Express the confidence interval $99.5 \pm 49.2$ in the form of a trilinear inequality:
$\square < \mu < \square$
A confidence interval represents a range of values within which we estimate a population parameter, such as the mean \(\mu\), to lie with a certain degree of confidence. The notation \(Point\ Estimate \pm Margin\ of\ Error\) provides a symmetric range around the point estimate. In our case, the confidence interval is given as \(99.5 \pm 49.2\). Here, the point estimate is \(99.5\) and the margin of error is \(49.2\).
To convert this range into the trilinear inequality form \(\text{Lower Bound} < \mu < \text{Upper Bound}\), we must calculate the boundaries.
The lower bound (\(L\)) is the point estimate minus the margin of error:
\[L = 99.5 - 49.2 = 50.3\]
The upper bound (\(U\)) is the point estimate plus the margin of error:
\[U = 99.5 + 49.2 = 148.7\]
With the lower bound of \(50.3\) and the upper bound of \(148.7\), we can now express the confidence interval as a trilinear inequality. The population mean \(\mu\) is bounded between these two values:
\[50.3 < \mu < 148.7\]
Understanding confidence intervals is essential in fields like medicine or manufacturing where you might estimate the average effectiveness of a drug or the average diameter of a bolt. For example, if a study reports the average heart rate increase after exercise as \(25 \pm 5\) beats per minute, it means the range is \(20 < \mu < 30\), which helps researchers assess the safety and impact of the activity accurately.