Textbook Question
Finding a Critical F-Value for a Right-Tailed Test In Exercises 5–8, find the critical F-value for a right-tailed test using the level of significance α and degrees of freedom d.f.N and d.f.D.
α=0.10, d.f.N=10, d.f.D=15
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Ch. 10 - Chi-Square Tests and the F-Distribution
Larson8th EditionElementary Statistics: Picturing the WorldISBN: 9780137493470
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Table of contents
- Ch. 1 - Introduction to Statistics87
- Ch. 2 - Descriptive Statistics176
- Ch. 3 - Probability184
- Ch. 4 - Discrete Probability Distributions102
- Ch. 5 - Normal Probability Distributions183
- Ch. 6 - Confidence Intervals154
- Ch. 7 - Hypothesis Testing with One Sample165
- Ch. 8 - Hypothesis Testing with Two Samples134
- Ch. 9 - Correlation and Regression87
- Ch. 10 - Chi-Square Tests and the F-Distribution87
Larson8th EditionElementary Statistics: Picturing the WorldISBN: 9780137493470Not the one you use?Change textbook
Chapter 10, Problem 10.3.4
Explain how to determine the values of d.f.N and d.f.D when performing a two-sample F-test.
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To determine the degrees of freedom for the numerator (d.f.N) in a two-sample F-test, identify the sample size of the first group (n₁). Subtract 1 from this sample size: d.f.N = n₁ - 1.
To determine the degrees of freedom for the denominator (d.f.D) in a two-sample F-test, identify the sample size of the second group (n₂). Subtract 1 from this sample size: d.f.D = n₂ - 1.
The F-test compares the variances of two independent samples, so the degrees of freedom are based on the sample sizes of the two groups being compared.
Ensure that the larger variance is placed in the numerator of the F-ratio to maintain consistency with the F-distribution's properties.
Finally, use the calculated d.f.N and d.f.D to find the critical value or p-value from the F-distribution table, depending on the significance level (α) of the test.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Degrees of Freedom (d.f.)
Degrees of freedom refer to the number of independent values or quantities that can vary in an analysis without violating any constraints. In the context of statistical tests, they are crucial for determining the distribution of the test statistic. For a two-sample F-test, the degrees of freedom help in assessing the variability between the sample means and the overall variability within the samples.
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Two-Sample F-Test
A two-sample F-test is a statistical method used to compare the variances of two independent samples to determine if they are significantly different. This test is based on the ratio of the variances of the two samples, which follows an F-distribution under the null hypothesis that the variances are equal. Understanding how to calculate and interpret this test is essential for analyzing the equality of variances.
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Calculating d.f.N and d.f.D
In a two-sample F-test, d.f.N (numerator degrees of freedom) is calculated as the number of groups minus one, while d.f.D (denominator degrees of freedom) is the total number of observations across all groups minus the number of groups. Specifically, if you have two samples with sizes n1 and n2, then d.f.N = 1 and d.f.D = n1 + n2 - 2. These values are essential for determining the critical value from the F-distribution.
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Related Practice
Textbook Question
Conditional Relative Frequencies In Exercises 37–42, use the contingency table from Exercises 33–36, and the information below.
Relative frequencies can also be calculated based on the row totals (by dividing each row entry by the row’s total) or the column totals (by dividing each column entry by the column’s total). These frequencies are conditional relative frequencies and can be used to determine whether an association exists between two categories in a contingency table.
What percent of U.S. adults ages 25 and over who are employed have a degree?
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Textbook Question
List five properties of the F-distribution.
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Textbook Question
In Exercises 13–18, test the claim about the difference between two population variances σ₁² and σ₂² at the level of significance α. Assume the samples are random and independent, and the populations are normally distributed.
Claim: σ₁² > σ₂²; α = 0.05.
Sample statistics: s₁² = 44.6, n₁ = 16 and s₂² = 39.3, n₂ = 12
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Textbook Question
In Exercises 13–18, test the claim about the difference between two population variances σ₁² and σ₂² at the level of significance α. Assume the samples are random and independent, and the populations are normally distributed.
Claim: σ₁² ≤ σ₂²; α = 0.01.
Sample statistics: s₁² = 842, n₁ = 11 and s₂² = 836, n₂ = 10
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Textbook Question
Performing a Chi-Square Independence Test In Exercises 13–28, perform the indicated chi-square independence test by performing the steps below.
a. Identify the claim and state H₀ and Hₐ
b. Determine the degrees of freedom, find the critical value, and identify the rejection region.
c. Find the chi-square test statistic.
d. Decide whether to reject or fail to reject the null hypothesis.
e. Interpret the decision in the context of the original claim.
Achievement and School Location The contingency table shows the results of a random sample of students by the location of school and the number of those students achieving basic skill levels in three subjects. At α=0.01, test the hypothesis that the variables are independent. (Adapted from HUD State of the Cities Report)
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