# How do I rewrite the confidence interval (0.0268, 0.133) in the form of hatp - E < p < hatp + E?

Dec 7, 2016

$0.0799 - 0.0531 < p < 0.0799 + 0.0531$.

#### Explanation:

If I understand correctly, we simply need to take the confidence bounds $\left(0.0268 , 0.133\right)$ and convert them to a "central value" $\hat{p}$, plus/minus an error margin $E$.

$\hat{p}$ will be halfway between our lower- and upper-bound, and so we take the average of the two bounds:

$\hat{p} = \frac{0.0268 + 0.133}{2} = \frac{0.1598}{2} = 0.0799$

The error margin $E$ will just be the distance between this $\hat{p}$ value and one of the original bounds:

$E = \left\mid \hat{p} - 0.0268 \right\mid$ or $E = \left\mid \hat{p} - 0.133 \right\mid$
$\textcolor{w h i t e}{E} = \left\mid 0.0799 - 0.0268 \right\mid = \left\mid 0.0799 - 0.133 \right\mid$
$\textcolor{w h i t e}{E} = \left\mid 0.0531 \right\mid \textcolor{w h i t e}{X X X \Xi i i} = \left\mid - 0.0531 \right\mid$
$\textcolor{w h i t e}{E} = 0.0531$

So in $\hat{p} - E < p < \hat{p} + E$ form, our confidence interval is

$0.0799 - 0.0531 < p < 0.0799 + 0.0531$
or
$p \in \left(0.0799 \pm 0.0531\right) .$

I hope this is the answer you're looking for!