Verified Answer :

  • The given confidence interval is expressed in the form \(\bar{x} \pm E\), where \(\bar{x} = 11.1\) and \(E = 52.6\).
  • The interval is calculated as \((L, U) = (\bar{x} - E, \bar{x} + E)\).
  • Substituting the values, we get \((11.1 - 52.6, 11.1 + 52.6)\).
  • The resulting interval in standard notation is \(\boxed{(-41.5, 63.7)}\).

Explanation

  1. Analyze the confidence interval notation

A confidence interval provides a range of values within which we expect a population parameter, such as the mean \(\mu\), to lie with a certain degree of confidence. The notation \(\bar{x} \pm E\) is a common way to express this interval, where \(\bar{x}\) is the point estimate (the sample mean) and \(E\) is the margin of error. In our case, the given interval is \(11.1 \pm 52.6\). This means:

  1. The point estimate \(\bar{x}\) is \(11.1\).
  2. The margin of error \(E\) is \(52.6\).
  1. Calculate the lower and upper bounds

To express the confidence interval in the standard interval form \((L, U)\), we calculate the lower bound \(L\) and the upper bound \(U\) using the point estimate and the margin of error:

  • The lower bound \(L\) is defined as \(\bar{x} - E\).
  • The upper bound \(U\) is defined as \(\bar{x} + E\).

We substitute the given values into these expressions:

  • \(L = 11.1 - 52.6 = -41.5\)
  • \(U = 11.1 + 52.6 = 63.7\)
  1. Finalize the interval notation

Now that we have found the lower and upper bounds, we can write the confidence interval in the requested interval notation \((L, U)\). Substituting our calculated values, the interval is \((-41.5, 63.7)\).

Examples

In real-world data science, calculating a confidence interval is crucial for understanding the uncertainty of a sample statistic. For example, if you are surveying the average time customers spend in a store, a result of \(11.1 \pm 52.6\) minutes suggests that while the sample average is \(11.1\) minutes, the true population mean could plausibly range from \(-41.5\) to \(63.7\) minutes, indicating a high degree of variability in the collected data. This interval helps analysts convey the reliability of their estimates.

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