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Ch. 6 - Confidence Intervals
Larson - Elementary Statistics: Picturing the World 8th Edition
Larson8th EditionElementary Statistics: Picturing the WorldISBN: 9780137493470Not the one you use?Change textbook
All textbooksLarson 8th EditionCh. 6 - Confidence IntervalsProblem 6.1.26
Chapter 6, Problem 6.1.26

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

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

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.
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Introduction to Confidence Intervals

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.
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Finding the Minimum Sample Size Needed for a Confidence Interval

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.
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Related Practice
Textbook Question

LGBT Identification In a survey of 15,349 U.S. adults, 860 identify as lesbian, gay, bisexual, or transgender. Construct a 95% confidence interval for the population proportion of U.S. adults who identify as lesbian, gay, bisexual, or transgender. (Adapted from Gallup)

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Textbook Question

In Exercises 13–16, find the margin of error for the values of c, σ and n.

c = 0.975, σ = 4.6, n = 100

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Textbook Question

Finding Critical Values for χ2 In Exercises 3–8, find the critical values χR2 and χL2 for the level of confidence c and sample size n.

c = 0.99, n = 30

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Textbook Question

When estimating the population mean, why not construct a 99% confidence interval every time?

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Textbook Question

Finding the Margin of Error In Exercises 33 and 34, use the confidence interval to find the estimated margin of error. Then find the sample mean. Book Prices A store manager reports a confidence interval of (244.07, 280.97) when estimating the mean price (in dollars) for the population of textbooks.

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Textbook Question

In Exercises 5–8, find the critical value zc necessary to construct a confidence interval at the level of confidence c.

c = 0.80

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