Question

In the Framingham Offspring study 3630 subjects were on anti-hypertensive medication out of 4657 total subjects. What is the length of the 99 percent confidence interval for this data.

Answer 1

We have

$n=4657\\\hat{p}=\frac{3630}{4657}=0.779472$

The confidence interval for the proportion is

$\left( \hat{p}-z_{1-\frac{\alpha}{2}}\sqrt{\frac{\hat{p}\left( 1-\hat{p} \right)}{n}},\hat{p}+z_{1-\frac{\alpha}{2}}\sqrt{\frac{\hat{p}\left( 1-\hat{p} \right)}{n}} \right)$

from which the length of the interval is

$2z_{\frac{\alpha}{2}}\sqrt{\frac{\hat{p}\left( 1-\hat{p} \right)}{n}}=2\cdot 2.5758\sqrt{\frac{0.7795\left( 1-0.7795 \right)}{4657}}=0.0312969$