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 10.3.9

Chapter 10, Problem 10.3.9

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=0.01, d.f.N=6, d.f.D=7

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Step 1: Understand the problem. You are tasked with finding the critical F-value for a two-tailed test. The level of significance (α) is 0.01, and the degrees of freedom for the numerator (d.f.N) is 6, while the degrees of freedom for the denominator (d.f.D) is 7.

Step 2: Recognize that for a two-tailed test, the level of significance (α) is split equally between the two tails of the F-distribution. This means each tail will have an area of α/2 = 0.01/2 = 0.005.

Step 3: Use an F-distribution table or statistical software to find the critical F-values. For the upper critical value, locate the value corresponding to α/2 = 0.005, d.f.N = 6, and d.f.D = 7. For the lower critical value, take the reciprocal of the upper critical value (1/F_upper).

Step 4: If using an F-table, find the row corresponding to d.f.N = 6 and the column corresponding to d.f.D = 7 under the α/2 = 0.005 column. This gives the upper critical F-value. For the lower critical value, calculate 1/F_upper.

Step 5: Summarize the results. The critical F-values for the two-tailed test are the lower critical value (1/F_upper) and the upper critical value (F_upper). These values define the rejection regions for the 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 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 is right-skewed, and it is important for hypothesis testing in ANOVA and regression analysis.

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 = 0.01, the critical values will be located in the extreme 0.5% of each tail of the F-distribution, indicating the regions where the null hypothesis can be rejected.

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 or treatment variances), while d.f.D represents the degrees of freedom associated with the denominator (the error or residual variances). These values are crucial for determining the critical F-value from the F-distribution table.

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Related Practice

Textbook Question

"Performing a Two-Sample F-Test In Exercises 19–26, (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, and the populations are normally distributed.

Carbon Monoxide Emissions An automobile manufacturer claims that the variance of the carbon monoxide emissions for a make and model of one of its vehicles is less than the variance of the carbon monoxide emissions for a top competitor’s equivalent vehicle. A sample of the carbon monoxide emissions of 19 of the manufacturer’s specified vehicles has a variance of 0.008. A sample of the carbon monoxide emissions of 21 of its competitor’s equivalent vehicles has a variance of 0.045. At α=0.10, can you support the manufacturer’s claim? (Adapted from U.S. Environmental Protection Agency)"

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Textbook Question

Life of Appliances Company A claims that the variance of the lives of its appliances is less than the variance of the lives of Company B’s appliances. A sample of the lives of 20 of Company A’s appliances has a variance of 1.8. A sample of the lives of 25 of Company B’s appliances has a variance of 3.9. At α=0.025, can you support Company A’s claim?"

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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.

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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] Well-Being Index The well-being index is a way to measure how people are faring physically, emotionally, socially, and professionally, as well as to rate the overall quality of their lives and their outlooks for the future. The table shows the well-being index scores for a sample of states from four regions of the United States. At α=0.10, can you reject the claim that the mean score is the same for all regions? (Adapted from Gallup and Healthways)

138

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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 10. At α=0.01, test the hypothesis that the variables are dependent.

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Textbook Question

Performing a Chi-Square Goodness-of-Fit Test

In Exercises 7–16, (e) interpret the decision in the context of the original claim.

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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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=0.01, d.f.N=6, d.f.D=7

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Key Concepts

F-Distribution

Critical Value

Degrees of Freedom

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=0.01, d.f.N=6, d.f.D=7