem the variable

$latex \frac{\hat p - E\hat p}{\sigma_{\hat p}} = \frac{\hat p - p}{\sqrt{p(1-p)/n}} \qquad (\textrm{Eq. 8.6})$

is approximately normal if $latex n$ is large enough (the rule of thumb is $latex np(1-p) > 9$). The problem with Eq. (8.6) is that in the denominator we have an unknown $latex p$. Following the general statistical idea, we replace it by $latex \hat p$ and obtain

$latex z_n = \frac{\hat p - p}{\sqrt{\hat p(1-\hat p)/n}}. \qquad (\textrm{Eq. 8.7})$

Exercise 8.7. Using Eq. (8.7) derive the confidence interval for $latex p$. What is the expression for the margin of error?

Previous: Confidence interval for the population mean: unknown variance | Next: Matched pairs

Want the whole book (all 14 chapters, exercises, and the index) as a single PDF? Available on Lulu for $1.99.