To express the confidence interval [tex]\((0.011, 0.101)\)[/tex] in the form of [tex]\(\hat{p} - E < p < \hat{p} + E\)[/tex], we need to determine the point estimate [tex]\(\hat{p}\)[/tex] and the margin of error [tex]\(E\)[/tex].
1. Find the point estimate [tex]\(\hat{p}\)[/tex]:
The point estimate [tex]\(\hat{p}\)[/tex] is the midpoint (or average) of the lower bound (0.011) and the upper bound (0.101) of the interval.
[tex]\[
\hat{p} = \frac{0.011 + 0.101}{2} = 0.056
\][/tex]
2. Determine the margin of error [tex]\(E\)[/tex]:
The margin of error [tex]\(E\)[/tex] is the distance from the point estimate to either the lower bound or the upper bound of the interval.
[tex]\[
E = \frac{0.101 - 0.011}{2} = 0.045
\][/tex]
Putting it together, we can express the confidence interval in the desired form:
[tex]\[
0.056 - 0.045 < p < 0.056 + 0.045
\][/tex]
Thus, the confidence interval [tex]\((0.011, 0.101)\)[/tex] can be expressed in the form [tex]\(\hat{p} - E < p < \hat{p} + E\)[/tex] as:
[tex]\[
0.011 < p < 0.101
\][/tex]
Here, [tex]\(\hat{p} = 0.056\)[/tex] and [tex]\(E = 0.045\)[/tex].
