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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 6.2.15
Chapter 6, Problem 6.2.15

In Exercises 15 and 16, find the t-value for the given values of xbar, μ, s and n.
xbar = 70.3, μ = 64.8, s = 7.1, n = 16

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Step 1: Recall the formula for the t-value in a one-sample t-test: t = \(\frac{x̄ - μ}{s / \sqrt{n}\)}, where x̄ is the sample mean, μ is the population mean, s is the sample standard deviation, and n is the sample size.
Step 2: Substitute the given values into the formula: x̄ = 70.3, μ = 64.8, s = 7.1, and n = 16. The formula becomes: t = \(\frac{70.3 - 64.8}{7.1 / \sqrt{16}\)}.
Step 3: Simplify the denominator by calculating the standard error of the mean: \(\text{Standard Error}\) = \(\frac{s}{\sqrt{n}\)} = \(\frac{7.1}{\sqrt{16}\)}. Compute the square root of 16 and divide 7.1 by the result.
Step 4: Subtract the population mean (μ) from the sample mean (x̄): 70.3 - 64.8. This gives the numerator of the t-value formula.
Step 5: Divide the result from Step 4 (numerator) by the result from Step 3 (denominator) to calculate the t-value: t = \(\frac{\text{Numerator}\)}{\(\text{Denominator}\)}.

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Key Concepts

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

t-value

The t-value is a statistic used in hypothesis testing to determine if there is a significant difference between the sample mean and the population mean. It is calculated by taking the difference between the sample mean (x̄) and the population mean (μ), and dividing it by the standard error of the mean. The t-value helps assess how far the sample mean is from the population mean in terms of standard deviations.
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Standard Error of the Mean (SEM)

The Standard Error of the Mean (SEM) quantifies how much the sample mean (x̄) is expected to vary from the true population mean (μ). It is calculated by dividing the sample standard deviation (s) by the square root of the sample size (n). A smaller SEM indicates that the sample mean is a more accurate estimate of the population mean, which is crucial for calculating the t-value.
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Calculating the Mean

Degrees of Freedom

Degrees of freedom (df) refer to the number of independent values that can vary in a statistical calculation. In the context of a t-test, the degrees of freedom are typically calculated as n - 1, where n is the sample size. This concept is important because it affects the shape of the t-distribution used to determine critical values and p-values in hypothesis testing.
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Related Practice
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 = 15

Textbook Question

In Exercise 28, the population mean weekly time spent on homework by students is 7.8 hours. Does the t-value fall between -t0.99 and t0.99?

Textbook Question

Why Check It? Why is it necessary to check that np^ ≥ 5 and nq^ ≥ 5?

Textbook Question

Describe how the t-distribution curve changes as the sample size increases.

Textbook Question

Constructing Confidence Intervals In Exercises 13 and 14, construct a 99% confidence interval for the population proportion. Interpret the results.

New Year’s Resolutions In a survey of 1790 U.S. adults in a recent year, 816 have a New Year’s resolution related to their health. (Adapted from Finder)

Textbook Question

Constructing Confidence Intervals In Exercises 27 and 28, use the figure, which shows the results of a survey in which 1021 U.S. adults were asked whether they see each of the possible threats to the vital interests of the United States as a critical threat in the next 10 years. (Source: Gallup)

Critical Threats Construct a 95% confidence interval for the population proportion of U.S. adults who gave each response.