To express the confidence interval [tex]\((0.066, 0.130)\)[/tex] in the form of [tex]\(\hat{p} - E < p < \hat{p} + E\)[/tex], we need to identify the point estimate [tex]\(\hat{p}\)[/tex] and the margin of error [tex]\(E\)[/tex].
1. Calculate the point estimate [tex]\(\hat{p}\)[/tex]:
The point estimate [tex]\(\hat{p}\)[/tex] is the midpoint of the given confidence interval. It can be found by averaging the lower and upper bounds of the interval:
[tex]\[
\hat{p} = \frac{0.066 + 0.130}{2}
\][/tex]
2. Calculate the margin of error [tex]\(E\)[/tex]:
The margin of error [tex]\(E\)[/tex] is half the width of the confidence interval. It can be found by subtracting the lower bound from the upper bound and dividing by 2:
[tex]\[
E = \frac{0.130 - 0.066}{2}
\][/tex]
Given the results:
[tex]\[
\hat{p} = 0.098
\][/tex]
[tex]\[
E = 0.032
\][/tex]
Thus, the confidence interval [tex]\((0.066, 0.130)\)[/tex] can be expressed in the form [tex]\(\hat{p} - E < p < \hat{p} + E\)[/tex] as follows:
[tex]\[
0.098 - 0.032 < p < 0.098 + 0.032
\][/tex]
Simplifying the expressions, we get:
[tex]\[
0.066 < p < 0.130
\][/tex]
Therefore, the confidence interval [tex]\((0.066, 0.130)\)[/tex] expressed in the form [tex]\(\hat{p} - E < p < \hat{p} + E\)[/tex] is:
[tex]\[
0.066 < p < 0.130
\][/tex]
