Textbook Question
List the three conditions that must be met in order to use a two-sample F-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 10.2.5
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.
Verified step by step guidance1
Understand the context of the chi-square independence test: This test is used to determine whether two categorical variables are independent or dependent. Independence means the variables do not influence each other, while dependence means they are related.
Recall the relationship between observed and expected frequencies: In a chi-square test, the observed frequencies are the actual counts from the data, and the expected frequencies are the counts predicted under the assumption of independence.
Analyze the statement: If the two variables are dependent, it implies that the observed frequencies will differ significantly from the expected frequencies. This is because dependence indicates a relationship that deviates from the assumption of independence.
Identify the error in the statement: The statement claims that if the variables are dependent, there will be little difference between observed and expected frequencies. This is false because dependence typically results in larger differences between observed and expected frequencies.
Rewrite the statement as true: If the two variables in a chi-square independence test are dependent, then you can expect significant differences between the observed frequencies and the expected frequencies.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Chi-Square Independence Test
The chi-square independence test is a statistical method used to determine if there is a significant association between two categorical variables. It compares the observed frequencies of occurrences in each category to the expected frequencies, which are calculated under the assumption that the variables are independent. A significant difference suggests that the variables are related.
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06:28
Independence Test
Observed vs. Expected Frequencies
Observed frequencies are the actual counts collected from data, while expected frequencies are the counts we would expect if there were no association between the variables. In a chi-square test, if the variables are independent, the observed and expected frequencies should be similar. A large discrepancy indicates a potential dependence between the variables.
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08:18Contingency Tables & Expected Frequencies
Dependence of Variables
When two variables are dependent, the value of one variable affects or is related to the value of the other. In the context of a chi-square test, if the variables are dependent, we would expect significant differences between observed and expected frequencies, indicating that the distribution of one variable changes based on the level of the other variable.
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05:17Multiplication Rule: Dependent Events
Related Practice
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.
Attitudes about Safety The contingency table shows the results of a random sample of students by type of school and their attitudes on safety steps taken by the school staff. At α=0.01, can you conclude that attitudes about the safety steps taken by the school staff are related to the type of school? (Adapted from Horatio Alger Association)
Textbook Question
Contingency Tables and Relative Frequencies In Exercises 33–36, use the information below.
The frequencies in a contingency table can be written as relative frequencies by dividing each frequency by the sample size. The contingency table below shows the number of U.S. adults (in millions) ages 25 and over by employment status and educational attainment. (Adapted from U.S. Census Bureau)
Explain why you cannot perform the chi-square independence test on these data.
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 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.