In Exercises 25–28, use the confidence interval to find the margin of error and the sample mean.
(21.61, 30.15)

Verified step by step guidance

1. Identify the given confidence interval, which is (21.61, 30.15). The lower bound is 21.61, and the upper bound is 30.15.
2. To find the margin of error (E), use the formula: $E\; =\; \backslash frac\{\backslash text\{Upper\; Bound\}\; -\; \backslash text\{Lower\; Bound\}\}\{2\}$. Substitute the values of the upper and lower bounds into this formula.
3. To find the sample mean ($\backslash bar\{x\}$), use the formula: $\backslash bar\{x\}\; =\; \backslash frac\{\backslash text\{Upper\; Bound\}\; +\; \backslash text\{Lower\; Bound\}\}\{2\}$. Substitute the values of the upper and lower bounds into this formula.
4. Perform the subtraction and division in the margin of error formula to calculate the value of E.
5. Perform the addition and division in the sample mean formula to calculate the value of $\backslash bar\{x\}$.

Key Concepts

Confidence Interval

A confidence interval is a range of values, derived from sample statistics, that is likely to contain the true population parameter. It is expressed as an interval (e.g., (21.61, 30.15)) and is associated with a confidence level, typically 95% or 99%, indicating the degree of certainty that the interval contains the parameter.

Margin of Error

The margin of error quantifies the uncertainty associated with a sample estimate. It is calculated as half the width of the confidence interval, representing the maximum expected difference between the sample statistic and the population parameter. In this case, it can be found by subtracting the lower limit from the upper limit of the interval and dividing by two.

Sample Mean

The sample mean is the average of a set of sample observations and serves as a point estimate of the population mean. It is calculated by summing all sample values and dividing by the number of observations. In the context of a confidence interval, the sample mean is typically the midpoint of the interval, providing a central value around which the margin of error is applied.