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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2:30 minutes

Problem 10.3.3

Textbook Question

List the three conditions that must be met in order to use a two-sample F-test.

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The first condition is that the two samples must be independent. This means that the selection of one sample should not influence or affect the selection of the other sample.

The second condition is that the populations from which the samples are drawn must be normally distributed. This ensures that the F-test is valid and reliable.

The third condition is that the variances of the two populations must be equal under the null hypothesis. This is a key assumption of the two-sample F-test.

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Key Concepts

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

Independence of Samples

The two samples being compared in a two-sample F-test must be independent of each other. This means that the selection of one sample does not influence the selection of the other. Independence is crucial because it ensures that the variability in one sample does not affect the variability in the other, allowing for a valid comparison of their variances.

Normality

Both samples should be drawn from populations that are normally distributed. This assumption is important because the F-test relies on the properties of the F-distribution, which is derived under the assumption of normality. If the sample sizes are large, the Central Limit Theorem may mitigate this requirement, but for smaller samples, normality is essential.

Homogeneity of Variances

The variances of the two populations from which the samples are drawn must be equal, a condition known as homogeneity of variances. This assumption is critical for the validity of the F-test, as it compares the ratio of the variances of the two samples. If this condition is violated, the results of the F-test may be misleading, and alternative methods should be considered.

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Related Practice

Textbook Question

Performing a One-Way ANOVA Test In Exercises 5–14, (a) identify the claim and state H0 and Ha, (b) find the critical value and identify the rejection region, (c) find the test statistic F, (d) decide whether to reject or fail to reject the null hypothesis, and (e) interpret the decision in the context of the original claim. Assume the samples are random and independent, the populations are normally distributed, and the population variances are equal.

[APPLET] Statistician Salaries The table shows the salaries of a sample of entry level statisticians from six large metropolitan areas. At α=0.05, can you conclude that the mean salary is different in at least one of the areas? (Adapted from Salary.com)

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Textbook Question

In Exercises 21 and 22, (a) identify the claim and state H₀ and Hₐ, (b) find the critical value and identify the rejection region, (c) find the test statistic F, (d) decide whether to reject or fail to reject the null hypothesis, and (e) interpret the decision in the context of the original claim. Assume the samples are random and independent, the populations are normally distributed, and the population variances are equal.

[APPLET] The table shows the annual incomes (in dollars) for a sample of families from four regions of the United States. At α=0.05, can you conclude that the mean annual income of families is different in at least one of the regions? (Adapted from U.S. Census Bureau)

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Textbook Question

Explain how to find the critical value for an F-test.

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Textbook Question

List five properties of the F-distribution.

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Textbook Question

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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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=0.05, d.f.N=9, d.f.D=16"

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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=0.01, 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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