Express the confidence interval $683.7 < \mu < 957.9$ in the form of $\bar{x} \pm E$.
A confidence interval provides a range of values within which we expect the population mean \(\mu\) to lie. In this problem, we are given an interval \((683.7, 957.9)\) and asked to express it in the standard point estimate form, \(\bar{x} \pm E\), where \(\bar{x}\) is the sample mean (the midpoint of the interval) and \(E\) is the margin of error (the distance from the mean to the endpoints).
To find the sample mean \(\bar{x}\), we calculate the midpoint of the lower bound \(L = 683.7\) and the upper bound \(U = 957.9\):
\[\bar{x} = \frac{L + U}{2} = \frac{683.7 + 957.9}{2} = \frac{1641.6}{2} = 820.8\]
So, the sample mean \(\bar{x}\) is \(820.8\).
The margin of error \(E\) represents the distance from the midpoint to either endpoint. We calculate this as half the total width of the interval:
\[E = \frac{U - L}{2} = \frac{957.9 - 683.7}{2} = \frac{274.2}{2} = 137.1\]
Thus, the margin of error \(E\) is \(137.1\).
Having found the sample mean \(\bar{x} = 820.8\) and the margin of error \(E = 137.1\), we can express the given interval as:
\[\bar{x} \pm E = 820.8 \pm 137.1\]
This form tells us that the population mean is estimated to be \(820.8\), with a margin of error of \(137.1\).
In real-world data science, such as calculating the average time a customer spends on a website, if a survey shows a 95% confidence interval for the average time as \( (3.5, 4.5) \) minutes, we report it as \(4.0 \pm 0.5\) minutes. This clearly tells the stakeholders that our best estimate is 4 minutes, with a potential variation of 0.5 minutes in either direction due to sampling error.