Question

express the confidence interval 0.444 < p < 0.666 in the form \hat{p} \pm e. \hat{p} \pm e = \square \pm \square

Explanation:

Step1: Find the sample proportion (\(\hat{p}\))

The confidence interval is in the form \((\text{lower bound}, \text{upper bound})\), so \(\hat{p}\) is the midpoint of the lower and upper bounds. The formula for the midpoint is \(\hat{p}=\frac{\text{lower bound}+\text{upper bound}}{2}\).
Given lower bound \( = 0.444\) and upper bound \( = 0.666\), we have:
\[
\hat{p}=\frac{0.444 + 0.666}{2}=\frac{1.11}{2}=0.555
\]

Step2: Find the margin of error (\(E\))

The margin of error \(E\) is the distance from \(\hat{p}\) to either the lower bound or the upper bound. We can calculate it as \(E=\hat{p}-\text{lower bound}\) (or \(E = \text{upper bound}-\hat{p}\)).
Using \(\hat{p}=0.555\) and lower bound \( = 0.444\):
\[
E=0.555 - 0.444 = 0.111
\]

Answer:

\(0.555\pm0.111\)