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 10.3.16

Chapter 10, Problem 10.3.16

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

Verified step by step guidance

Step 1: Identify the null and alternative hypotheses. The null hypothesis (H₀) states that the population variances are equal (σ₁² = σ₂²), while the alternative hypothesis (H₁) states that the population variances are not equal (σ₁² ≠ σ₂²). This is a two-tailed test.

Step 2: Calculate the test statistic using the F-distribution formula: F = (s₁² / s₂²), where s₁² and s₂² are the sample variances. Substitute the given values: s₁² = 245 and s₂² = 112.

Step 3: Determine the degrees of freedom for the numerator (df₁ = n₁ - 1) and the denominator (df₂ = n₂ - 1). Use the sample sizes n₁ = 31 and n₂ = 28 to calculate df₁ and df₂.

Step 4: Find the critical values for the F-distribution at the significance level α = 0.05. Since this is a two-tailed test, divide α by 2 for each tail (α/2 = 0.025). Use an F-distribution table or statistical software to find the critical values for df₁ and df₂.

Step 5: Compare the calculated F-value to the critical values. If the F-value falls outside the range defined by the critical values, reject the null hypothesis (H₀). Otherwise, fail to reject the null hypothesis. Conclude whether there is sufficient evidence to support the claim that σ₁² ≠ σ₂².

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Key Concepts

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

Hypothesis Testing

Hypothesis testing is a statistical method used to make decisions about population parameters based on sample data. In this context, we are testing the claim that the variances of two populations are different (σ₁² ≠ σ₂²). The process involves formulating a null hypothesis (H₀) and an alternative hypothesis (H₁), calculating a test statistic, and comparing it to a critical value to determine whether to reject H₀.

F-Test for Variances

The F-test is a statistical test used to compare the variances of two populations. It is based on the ratio of the sample variances (s₁² and s₂²) and follows an F-distribution under the null hypothesis that the population variances are equal. The calculated F-statistic is compared to a critical value from the F-distribution table, which depends on the degrees of freedom of the samples and the significance level (α).

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. In this case, α is set at 0.05, indicating a 5% risk of concluding that the variances are different when they are not. This threshold helps determine the critical value for the F-test, guiding the decision-making process in hypothesis testing.

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Textbook Question

"Using Technology to Perform a Two-Way ANOVA Test In Exercises 15–18, use technology and the block design to perform a two-way ANOVA test. Use α=0.10. Interpret the results. Assume the samples are random and independent, the populations are normally distributed, and the population variances are equal.

[APPLET] Laptop Repairs The manager of a computer repair service wants to determine whether there is a difference in the time it takes four technicians to repair different brands of laptops. The block design shows the times (in minutes) it took for each technician to repair three laptops of each brand.

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=0.10, d.f.N=24, d.f.D=28"

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)