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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 10.3.10
Chapter 10, Problem 10.3.10

"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=0.10, d.f.N=24, d.f.D=28"

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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 0.10, and the degrees of freedom for the numerator (d.f.N) and denominator (d.f.D) are 24 and 28, respectively.
Step 2: Recall the properties of the F-distribution. The F-distribution is used to compare variances, and the critical F-value is determined based on the level of significance (α), the degrees of freedom for the numerator (d.f.N), and the degrees of freedom for the denominator (d.f.D). For a two-tailed test, the significance level is split equally between the two tails (α/2 for each tail).
Step 3: Use an F-distribution table or statistical software to find the critical F-values. For a two-tailed test, you will need to find two critical F-values: one for the upper tail and one for the lower tail. The upper critical F-value corresponds to the right tail with α/2 = 0.05, and the lower critical F-value corresponds to the left tail with α/2 = 0.05.
Step 4: Locate the critical F-values in the F-distribution table. Find the row corresponding to d.f.N = 24 and the column corresponding to d.f.D = 28 for the upper tail (α/2 = 0.05). For the lower tail, take the reciprocal of the upper critical F-value because the F-distribution is not symmetric.
Step 5: Verify your results. Ensure that the critical F-values you found are consistent with the level of significance (α = 0.10) and the degrees of freedom provided. If using statistical software, double-check the input values for accuracy.

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Key Concepts

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

Critical F-Value

The critical F-value is a threshold used in hypothesis testing to determine whether to reject the null hypothesis. It is derived from the F-distribution, which is used when comparing variances between two or more groups. In a two-tailed test, the critical F-value is found at both ends of the distribution, corresponding to the specified level of significance (α).
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Degrees of Freedom

Degrees of freedom (d.f.) refer to the number of independent values that can vary in an analysis without violating any constraints. In the context of an 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.
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Level of Significance (α)

The level of significance (α) is the probability of rejecting the null hypothesis when it is actually true, commonly set at values like 0.05 or 0.10. It defines the threshold for determining whether the observed data is statistically significant. In a two-tailed test, this significance level is split between both tails of the distribution, affecting the critical values used in hypothesis testing.
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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 have a degree are not in the civilian labor force?

Textbook Question

Finding Expected Frequencies

In Exercises 3–6, find the expected frequency for the values of n and pᵢ.


n=230, pᵢ=0.25

Textbook Question

Finding Expected Frequencies

In Exercises 3–6, find the expected frequency for the values of n and pᵢ.


n=415, pᵢ=0.08

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Textbook Question

State the null and alternative hypotheses for a one-way ANOVA test.

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.


Alcohol-Related Accidents The contingency table shows the results of a random sample of fatally injured passenger vehicle drivers (with blood alcohol concentrations greater than or equal to 0.08) by age and gender. At α=0.05, can you conclude that age is related to gender in such alcohol-related accidents? (Adapted from Insurance Institute for Highway Safety)

Textbook Question

In Exercises 13–18, test the claim about the difference between two population variances σ₁² and σ₂² at the level of significance α. Assume the samples are random and independent, and the populations are normally distributed.


Claim: σ₁² ≠ σ₂²; α = 0.05.

Sample statistics: s₁² = 245, n₁ = 31 and s₂² = 112, n₂ = 28

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