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

Larson 8th Edition

Ch. 10 - Chi-Square Tests and the F-Distribution

Problem 10.3.15

Chapter 10, Problem 10.3.15

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 = 0.01.
Sample statistics: s₁² = 842, n₁ = 11 and s₂² = 836, n₂ = 10

Verified step by step guidance

Step 1: Identify the null hypothesis (H₀) and the alternative hypothesis (H₁). The claim is that σ₁² ≤ σ₂². Therefore, the null hypothesis is H₀: σ₁² ≤ σ₂², and the alternative hypothesis is H₁: σ₁² > σ₂². This is a one-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 = 842 and s₂² = 836.

Step 3: Determine the degrees of freedom for the numerator (df₁) and denominator (df₂). For the numerator, df₁ = n₁ - 1, and for the denominator, df₂ = n₂ - 1. Substitute the sample sizes n₁ = 11 and n₂ = 10 to calculate df₁ and df₂.

Step 4: Find the critical value of F from the F-distribution table at the significance level α = 0.01 for a one-tailed test, using the calculated degrees of freedom (df₁ and df₂).

Step 5: Compare the calculated F value to the critical F value. If the calculated F value is greater than the critical F value, reject the null hypothesis H₀. Otherwise, fail to reject H₀. Interpret the result in the context of the claim.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.

Video duration:

Was this helpful?

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. It involves formulating a null hypothesis (H0) and an alternative hypothesis (H1), then using sample statistics to determine whether to reject H0. In this case, the null hypothesis states that the variance of the first population is less than or equal to that of the second (σ₁² ≤ σ₂²).

F-Test for Variances

The F-test is a statistical test used to compare two population variances. It calculates the ratio of the two sample variances (s₁²/s₂²) and compares it to a critical value from the F-distribution based on the degrees of freedom of the samples. This test helps determine if there is a significant difference between the variances, which is essential for validating the claim in the question.

Level of Significance (α)

The level of significance, denoted as α, is the probability of rejecting the null hypothesis when it is actually true (Type I error). In this scenario, α is set at 0.01, indicating a 1% risk of concluding that a difference exists when there is none. This threshold helps determine the critical value for the F-test and influences the strength of evidence required to support the claim.

Recommended video:

03:33

Finding Binomial Probabilities Using TI-84 Example 1

Related Practice

Textbook Question

List five properties of the F-distribution.

Textbook Question

Explain how to determine the values of d.f.N and d.f.D when performing a two-sample F-test.

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 = 44.6, n₁ = 16 and s₂² = 39.3, n₂ = 12

views

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.

Achievement and School Location The contingency table shows the results of a random sample of students by the location of school and the number of those students achieving basic skill levels in three subjects. At α=0.01, test the hypothesis that the variables are independent. (Adapted from HUD State of the Cities Report)

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.

U.S. History Assessment Tests A state school administrator claims that the standard deviations of U.S. history assessment test scores for eighth-grade students are the same in Districts 1 and 2. A sample of 10 test scores from District 1 has a standard deviation of 30.9 points, and a sample of 13 test scores from District 2 has a standard deviation of 27.2 points. At α=0.01, can you reject the administrator’s claim? (Adapted from National Center for Education Statistics)

Textbook Question

List the three conditions that must be met in order to use a two-sample F-test.

views