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 10.3.1

Chapter 10, Problem 10.3.1

Explain how to find the critical value for an F-test.

Verified step by step guidance

Understand the purpose of the F-test: The F-test is used to compare the variances of two populations or to test the overall significance of a regression model. The critical value helps determine whether to reject the null hypothesis.

Identify the degrees of freedom: For an F-test, you need two degrees of freedom. The first degree of freedom (df1) corresponds to the numerator (variance of the first sample or group), and the second degree of freedom (df2) corresponds to the denominator (variance of the second sample or group).

Choose the significance level (α): Decide on the level of significance for the test, typically 0.05 or 0.01. This represents the probability of rejecting the null hypothesis when it is actually true.

Use an F-distribution table or software: Locate the critical value in an F-distribution table by finding the intersection of df1 and df2 at the chosen significance level. Alternatively, statistical software or a calculator can be used to compute the critical value directly.

Interpret the critical value: The critical value divides the rejection region from the non-rejection region in the F-distribution. If the calculated F-statistic exceeds the critical value, you reject the null hypothesis; otherwise, you fail to reject it.

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Key Concepts

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

F-test

The F-test is a statistical test used to compare the variances of two or more groups. It helps determine if the group means are significantly different from each other by analyzing the ratio of variances. The F-test is commonly used in ANOVA (Analysis of Variance) and regression analysis.

Critical Value

The critical value in hypothesis testing is a threshold that determines whether to reject the null hypothesis. It is derived from the chosen significance level (alpha) and the distribution of the test statistic. For an F-test, the critical value is obtained from the F-distribution table based on the degrees of freedom for the numerator and denominator.

Degrees of Freedom

Degrees of freedom (df) refer to the number of independent values that can vary in an analysis without violating any constraints. In the context of an F-test, the degrees of freedom are calculated based on the sample sizes of the groups being compared. Specifically, df for the numerator is related to the number of groups minus one, while the denominator df is related to the total sample size minus the number of groups.

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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.

Annual Salaries An employment information service claims that the standard deviation of the annual salaries for public relations managers is less in Louisiana than in Florida. You select a sample of public relations managers from each state. The results of each survey are shown in the figure. At α=0.05, can you support the service’s claim? (Adapted from America’s Career InfoNet)

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Textbook Question

Testing for Normality Using a chi-square goodness-of-fit test, you can decide, with some degree of certainty, whether a variable is normally distributed. In all chi-square tests for normality, the null and alternative hypotheses are as listed below.

H₀: The variable has a normal distribution.

Hₐ: The variable does not have a normal distribution.

To determine the expected frequencies when performing a chi-square test for normality, first estimate the mean and standard deviation of the frequency distribution. Then, use the mean and standard deviation to compute the z-score for each class boundary. Then, use the z-scores to calculate the area under the standard normal curve for each class. Multiplying the resulting class areas by the sample size yields the expected frequency for each class.In Exercises 17 and 18, (a) find the expected frequencies, (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.

In Exercises 17 and 18, (a) find the expected frequencies, (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.

Test Scores At α=0.01, test the claim that the 200 test scores shown in the frequency distribution are normally distributed.

280

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Textbook Question

"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=0.05, d.f.N=9, d.f.D=16"

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Textbook Question

"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=0.01, d.f.N=2, d.f.D=11"

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Textbook Question

True or False? In Exercises 5 and 6, determine whether the statement is true or false. If it is false, rewrite it as a true statement.

When the test statistic for the chi-square independence test is large, you will, in most cases, reject the null hypothesis.

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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 = 310, n₁ = 7 and s₂² = 297, n₂ = 8"

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