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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 10.3.7
Chapter 10, Problem 10.3.7

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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1
Identify the parameters for the F-distribution: the level of significance (α = 0.10), the degrees of freedom for the numerator (d.f.N = 10), and the degrees of freedom for the denominator (d.f.D = 15).
Understand that this is a right-tailed test, so the critical F-value corresponds to the point in the F-distribution where the area to the right equals the level of significance (α = 0.10).
Use an F-distribution table or statistical software to locate the critical F-value. In the table, find the row corresponding to d.f.N = 10 and the column corresponding to d.f.D = 15 under the α = 0.10 column.
If using statistical software (e.g., R, Python, or a calculator), input the parameters into the appropriate function for the F-distribution. For example, in R, you can use the function `qf(1 - α, d.f.N, d.f.D)` to find the critical F-value.
Record the critical F-value obtained from the table or software. This value will be used as the threshold for determining whether to reject the null hypothesis in the right-tailed test.

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Key Concepts

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

F-Distribution

The F-distribution is a probability distribution that arises frequently in statistics, particularly in the context of variance analysis. It is used to compare the variances of two populations and is defined by two sets of degrees of freedom: one for the numerator and one for the denominator. The shape of the F-distribution is right-skewed, meaning it has a long tail on the right side, which is important for hypothesis testing in ANOVA and regression analysis.
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Critical Value

A critical value is a threshold that determines the boundary for rejecting the null hypothesis in hypothesis testing. It is derived from the chosen level of significance (α), which represents the probability of making a Type I error. In a right-tailed test, the critical value is the point beyond which the test statistic is considered significant, indicating that the observed data is unlikely under the null hypothesis.
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Degrees of Freedom

Degrees of freedom (d.f.) refer to the number of independent values or quantities that can vary in a statistical calculation. In the context of the F-test, there are two types of degrees of freedom: d.f.N (numerator) and d.f.D (denominator), which correspond to the number of groups being compared and the total number of observations, respectively. These values are crucial for determining the critical F-value from the F-distribution table.
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Related Practice
Textbook Question

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

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)


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)


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?

Textbook Question

List five properties of the F-distribution.

Textbook Question

Explain how to determine the values of d.f.N and d.f.D when performing a two-sample F-test.