Express the confidence interval $(0.488, 0.61)$ in the form of (point estimate) $\pm$ (margin of error):
$\square \pm \square$
We are given a confidence interval as an open interval: \((0.488, 0.61)\). Our task is to express this interval in the form of a point estimate \(\pm\) margin of error, where the point estimate is the midpoint of the interval and the margin of error is the distance from the midpoint to either endpoint.
The point estimate, denoted as \(\hat{p}\), is the midpoint of the confidence interval \((L, U)\). We calculate this using the formula:
\[\hat{p} = \frac{L + U}{2}\]
Given \(L = 0.488\) and \(U = 0.61\), we have:
\[\hat{p} = \frac{0.488 + 0.61}{2} = \frac{1.098}{2} = 0.549\]
The margin of error, denoted as \(E\), is the distance from the point estimate to either endpoint of the interval. We calculate this using the formula:
\[E = \frac{U - L}{2}\]
Using our values for \(L\) and \(U\):
\[E = \frac{0.61 - 0.488}{2} = \frac{0.122}{2} = 0.061\]
With the point estimate \(\hat{p} = 0.549\) and the margin of error \(E = 0.061\), we can express the confidence interval \((0.488, 0.61)\) as:
\[0.549 \pm 0.061\]
This form clearly shows the center of the interval and the width of the variation on either side. Thus, the final answer is \(\boxed{0.549 \pm 0.061}\).
Confidence intervals are essential in statistics for estimating population parameters. For example, if a poll shows that a candidate's support is between 48.8% and 61%, you can report this as 54.9% \(\pm\) 6.1%. This helps communicate both the most likely value and the range of uncertainty associated with the estimate.