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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.RE.12
Chapter 10, Problem 10.RE.12

In Exercises 9–12, 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=0.05,d.f.N=20,d.f.D=25

Verified step by step guidance
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Step 1: Understand the problem. You are tasked with finding the critical F-value for a right-tailed test using the given level of significance (α = 0.05) and degrees of freedom for the numerator (d.f.N = 20) and denominator (d.f.D = 25). The critical F-value is the value that separates the rejection region from the non-rejection region in an F-distribution.
Step 2: Recall the formula and concept. The F-distribution is used in hypothesis testing, particularly in ANOVA and regression analysis. The critical F-value depends 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).
Step 3: Use an F-distribution table or statistical software. Locate the row corresponding to d.f.N = 20 and the column corresponding to d.f.D = 25 in the F-distribution table for α = 0.05. Alternatively, use statistical software like R, Python, or Excel to compute the critical F-value.
Step 4: Interpret the result. The critical F-value represents the threshold above which the test statistic would lead to rejecting the null hypothesis in a right-tailed test. Ensure you understand its role in hypothesis testing.
Step 5: Verify your result. Double-check the table or software output to ensure accuracy. If using software, confirm that the inputs (α, d.f.N, d.f.D) are correctly entered.

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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 in an F-test. It is derived from the F-distribution, which is used to compare variances between two groups. The critical value is based on the chosen significance level (α) and the degrees of freedom for the numerator (d.f.N) and denominator (d.f.D). If the calculated F-statistic exceeds this critical value, the null hypothesis is 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 an F-test, d.f.N represents the degrees of freedom associated with the numerator (typically the group with more variance), while d.f.D represents the degrees of freedom for the denominator (the group with less variance). These values are crucial for determining the shape of the F-distribution and finding the critical F-value.
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Significance Level (α)

The significance level (α) is the probability of rejecting the null hypothesis when it is actually true, also known as a Type I error. It is a threshold set by the researcher, commonly at 0.05, which indicates a 5% risk of concluding that a difference exists when there is none. In hypothesis testing, the significance level helps determine the critical value needed to assess the results of the test, guiding the decision-making process regarding the null hypothesis.
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Related Practice
Textbook Question

Performing a Chi-Square Goodness-of-Fit Test

In Exercises 7–16, (a) identify the claim and state H₀ and Hₐ.


Ways to Pay A financial analyst claims that the distribution of people’s preferences on how to pay for goods is different from the distribution shown in the figure. You randomly select 600 people and record their preferences on how to pay for goods. The table shows the results. At α=0.01, test the financial analyst’s claim. (Adapted from Travis Credit Union)


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Textbook Question

In each exercise,

e. interpret the decision in the context of the original claim.

[APPLET] In Exercises 1–3, use the data, which list the hourly wages (in dollars) for randomly selected surgical technologists from three states. Assume the wages are normally distributed and that the samples are independent. (Adapted from U.S. Bureau of Labor Statistics)

Maine: 22.76, 27.60, 25.08, 17.01, 30.15, 27.09, 20.95, 25.52, 20.11, 23.67, 24.32

Oklahoma: 24.64, 21.66, 19.38, 18.19, 23.14, 20.58, 19.53, 30.77, 27.46, 23.80

Massachusetts: 27.07, 24.71, 32.80, 28.34, 33.45, 33.36, 36.81, 30.04, 29.01, 24.30, 29.22, 29.50

Are the mean hourly wages of surgical technologists the same for all three states? Use α=0.01. Assume that the population variances are equal.

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Textbook Question

In Exercises 13–16, 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=0.01,d.f.N=11,d.f.D=13

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Textbook Question

In each exercise,

d. decide whether to reject or fail to reject the null hypothesis, and

[APPLET] In Exercises 1–3, use the data, which list the hourly wages (in dollars) for randomly selected surgical technologists from three states. Assume the wages are normally distributed and that the samples are independent. (Adapted from U.S. Bureau of Labor Statistics)

Maine: 22.76, 27.60, 25.08, 17.01, 30.15, 27.09, 20.95, 25.52, 20.11, 23.67, 24.32

Oklahoma: 24.64, 21.66, 19.38, 18.19, 23.14, 20.58, 19.53, 30.77, 27.46, 23.80

Massachusetts: 27.07, 24.71, 32.80, 28.34, 33.45, 33.36, 36.81, 30.04, 29.01, 24.30, 29.22, 29.50

Are the mean hourly wages of surgical technologists the same for all three states? Use α=0.01. Assume that the population variances are equal.

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Textbook Question

In Exercises 1–4, (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.


A sports website claims that the opinions of golfers about what irritates them the most on the golf course are distributed as shown in the pie chart. You randomly select 1018 golfers and ask them what irritates them the most on the golf course. The table shows the results. At α=0.05, test the sports website’s claim. (Adapted from GOLF.com)


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Textbook Question

In Exercises 13–16, 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=0.01,d.f.N=40,d.f.D=60

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