Table of contents

1. Intro to Stats and Collecting Data55m
2. Describing Data with Tables and Graphs1h 55m
3. Describing Data Numerically1h 45m
4. Probability2h 16m
5. Binomial Distribution & Discrete Random Variables2h 33m
6. Normal Distribution and Continuous Random Variables1h 38m
7. Sampling Distributions & Confidence Intervals: Mean1h 53m
8. Sampling Distributions & Confidence Intervals: Proportion1h 12m
- Sampling Distribution of Sample Proportion
  29m
- Confidence Intervals for Population Proportion
  42m
9. Hypothesis Testing for One Sample2h 19m
10. Hypothesis Testing for Two Samples3h 22m
11. Correlation1h 6m
12. Regression1h 4m
13. Chi-Square Tests & Goodness of Fit1h 20m
14. ANOVA1h 0m

14. ANOVA

Introduction to ANOVA

Statistics

14. ANOVA

Introduction to ANOVA

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2:15 minutes

Problem 10.3.12

Textbook Question

"Finding a Critical F-Value for a Two-Tailed Test In Exercises 9–12, find the critical F-value for a two-tailed test using the level of significance α and degrees of freedom d.f.N and d.f.D.

α=0.05, d.f.N=27, d.f.D=19"

Verified step by step guidance

Step 1: Understand the problem. We are tasked with finding the critical F-value for a two-tailed test. The level of significance (α) is 0.05, and the degrees of freedom for the numerator (d.f.N) and denominator (d.f.D) are 27 and 19, respectively.

Step 2: Recognize that for a two-tailed test, the significance level (α) is split equally between the two tails of the F-distribution. This means each tail will have an area of α/2 = 0.05/2 = 0.025.

Step 3: Use an F-distribution table or statistical software to find the critical F-value. Look up the F-value corresponding to the upper tail with α/2 = 0.025, d.f.N = 27 (numerator degrees of freedom), and d.f.D = 19 (denominator degrees of freedom).

Step 4: Remember that the F-distribution is not symmetric. For the lower tail critical value, you can use the reciprocal property of the F-distribution: F_lower = 1 / F_upper, where F_upper is the critical value from Step 3.

Step 5: Combine the results. The critical F-values for the two-tailed test will be the lower tail critical value (F_lower) and the upper tail critical value (F_upper). These values define the rejection regions for the null hypothesis.

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Key Concepts

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

F-Distribution

The F-distribution is a probability distribution that arises frequently in statistics, particularly in the context of variance analysis. It is used to compare variances between two populations and is defined by two sets of degrees of freedom: one for the numerator (d.f.N) and one for the denominator (d.f.D). The shape of the F-distribution varies based on these degrees of freedom, and it is always right-skewed.

Critical Value

A critical value is a threshold that determines the boundary for rejecting the null hypothesis in hypothesis testing. For a two-tailed test, critical values are found at both ends of the distribution, corresponding to the chosen level of significance (α). In this case, with α = 0.05, the critical values will be located at the 2.5th percentile and the 97.5th percentile of the F-distribution, indicating the regions where the null hypothesis can be rejected.

Degrees of Freedom

Degrees of freedom (d.f.) refer to the number of independent values or quantities that can vary in an analysis without violating any constraints. In the context of the F-test, d.f.N represents the degrees of freedom associated with the numerator (the group being tested), while d.f.D represents the degrees of freedom associated with the denominator (the error term). These values are crucial for determining the critical F-value and interpreting the results of the test.

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Related Practice

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=0.01, d.f.N=2, d.f.D=11"

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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=0.025, d.f.N=7, d.f.D=3"

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Textbook Question

"Finding a Critical F-Value for a Two-Tailed Test In Exercises 9–12, find the critical F-value for a two-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=24, d.f.D=28"

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Textbook Question

"Finding a Critical F-Value for a Two-Tailed Test In Exercises 9–12, find the critical F-value for a two-tailed test using the level of significance α and degrees of freedom d.f.N and d.f.D.

α=0.05, d.f.N=60, d.f.D=40"

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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 = 0.10.

Sample statistics: s₁² = 773, n₁ = 5 and s₂² = 765, n₂ = 6"

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Textbook Question

Claim: σ₁² = σ₂²; α = 0.05.

Sample statistics: s₁² = 310, n₁ = 7 and s₂² = 297, n₂ = 8"

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Textbook Question

"In Exercises 9–12, 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=0.05,d.f.N=6,d.f.D=50"

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Textbook Question