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

Larson 8th Edition

Ch. 10 - Chi-Square Tests and the F-Distribution

Problem 10.2.34

Chapter 10, Problem 10.2.34

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.

Verified step by step guidance

Step 1: Understand the chi-square independence test. This test is used to determine whether there is a significant association between two categorical variables. It requires raw frequency counts, not relative frequencies, as input data.

Step 2: Analyze the contingency table provided. The table shows the number of U.S. adults (in millions) categorized by employment status and educational attainment. These values are raw frequencies, not relative frequencies.

Step 3: Consider the requirement for the chi-square test. One key assumption is that the expected frequency in each cell of the table must be at least 5. If any cell has an expected frequency less than 5, the test cannot be performed reliably.

Step 4: Examine the data in the table. Some cells, such as 'Unemployed' for 'Not a high school graduate' (0.8 million) and 'Unemployed' for 'Some college, no degree' (1.1 million), have frequencies less than 5. This violates the assumption of the chi-square test.

Step 5: Conclude that the chi-square independence test cannot be performed on these data because some cells have frequencies less than 5, which makes the test invalid under its assumptions.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.

Video duration:

Was this helpful?

Key Concepts

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

Contingency Tables

A contingency table is a type of data representation that displays the frequency distribution of variables. It allows for the examination of the relationship between two categorical variables by showing how the frequencies of one variable are distributed across the categories of another. In this case, the table illustrates the employment status of U.S. adults based on their educational attainment.

Relative Frequencies

Relative frequencies are calculated by dividing the frequency of a specific category by the total number of observations, providing a proportion that reflects the size of that category relative to the whole. This transformation is useful for comparing categories on a common scale, especially when sample sizes differ. In the context of the contingency table, relative frequencies would help in understanding the distribution of employment status across educational levels.

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. However, this test requires that the expected frequency in each cell of the contingency table be sufficiently large (typically at least 5). If any expected frequencies are too low, the test may not be valid, which is likely the case with the provided data, as some categories may have very few observations.

Recommended video:

Guided course

06:28

Independence Test

Related Practice

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)

views

Textbook Question

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

views

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)

views

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.

views

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.

views

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.

123

views