Textbook Question
What conditions are necessary in order to use a one-way ANOVA test?
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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 10.3.13
"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"
Verified step by step guidance1
Step 1: Understand the problem. The goal is to test the claim that the variance of population 1 (σ₁²) is greater than the variance of population 2 (σ₂²) using a significance level of α = 0.10. This involves conducting an F-test for comparing two variances.
Step 2: State the null and alternative hypotheses. The null hypothesis (H₀) is that the variances are equal: H₀: σ₁² = σ₂². The alternative hypothesis (H₁) is that the variance of population 1 is greater than the variance of population 2: H₁: σ₁² > σ₂².
Step 3: Calculate the test statistic. The formula for the F-test statistic is F = (s₁² / s₂²), where s₁² and s₂² are the sample variances. Substitute the given values: s₁² = 773 and s₂² = 765.
Step 4: Determine the degrees of freedom for each sample. For sample 1, degrees of freedom (df₁) = n₁ - 1 = 5 - 1 = 4. For sample 2, degrees of freedom (df₂) = n₂ - 1 = 6 - 1 = 5.
Step 5: Compare the calculated F-statistic to the critical value from the F-distribution table at α = 0.10 with df₁ = 4 and df₂ = 5. If the calculated F-statistic exceeds the critical value, reject the null hypothesis; otherwise, fail to reject the null hypothesis.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Hypothesis Testing
Hypothesis testing is a statistical method used to make decisions about population parameters based on sample data. In this context, we formulate a null hypothesis (H0: σ₁² ≤ σ₂²) and an alternative hypothesis (H1: σ₁² > σ₂²) to test the claim regarding the variances. The outcome of the test will help determine if there is enough evidence to reject the null hypothesis in favor of the alternative.
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F-Test for Variances
The F-test is a statistical test used to compare the variances of two populations. It involves calculating the F-statistic, which is the ratio of the two sample variances (s₁²/s₂²). This statistic follows an F-distribution under the null hypothesis, and we compare it to a critical value from the F-distribution table based on the degrees of freedom to determine if the variances are significantly different.
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Level of Significance (α)
The level of significance, denoted as α, is the probability of rejecting the null hypothesis when it is actually true (Type I error). In this case, α is set at 0.10, meaning there is a 10% risk of concluding that σ₁² is greater than σ₂² when it is not. This threshold helps to determine the critical value for the F-test and guides the decision-making process in hypothesis testing.
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Related Practice
Textbook Question
Performing a Chi-Square Goodness-of-Fit Test
In Exercises 7–16, (a) identify the claim and state H₀ and Hₐ, (b) 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, and (e) interpret the decision in the context of the original claim.
Coffee A researcher claims that the numbers of cups of coffee U.S. adults drink per day are distributed as shown in the figure. You randomly select 1600 U.S. adults and ask them how many cups of coffee they drink per day. The table shows the results. At α=0.05, test the researcher’s claim. (Adapted from Gallup)
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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 have a degree are not in the civilian labor force?
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Textbook Question
Finding Expected Frequencies
In Exercises 3–6, find the expected frequency for the values of n and pᵢ.
n=230, pᵢ=0.25
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Textbook Question
Finding Expected Frequencies
In Exercises 3–6, find the expected frequency for the values of n and pᵢ.
n=415, pᵢ=0.08
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Textbook Question
Describe the difference between the variance between samples MSB and the variance within samples MSW.
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