Question

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

Answer 1

`0.0799-0.0531< p < 0.0799+0.0531`.

Explanation:

If I understand correctly, we simply need to take the confidence bounds `(0.0268, 0.133)` and convert them to a "central value" `hatp`, plus/minus an error margin `E`.

`hatp` will be halfway between our lower- and upper-bound, and so we take the average of the two bounds:

`hatp=(0.0268+0.133)/2=0.1598/2=0.0799`

The error margin `E` will just be the distance between this `hatp` value and one of the original bounds:

`E=abs(hatp-0.0268)` or `E=abs(hatp-0.133)`
`color(white)E=abs(0.0799-0.0268)=abs(0.0799-0.133)`
`color(white)E=abs(0.0531)color(white)(XXXXiii)=abs(-0.0531)`
`color(white)E=0.0531`

So in `hatp-E < p < hatp + E` form, our confidence interval is

`0.0799-0.0531< p < 0.0799+0.0531`
or
`p in (0.0799+-0.0531).`

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