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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 10.3.12
Chapter 10, Problem 10.3.12

"Finding a Critical F-Value for a Two-Tailed Test In Exercises 9–12, find the critical F-value for a two-tailed test using the level of significance α and degrees of freedom d.f.N and d.f.D.


α=0.05, d.f.N=27, d.f.D=19"

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Step 1: Understand the problem. We are tasked with finding the critical F-value for a two-tailed test. The level of significance (α) is 0.05, and the degrees of freedom for the numerator (d.f.N) and denominator (d.f.D) are 27 and 19, respectively.
Step 2: Recognize that for a two-tailed test, the significance level (α) is split equally between the two tails of the F-distribution. This means each tail will have an area of α/2 = 0.05/2 = 0.025.
Step 3: Use an F-distribution table or statistical software to find the critical F-value. Look up the F-value corresponding to the upper tail with α/2 = 0.025, d.f.N = 27 (numerator degrees of freedom), and d.f.D = 19 (denominator degrees of freedom).
Step 4: Remember that the F-distribution is not symmetric. For the lower tail critical value, you can use the reciprocal property of the F-distribution: F_lower = 1 / F_upper, where F_upper is the critical value from Step 3.
Step 5: Combine the results. The critical F-values for the two-tailed test will be the lower tail critical value (F_lower) and the upper tail critical value (F_upper). These values define the rejection regions for the null hypothesis.

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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 variances between two populations and is defined by two sets of degrees of freedom: one for the numerator (d.f.N) and one for the denominator (d.f.D). The shape of the F-distribution varies based on these degrees of freedom, and it is always right-skewed.
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Critical Value

A critical value is a threshold that determines the boundary for rejecting the null hypothesis in hypothesis testing. For a two-tailed test, critical values are found at both ends of the distribution, corresponding to the chosen level of significance (α). In this case, with α = 0.05, the critical values will be located at the 2.5th percentile and the 97.5th percentile of the F-distribution, indicating the regions where the null hypothesis can be rejected.
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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 an analysis without violating any constraints. In the context of the F-test, d.f.N represents the degrees of freedom associated with the numerator (the group being tested), while d.f.D represents the degrees of freedom associated with the denominator (the error term). These values are crucial for determining the critical F-value and interpreting the results of the test.
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Related Practice
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 not high school graduates are unemployed?

Textbook Question

Describe the hypotheses for a two-way ANOVA test.

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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 11. At α=0.10, test the hypothesis that the variables are independent.

Textbook Question

Performing a Chi-Square Goodness-of-Fit Test

In Exercises 7–16, (a) identify the claim and state H₀ and Hₐ, (b) 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, and (e) interpret the decision in the context of the original claim.


Births by Day of the Week A doctor claims that the number of births by day of the week is uniformly distributed. To test this claim, you randomly select 700 births from a recent year and record the day of the week on which each takes place. The table shows the results. At α=0.10, test the doctor’s claim. (Adapted from National Center for Health Statistics)


Textbook Question

"Finding a Critical F-Value for a Two-Tailed Test In Exercises 9–12, find the critical F-value for a two-tailed test using the level of significance α and degrees of freedom d.f.N and d.f.D.


α=0.05, d.f.N=60, d.f.D=40"

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)


What percent of U.S. adults ages 25 and over (a) are employed and are only high school graduates, (b) are not in the civilian labor force, and (c) are not high school graduates?