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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 6.1.16
Chapter 6, Problem 6.1.16

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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Step 1: Understand the formula for the margin of error (ME). The formula is ME = Z * (σ / √n), where Z is the critical value corresponding to the confidence level (c), σ is the population standard deviation, and n is the sample size.
Step 2: Determine the critical value (Z) for the given confidence level (c = 0.975). Use a Z-table or statistical software to find the Z-value that corresponds to the middle 97.5% of the standard normal distribution.
Step 3: Calculate the standard error (SE) using the formula SE = σ / √n. Substitute the given values of σ = 4.6 and n = 100 into the formula.
Step 4: Multiply the critical value (Z) by the standard error (SE) to compute the margin of error (ME). This step combines the results from Step 2 and Step 3.
Step 5: Interpret the margin of error in the context of the problem. The margin of error represents the range within which the true population parameter is expected to lie with the given confidence level.

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Key Concepts

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

Margin of Error

The margin of error quantifies the uncertainty in a sample estimate. It indicates the range within which the true population parameter is expected to lie, given a certain confidence level. A smaller margin of error suggests a more precise estimate, while a larger margin indicates more variability. It is commonly used in survey results and statistical inference.
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Confidence Level (c)

The confidence level represents the probability that the margin of error will contain the true population parameter. In this case, a confidence level of 0.975 means that if the same sampling method were repeated multiple times, approximately 97.5% of the calculated margins of error would capture the true value. It reflects the degree of certainty in the results.
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Standard Deviation (σ) and Sample Size (n)

Standard deviation (σ) measures the dispersion of data points around the mean, indicating how spread out the values are. A larger standard deviation suggests more variability in the data. Sample size (n) refers to the number of observations in a sample; larger sample sizes generally lead to more reliable estimates and smaller margins of error, as they better represent the population.
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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

Bisexual Idenfitication In a survey of 692 lesbian, gay, bisexual, or transgender U.S adults, 378 said that they consider themselves bisexual. Construct a 90% confidence interval for the population proportion of lesbian, gay, bisexual, or transgender U.S. adults who consider themselves bisexual. (Adapted from Gallup)

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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

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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Textbook Question

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

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Textbook Question

In Exercises 29–32, determine the minimum sample size n needed to estimate for the values of c, σ, and E.

c = 0.80, σ = 4.1, E = 2.

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