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

Larson - Elementary Statistics: Picturing the World 8th Edition

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Larson 8th Edition

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

Problem 10.2.4

Chapter 10, Problem 10.2.4

Explain why the chi-square independence test is always a right-tailed test.

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The chi-square independence test is used to determine whether there is a significant association between two categorical variables. The test statistic is calculated based on the differences between observed and expected frequencies in a contingency table.

The chi-square test statistic formula is:

χ² = Σ((Oᵢ - Eᵢ)² / Eᵢ)

, where Oᵢ represents the observed frequency and Eᵢ represents the expected frequency for each cell in the table.

The chi-square distribution is inherently non-negative because the test statistic involves squaring the differences between observed and expected frequencies. This ensures that the test statistic is always greater than or equal to zero.

In hypothesis testing, the null hypothesis assumes no association between the variables, meaning the observed frequencies are close to the expected frequencies. Large values of the chi-square statistic indicate greater deviations from the null hypothesis, suggesting a stronger association between the variables.

Since we are interested in detecting significant deviations from the null hypothesis (large chi-square values), the chi-square independence test is always conducted as a right-tailed test, where the critical region lies in the upper tail of the chi-square distribution.

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Key Concepts

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

Chi-Square Test

The chi-square test is a statistical method used to determine if there is a significant association between categorical variables. It compares the observed frequencies in each category to the frequencies expected under the null hypothesis of independence. The test statistic follows a chi-square distribution, which is crucial for interpreting the results.

Null Hypothesis

In hypothesis testing, the null hypothesis represents a statement of no effect or no association between variables. For the chi-square independence test, the null hypothesis posits that the two categorical variables are independent of each other. The goal of the test is to assess whether the observed data provides sufficient evidence to reject this null hypothesis.

Right-Tailed Test

A right-tailed test is a type of hypothesis test where the critical region for rejecting the null hypothesis is located in the right tail of the distribution. In the context of the chi-square independence test, if the test statistic is significantly large, it indicates a strong association between the variables, leading to rejection of the null hypothesis. This is why the chi-square test is always right-tailed, as it assesses whether the observed frequencies deviate significantly from the expected frequencies in a positive direction.

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Independence Test

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] Statistician Salaries The table shows the salaries of a sample of entry level statisticians from six large metropolitan areas. At α=0.05, can you conclude that the mean salary is different in at least one of the areas? (Adapted from Salary.com)

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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.

Ages and Goals You are investigating the relationship between the ages of U.S. adults and what aspect of career development they consider to be the most important. You randomly collect the data shown in the contingency table. At α=0.10, is there enough evidence to conclude that age is related to which aspect of career development is considered to be most important? (Adapted from The Harris Poll)

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Textbook Question

"Finding Left-Tailed Critical F-Values In this section, you only needed to calculate the right-tailed critical F-value for a two-tailed test. For other applications of the F-distribution, you will need to calculate the left-tailed critical F-value. To calculate the left-tailed critical F-value, perform the steps below.

1. Interchange the values for d.f.N and d.f.D.

2. Find the corresponding F-value in Table 7.

3. Calculate the reciprocal of the F-value to obtain the left-tailed critical F-value.

In Exercises 27 and 28, find the right- and left-tailed critical F-values for a two-tailed test using the level of significance α and degrees of freedom d.f.N and d.f.D.

α=0.10, d.f.N=20, d.f.D=15"

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Textbook Question

Finding a Critical F-Value for a Right-Tailed Test In Exercises 5–8, find the critical F-value for a right-tailed test using the level of significance α and degrees of freedom d.f.N and d.f.D.

α=0.10, d.f.N=10, d.f.D=15

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Textbook Question

Conditional Relative Frequencies In Exercises 37–42, use the contingency table from Exercises 33–36, and the information below.

Relative frequencies can also be calculated based on the row totals (by dividing each row entry by the row’s total) or the column totals (by dividing each column entry by the column’s total). These frequencies are conditional relative frequencies and can be used to determine whether an association exists between two categories in a contingency table.

What percent of U.S. adults ages 25 and over who are employed have a degree?

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Textbook Question

List five properties of the F-distribution.

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