Express the confidence interval $(347.7, 479.5)$ in the form of $\bar{x} \pm E$.
$\bar{x}$ | ± | $E$ |
□ | ± | □ |
A confidence interval represents a range of values within which we are fairly confident the true population mean lies. It is typically expressed in the format \(\bar{x} \pm E\), where \(\bar{x}\) is the sample mean (the center of the interval) and \(E\) is the margin of error (the distance from the center to either endpoint of the interval). We are given the interval \((347.7, 479.5)\). Let \(L = 347.7\) be the lower bound and \(U = 479.5\) be the upper bound.
The sample mean \(\bar{x}\) is the midpoint of the interval. We calculate it by taking the average of the lower and upper bounds:
\[\bar{x} = \frac{L + U}{2} = \frac{347.7 + 479.5}{2} = \frac{827.2}{2} = 413.6\]
So, the sample mean is \(413.6\).
The margin of error \(E\) is the distance from the midpoint to either endpoint. We calculate it by finding half the width of the interval:
\[E = \frac{U - L}{2} = \frac{479.5 - 347.7}{2} = \frac{131.8}{2} = 65.9\]
Thus, the margin of error is \(65.9\).
Combining these two values into the requested format \(\bar{x} \pm E\), we have \(413.6 \pm 65.9\). This means the interval starts at \(413.6 - 65.9 = 347.7\) and ends at \(413.6 + 65.9 = 479.5\), confirming our calculations are correct.
| \(\bar{x}\) | \(\pm\) | \(E\) |
| :--- | :--- | :--- |
| \(413.6\) | \(\pm\) | \(65.9\) |
Confidence intervals are essential in real-world data analysis, such as when researchers estimate the average height of a group or the expected revenue of a product. For instance, if a company estimates its next month's sales will fall between \$347,700 and \$479,500, they express this as \$413,600 \pm \$65,900 to clearly convey both their expected mean and the degree of uncertainty in their forecast. Understanding this conversion helps stakeholders quickly grasp both the central tendency and the risk associated with an estimate.