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Ch. 10 - Chi-Square Tests and the F-Distribution
Larson - Elementary Statistics: Picturing the World 8th Edition
Larson8th EditionElementary Statistics: Picturing the WorldISBN: 9780137493470당신이 사용하는 게 아니라요?교과서 변경
모든 교과서Larson 8th EditionCh. 10 - Chi-Square Tests and the F-Distribution문제 10.2.14
10장, 문제 10.2.14

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.

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Step 1: Identify the claim and state the null hypothesis (H₀) and the alternative hypothesis (Hₐ). The claim is that the variables are dependent. Therefore, H₀: The variables are independent, and Hₐ: The variables are dependent.
Step 2: Determine the degrees of freedom (df) using the formula df = (r - 1)(c - 1), 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 for the given significance level (α = 0.05) and degrees of freedom. Identify the rejection region as χ² > critical value.
Step 3: Calculate the chi-square test statistic using the formula χ² = Σ((O - E)² / E), where O represents the observed frequencies and E represents the expected frequencies. For each cell in the contingency table, compute the expected frequency using the formula E = (row total × column total) / grand total, and then substitute into the chi-square formula.
Step 4: Compare the calculated chi-square test statistic to the critical value. If the test statistic falls in the rejection region (χ² > critical value), 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 H₀ is rejected, conclude that there is sufficient evidence to support the claim that the variables are dependent. If H₀ is not rejected, conclude that there is insufficient evidence to support the claim that the variables are dependent.

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주요 개념

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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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가이드 코스
06:28
Independence Test

Null and Alternative Hypotheses

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 as they guide the analysis and interpretation of the test results.
추천 영상:
가이드 코스
06:21
Step 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 determines the rejection region for the null hypothesis. If the calculated Chi-Square statistic exceeds this critical value, the null hypothesis is rejected, indicating a significant association between the variables.
추천 영상:
05:50
Critical Values: t-Distribution
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교과서 질문

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)


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교과서 질문

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.

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교과서 질문

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.

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교과서 질문

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.


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교과서 질문

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.

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교과서 질문

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)

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