In Exercises 25–28, use the confidence interval to find the margin of error and the sample mean.
(3.144, 3.176)

Verified step by step guidance

1. Identify the given confidence interval, which is (3.144, 3.176). The lower bound is 3.144, and the upper bound is 3.176.
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 margin of error.
5. Perform the addition and division in the sample mean formula to calculate the sample mean.

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., (3.144, 3.176)) and is associated with a confidence level, indicating the probability that the interval contains the parameter. Understanding confidence intervals is crucial for estimating population parameters and assessing the precision of sample estimates.

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 the given interval (3.144, 3.176), the margin of error would be 0.016, indicating how much the sample mean could vary from the true mean.

Sample Mean

The sample mean is the average of a set of observations from a sample, serving 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 the confidence interval (3.144, 3.176), the sample mean can be found as the midpoint of the interval, which provides a central value around which the data is distributed.