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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: 9780137493470Not the one you use?Change textbook
All textbooksLarson 8th EditionCh. 10 - Chi-Square Tests and the F-DistributionProblem 10.2.16
Chapter 10, Problem 10.2.16

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)


Contingency table showing student attitudes on safety steps by school type: public and private, with corresponding counts.

Verified step by step guidance
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Step 1: Identify the claim and state the null hypothesis (H₀) and alternative hypothesis (Hₐ). The claim is that attitudes about the safety steps taken by the school staff are related to the type of school. The null hypothesis (H₀) states that attitudes about safety steps are independent of the type of school. The alternative hypothesis (Hₐ) states that attitudes about safety steps are related to the type of school.
Step 2: Determine the degrees of freedom (df), find the critical value, and identify the rejection region. Degrees of freedom are calculated using the formula df = (number of rows - 1) × (number of columns - 1). Here, df = (2 - 1) × (2 - 1) = 1. Using α = 0.01 and df = 1, find the critical value from the chi-square distribution table. The rejection region is where the test statistic exceeds the critical value.
Step 3: Calculate the expected frequencies for each cell in the contingency table using the formula E = (row total × column total) / grand total. For example, for the cell corresponding to 'Public' and 'Taken all steps necessary for student safety', calculate E = (91 × 104) / 189. Repeat this for all cells in the table.
Step 4: Compute the chi-square test statistic using the formula χ² = Σ((O - E)² / E), where O represents the observed frequency and E represents the expected frequency for each cell. Sum the values for all cells to find the test statistic.
Step 5: Compare the calculated chi-square test statistic to the critical value. If the test statistic exceeds the critical value, reject the null hypothesis (H₀). Otherwise, fail to reject the null hypothesis. Interpret the decision in the context of the original claim: if H₀ is rejected, conclude that attitudes about safety steps are related to the type of school; if H₀ is not rejected, conclude that attitudes about safety steps are independent of the type of school.

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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 in each category of a contingency table to the frequencies expected if the variables were independent. A significant result indicates that the variables are related, while a non-significant result suggests independence.
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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 relationship between the variables being studied. The alternative hypothesis (Hₐ) posits that there is a significant effect or relationship. In the context of the Chi-Square test, H₀ would state that attitudes about safety steps are independent of the type of school, while Hₐ would suggest they are related.
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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). The critical value is a threshold derived from the Chi-Square distribution table, which helps determine the rejection region for the null hypothesis. If the calculated Chi-Square statistic exceeds the critical value, the null hypothesis is rejected.
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Critical Values: t-Distribution
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.


Achievement and School Location The contingency table shows the results of a random sample of students by the location of school and the number of those students achieving basic skill levels in three subjects. At α=0.01, test the hypothesis that the variables are independent. (Adapted from HUD State of the Cities Report)


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.


U.S. History Assessment Tests A state school administrator claims that the standard deviations of U.S. history assessment test scores for eighth-grade students are the same in Districts 1 and 2. A sample of 10 test scores from District 1 has a standard deviation of 30.9 points, and a sample of 13 test scores from District 2 has a standard deviation of 27.2 points. At α=0.01, can you reject the administrator’s claim? (Adapted from National Center for Education Statistics)

Textbook Question

List the three conditions that must be met in order to use a two-sample F-test.

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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

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.