In Exercises 13 and 14, use the confidence interval to find the margin of error and the sample mean.
(14.7, 22.1)

Verified step by step guidance

1. Step 1: Understand the problem. The confidence interval is given as (14.7, 22.1). The goal is to find the margin of error and the sample mean. The confidence interval represents the range of values within which the true population parameter is expected to lie.
2. Step 2: Recall the formula for the margin of error. The margin of error (E) is half the width of the confidence interval. Mathematically, it is calculated as: $E\; =\; \backslash (\backslash frac\{\backslash text\{Upper\; Limit\}\backslash )\; -\; \backslash (\backslash text\{Lower\; Limit\}\backslash )\}\{2\}$.
3. Step 3: Recall the formula for the sample mean. The sample mean (\(\bar{x}\)) is the midpoint of the confidence interval. Mathematically, it is calculated as: $\backslash (\backslash bar\{x\}\backslash )\; =\; \backslash (\backslash frac\{\backslash text\{Upper\; Limit\}\backslash )\; +\; \backslash (\backslash text\{Lower\; Limit\}\backslash )\}\{2\}$.
4. Step 4: Substitute the given values into the formulas. For the margin of error, substitute 22.1 as the upper limit and 14.7 as the lower limit into the formula for E. For the sample mean, substitute the same values into the formula for \(\bar{x}\).
5. Step 5: Simplify the expressions to find the numerical values of the margin of error and the sample mean. This will give you the final results.

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., (14.7, 22.1)) and is associated with a confidence level, typically 95% or 99%. This means that if we were to take many samples and build intervals in this way, a certain percentage of them would contain the true 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 the given interval (14.7, 22.1), the margin of error would be (22.1 - 14.7) / 2 = 3.7.

Sample Mean

The sample mean is the average of a set of values obtained from a sample, serving as an 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, the sample mean can be found as the midpoint of the interval, which is (14.7 + 22.1) / 2 = 18.4.