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#.

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!

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Answer 2

Hunter Aldridge

To rewrite the confidence interval (0.0268, 0.133) in the form of hatp - E < p < hatp + E, where hatp is the point estimate and E is the margin of error:

Find the point estimate (hatp) by taking the average of the lower and upper bounds of the confidence interval: hatp = (0.0268 + 0.133) / 2 = 0.0799.
Find the margin of error (E) by subtracting the lower bound from the point estimate (or alternatively, subtracting the point estimate from the upper bound): E = 0.0799 - 0.0268 = 0.0531.
Rewrite the confidence interval in the desired form: 0.0799 - 0.0531 < p < 0.0799 + 0.0531.

Therefore, the confidence interval in the form of hatp - E < p < hatp + E is: 0.0268 < p < 0.133.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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

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