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
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 10.2.26
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 10. At α=0.01, test the hypothesis that the variables are dependent.
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 variables are dependent. The null hypothesis (H₀) states that the variables are independent, while the alternative hypothesis (Hₐ) states that the variables are dependent.
Step 2: Determine the degrees of freedom (df) using the formula: , where r is the number of rows and c is the number of columns in the contingency table. Then, find the critical value from the chi-square distribution table at α = 0.01 and identify the rejection region.
Step 3: Calculate the chi-square test statistic using the formula: , where Oᵢⱼ represents the observed frequency and Eᵢⱼ represents the expected frequency for each cell in the contingency table.
Step 4: Compare the calculated chi-square test statistic to the critical value. If the test statistic falls in the rejection region, reject the null hypothesis (H₀). 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 support the claim that the variables are dependent. If the null hypothesis is not rejected, conclude that there is insufficient evidence to support the claim that the variables are dependent.
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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 determining the direction of the test and interpreting the results.
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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 that defines the rejection region for the test statistic. If the calculated Chi-Square statistic exceeds this critical value, the null hypothesis is rejected, indicating a significant association between the variables.
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05:50Critical Values: t-Distribution
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] Well-Being Index The well-being index is a way to measure how people are faring physically, emotionally, socially, and professionally, as well as to rate the overall quality of their lives and their outlooks for the future. The table shows the well-being index scores for a sample of states from four regions of the United States. At α=0.10, can you reject the claim that the mean score is the same for all regions? (Adapted from Gallup and Healthways)
Textbook Question
True or False? In Exercises 5 and 6, determine whether the statement is true or false. If it is false, rewrite it as a true statement.
If the two variables in a chi-square independence test are dependent, then you can expect little difference between the observed frequencies and the expected frequencies.
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.
Use the contingency table and expected frequencies from Exercise 8. At α=0.05, test the hypothesis that the variables are dependent.
Textbook Question
Finding Expected Frequencies
In Exercises 7–12, (a) calculate the marginal frequencies and (b) find the expected frequency for each cell in the contingency table. Assume that the variables are independent.
Textbook Question
Performing a Chi-Square Goodness-of-Fit Test
In Exercises 7–16, (e) interpret the decision in the context of the original claim.
Ways to Pay A financial analyst claims that the distribution of people’s preferences on how to pay for goods is different from the distribution shown in the figure. You randomly select 600 people and record their preferences on how to pay for goods. The table shows the results. At α=0.01, test the financial analyst’s claim. (Adapted from Travis Credit Union)