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=0.10,d.f.N=15,d.f.D=27"
Ch. 10 - Chi-Square Tests and the F-Distribution
Larson8th EditionElementary Statistics: Picturing the WorldISBN: 9780137493470
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Table of contents
- Ch. 1 - Introduction to Statistics87
- Ch. 2 - Descriptive Statistics176
- Ch. 3 - Probability184
- Ch. 4 - Discrete Probability Distributions102
- Ch. 5 - Normal Probability Distributions183
- Ch. 6 - Confidence Intervals154
- Ch. 7 - Hypothesis Testing with One Sample165
- Ch. 8 - Hypothesis Testing with Two Samples134
- Ch. 9 - Correlation and Regression87
- Ch. 10 - Chi-Square Tests and the F-Distribution87
Larson8th EditionElementary Statistics: Picturing the WorldISBN: 9780137493470Not the one you use?Change textbook
Chapter 10, Problem 10.R.19
In Exercises 17–20, (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.
[APPLET] An instructor claims that the variance of SAT evidence-based reading and writing scores is different than the variance of SAT math scores. The table shows the SAT evidence-based reading and writing scores for 12 randomly selected students and the SAT math scores for 12 randomly selected students. At α=0.01, can you support the instructor’s claim?
Verified step by step guidance1
Step 1: Identify the claim and state the null hypothesis (H₀) and the alternative hypothesis (Hₐ). The claim is that the variance of SAT evidence-based reading and writing scores is different from the variance of SAT math scores. Thus, H₀: σ₁² = σ₂² (the variances are equal), and Hₐ: σ₁² ≠ σ₂² (the variances are not equal).
Step 2: Determine the critical value and rejection region. Since this is a two-tailed test and the significance level is α = 0.01, use the F-distribution table to find the critical values for the degrees of freedom (df₁ = n₁ - 1 and df₂ = n₂ - 1, where n₁ and n₂ are the sample sizes). The rejection region will be in both tails of the F-distribution.
Step 3: Calculate the test statistic F. First, compute the sample variances for both groups (reading and writing, and math). Use the formula for variance: s² = Σ(xᵢ - x̄)² / (n - 1). Then, calculate the test statistic F = s₁² / s₂², where s₁² is the larger sample variance and s₂² is the smaller sample variance.
Step 4: Compare the test statistic F to the critical values. If the test statistic falls in the rejection region, reject the null hypothesis H₀. Otherwise, fail to reject H₀.
Step 5: Interpret the decision in the context of the original claim. If H₀ is rejected, there is sufficient evidence to support the instructor's claim that the variances are different. If H₀ is not rejected, there is not enough evidence to support the claim.
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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 a population based on sample data. It involves formulating two competing hypotheses: the null hypothesis (H₀), which represents no effect or no difference, and the alternative hypothesis (Hₐ), which indicates the presence of an effect or difference. In this case, the instructor's claim about the variances of SAT scores forms the basis for these hypotheses.
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Step 1: Write Hypotheses
F-Test for Variances
The F-test is a statistical test used to compare the variances of two populations to determine if they are significantly different. It calculates the F-statistic, which is the ratio of the variances of the two samples. If the calculated F-statistic exceeds a critical value from the F-distribution table at a specified significance level (α), the null hypothesis is rejected, indicating that the variances are significantly different.
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Critical Value and Rejection Region
The critical value is a threshold that determines the boundary for rejecting the null hypothesis in hypothesis testing. It is derived from the chosen significance level (α), which indicates the probability of making a Type I error. The rejection region is the range of values for the test statistic that leads to the rejection of H₀. In this scenario, identifying the critical value and rejection region is essential to assess whether the instructor's claim about the variances can be supported.
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Related Practice
Textbook Question
In Exercises 21 and 22, (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] The table shows the monthly electric bills (in dollars) for a sample of households from four regions of the United States. At α=0.10, can you conclude that the mean monthly electric bill is different in at least one of the regions? (Adapted from U.S. Energy Information Administration)
Textbook Question
In Exercises 21 and 22, (c) find the test statistic F, Assume the samples are random and independent, the populations are normally distributed, and the population variances are equal.
[APPLET] The table shows the monthly electric bills (in dollars) for a sample of households from four regions of the United States. At α=0.10, can you conclude that the mean monthly electric bill is different in at least one of the regions? (Adapted from U.S. Energy Information Administration)
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Textbook Question
"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=0.01,d.f.N=12,d.f.D=10"
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Textbook Question
"In Exercises 17–20, (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.
A travel consultant claims that the standard deviations of hotel room rates for Sacramento, CA, and San Francisco, CA, are the same. A sample of 36 hotel room rates in Sacramento has a standard deviation of \$51 and a sample of 31 hotel room rates in San Francisco has a standard deviation of \$37. At α=0.10, can you reject the travel consultant’s claim? (Adapted from Expedia)"
Textbook Question
"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=0.10,d.f.N=5,d.f.D=12"