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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 10.2.11
Chapter 10, Problem 10.2.11

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.


A table displaying car type preferences by gender, with counts for compact, full-size, SUV, and truck/van categories.

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Step 1: Calculate the marginal frequencies for each row and column. Marginal frequencies are the totals for each row and column in the contingency table. Add the values across each row to find the row totals, and add the values down each column to find the column totals.
Step 2: Compute the grand total by summing all the values in the table. This represents the total number of observations in the dataset.
Step 3: Use the formula for expected frequency to calculate the expected frequency for each cell in the table. The formula is: E=(row total×column total)/grand total, where E is the expected frequency.
Step 4: Apply the formula to each cell in the table. For example, for the cell corresponding to 'Male' and 'Compact', use the row total for 'Male', the column total for 'Compact', and the grand total to calculate the expected frequency.
Step 5: Repeat the calculation for all cells in the table to find the expected frequencies for each cell. Ensure that the expected frequencies are rounded appropriately if needed.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Marginal Frequencies

Marginal frequencies are the totals of the rows and columns in a contingency table. They provide a summary of the data by showing the total counts for each category, allowing for a quick overview of the distribution of responses. For example, in the given table, the marginal frequency for males would be the sum of all car types chosen by males.
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Expected Frequencies

Expected frequencies are the theoretical counts that would occur in each cell of a contingency table if the variables were independent. They are calculated by multiplying the marginal totals of the corresponding row and column, then dividing by the total number of observations. This concept is crucial for conducting chi-square tests to determine if there is a significant association between the variables.
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Contingency Tables & Expected Frequencies

Independence of Variables

The independence of variables means that the occurrence of one variable does not affect the occurrence of another. In the context of a contingency table, if the variables (like gender and car type) are independent, the expected frequencies can be calculated using the marginal totals. This assumption is essential for accurately interpreting the results of statistical tests, such as the chi-square test.
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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)


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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.

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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.

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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.

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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.

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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)

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