To express the confidence interval [tex]\((0.007, 0.101)\)[/tex] in the form of [tex]\(\hat{p} - E < p < \hat{p} + E\)[/tex], we need to find the midpoint [tex]\(\hat{p}\)[/tex] and the margin of error [tex]\(E\)[/tex].
1. Midpoint ([tex]\(\hat{p}\)[/tex]) Calculation:
[tex]\[
\hat{p} = \frac{\text{lower bound} + \text{upper bound}}{2}
\][/tex]
Given the lower bound is [tex]\(0.007\)[/tex] and the upper bound is [tex]\(0.101\)[/tex]:
[tex]\[
\hat{p} = \frac{0.007 + 0.101}{2} = 0.054
\][/tex]
2. Margin of Error ([tex]\(E\)[/tex]) Calculation:
[tex]\[
E = \frac{\text{upper bound} - \text{lower bound}}{2}
\][/tex]
Given the lower bound is [tex]\(0.007\)[/tex] and the upper bound is [tex]\(0.101\)[/tex]:
[tex]\[
E = \frac{0.101 - 0.007}{2} = 0.047
\][/tex]
3. Forming the Interval:
Using the values of [tex]\(\hat{p}\)[/tex] and [tex]\(E\)[/tex]:
[tex]\[
\hat{p} - E = 0.054 - 0.047 = 0.007
\][/tex]
[tex]\[
\hat{p} + E = 0.054 + 0.047 = 0.101
\][/tex]
Therefore, the confidence interval [tex]\(0.007 < p < 0.101\)[/tex] can be expressed in the form:
[tex]\[
0.007 < p < 0.101
\][/tex]
In summary, the given confidence interval [tex]\((0.007, 0.101)\)[/tex] can be neatly expressed as [tex]\(\hat{p} - E < p < \hat{p} + E\)[/tex] with:
[tex]\[
\boxed{0.007} < p < \boxed{0.101}
\][/tex]
