In Exercises 5 and 6, use the confidence interval to find the margin of error and the sample mean.
(20.75, 24.10)

검증된 단계별 안내

Step 1: Understand the problem. The confidence interval is given as (20.75, 24.10). The goal is to find the margin of error and the sample mean.

Step 2: Recall the formula for the margin of error. The margin of error (E) is half the width of the confidence interval. Mathematically, this can be expressed as:

E = \(\frac{\text{Upper Limit}\) - \(\text{Lower Limit}\)}{2}

Step 3: Recall the formula for the sample mean. The sample mean is the midpoint of the confidence interval. Mathematically, this can be expressed as:

\(\text{Sample Mean}\) = \(\frac{\text{Upper Limit}\) + \(\text{Lower Limit}\)}{2}

Step 4: Substitute the given values into the formulas. For the margin of error, substitute 24.10 as the upper limit and 20.75 as the lower limit into the formula for E. Similarly, substitute these values into the formula for the sample mean.

Step 5: Simplify the expressions to calculate the margin of error and the sample mean. This will give you the final results for both values.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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., (20.75, 24.10)) 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 can be calculated by taking the midpoint of the confidence interval, which provides a central value around which the interval is constructed. In this example, the sample mean can be found by averaging the two endpoints of the interval.

In Exercises 5 and 6, use the confidence interval to find the margin of error and the sample mean.(20.75, 24.10)