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"
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 10.2.28
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.
Use the contingency table and expected frequencies from Exercise 11. At α=0.10, test the hypothesis that the variables are independent.
Verified step by step guidance1
Step 1: Identify the claim and state the null hypothesis (H₀) and the alternative hypothesis (Hₐ). The claim is that the two variables are independent. Therefore, H₀: The variables are independent, and Hₐ: The variables are not independent.
Step 2: Determine the degrees of freedom (df), find the critical value, and identify the rejection region. The degrees of freedom are calculated as df = (number of rows - 1) × (number of columns - 1). Use the chi-square distribution table to find the critical value for the given significance level (α = 0.10) and the calculated degrees of freedom. The rejection region is the set of chi-square values greater than the critical value.
Step 3: Calculate the chi-square test statistic. Use the formula χ² = Σ [(Oᵢ - Eᵢ)² / Eᵢ], where Oᵢ represents the observed frequency and Eᵢ represents the expected frequency for each cell in the contingency table. Compute the expected frequencies using the formula Eᵢ = (row total × column total) / grand total.
Step 4: Compare the calculated chi-square test statistic to the critical value. If the test statistic falls into the rejection region (greater than the critical value), reject the null hypothesis. Otherwise, fail to reject the null hypothesis.
Step 5: Interpret the decision in the context of the original claim. If the null hypothesis is rejected, conclude that there is sufficient evidence to suggest the variables are not independent. If the null hypothesis is not rejected, conclude that there is insufficient evidence to suggest the variables are not independent.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Chi-Square Test of Independence
The Chi-Square Test of Independence is a statistical method used to determine if there is a significant association between two categorical variables. It compares the observed frequencies in each category of a contingency table to the frequencies expected if the variables were independent. A significant result suggests that the variables are related, while a non-significant result indicates independence.
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06:28
Independence Test
Null and Alternative Hypotheses (H₀ and Hₐ)
In hypothesis testing, the null hypothesis (H₀) represents the default position that there is no effect or no association between the variables. The alternative hypothesis (Hₐ) posits that there is an effect or an association. Clearly stating these hypotheses is crucial for guiding the analysis and interpreting the results of the test.
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06:21Step 1: Write Hypotheses
Degrees of Freedom and Critical Value
Degrees of freedom in a Chi-Square test are calculated based on the number of categories in the variables being analyzed, typically as (rows - 1) * (columns - 1) for a contingency table. The critical value is a threshold derived from the Chi-Square distribution, which helps determine the rejection region for the null hypothesis. If the test statistic exceeds this critical value, the null hypothesis is rejected.
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05:50Critical Values: t-Distribution
Related Practice
Textbook Question
"Performing a Two-Sample F-Test In Exercises 19–26, (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, and the populations are normally distributed.
Carbon Monoxide Emissions An automobile manufacturer claims that the variance of the carbon monoxide emissions for a make and model of one of its vehicles is less than the variance of the carbon monoxide emissions for a top competitor’s equivalent vehicle. A sample of the carbon monoxide emissions of 19 of the manufacturer’s specified vehicles has a variance of 0.008. A sample of the carbon monoxide emissions of 21 of its competitor’s equivalent vehicles has a variance of 0.045. At α=0.10, can you support the manufacturer’s claim? (Adapted from U.S. Environmental Protection Agency)"
Textbook Question
"Performing a Two-Sample F-Test In Exercises 19–26, (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, and the populations are normally distributed.
Life of Appliances Company A claims that the variance of the lives of its appliances is less than the variance of the lives of Company B’s appliances. A sample of the lives of 20 of Company A’s appliances has a variance of 1.8. A sample of the lives of 25 of Company B’s appliances has a variance of 3.9. At α=0.025, can you support Company A’s claim?"
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=0.01, d.f.N=6, d.f.D=7
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.
Births by Day of the Week A doctor claims that the number of births by day of the week is uniformly distributed. To test this claim, you randomly select 700 births from a recent year and record the day of the week on which each takes place. The table shows the results. At α=0.10, test the doctor’s claim. (Adapted from National Center for Health Statistics)
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"