Sure! Let's break down the process step-by-step to express the confidence interval [tex]$(43.7 \%, 57.1 \%)$[/tex] in the form of [tex]$\widehat{p} \pm M E$[/tex].
1. Identify the bounds of the confidence interval:
- The lower bound is [tex]\( 43.7 \% \)[/tex]
- The upper bound is [tex]\( 57.1 \% \)[/tex]
2. Calculate the point estimate ([tex]\(\widehat{p}\)[/tex]):
- The point estimate, often denoted as [tex]\(\widehat{p}\)[/tex], is the midpoint of the confidence interval.
- To find it, we take the average of the lower and upper bounds:
[tex]\[
\widehat{p} = \frac{43.7 + 57.1}{2}
\][/tex]
The point estimate [tex]\(\widehat{p}\)[/tex] is approximately [tex]\( 50.40 \% \)[/tex].
3. Determine the margin of error (ME):
- The margin of error is the difference between the point estimate and the lower bound (or half the width of the interval):
[tex]\[
ME = \frac{57.1 - 43.7}{2}
\][/tex]
The margin of error [tex]\( ME \)[/tex] is approximately [tex]\( 6.70 \% \)[/tex].
4. Present the confidence interval in [tex]\(\widehat{p} \pm ME\)[/tex] form:
- We can now express the confidence interval as:
[tex]\[
50.40 \% \pm 6.70 \%
\][/tex]
Thus, the confidence interval [tex]$(43.7 \%, 57.1 \%)$[/tex] is expressed in the form of [tex]\(\widehat{p} \pm M E\)[/tex] as:
[tex]\[
50.40 \% \pm 6.70 \%
\][/tex]
