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=0.05,d.f.N=20,d.f.D=25
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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.RE.16
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=0.01,d.f.N=11,d.f.D=13
Verified step by step guidance1
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 11, while the degrees of freedom for the denominator (d.f.D) is 13.
Step 2: Recall that the F-distribution is used in hypothesis testing, and the critical F-value is determined based on the level of significance (α), the degrees of freedom for the numerator (d.f.N), and the degrees of freedom for the denominator (d.f.D). For a two-tailed test, the significance level is split equally between the two tails (α/2 for each tail).
Step 3: Use an F-distribution table or statistical software to find the critical F-value. Locate the row corresponding to d.f.N = 11 and the column corresponding to d.f.D = 13. Since this is a two-tailed test, you will need to find the critical F-value for α/2 = 0.005 in each tail.
Step 4: If using an F-distribution table, ensure you are looking at the correct table for the desired significance level (α/2 = 0.005). If the exact degrees of freedom are not listed, interpolation may be required to estimate the critical F-value.
Step 5: Once the critical F-value is identified, interpret it as the threshold beyond which the test statistic would fall in the rejection region for the null hypothesis. This value will be used to determine whether to reject or fail to reject the null hypothesis in the context of the hypothesis test.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Critical F-value
The critical F-value is a threshold used in hypothesis testing to determine whether to reject the null hypothesis. It is derived from the F-distribution, which is used when comparing variances between two groups. The critical value is based on the chosen significance level (α) and the degrees of freedom for the numerator (d.f.N) and denominator (d.f.D). If the calculated F-statistic exceeds this critical value, the null hypothesis is rejected.
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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 an F-test, d.f.N represents the degrees of freedom associated with the numerator (the group with more variability), while d.f.D represents the degrees of freedom for the denominator (the group with less variability). These values are crucial for determining the shape of the F-distribution and finding the critical F-value.
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Level of Significance (α)
The level of significance (α) is the probability of rejecting the null hypothesis when it is actually true, also known as a Type I error. It is a threshold set by the researcher, commonly at values like 0.01 or 0.05, which indicates the likelihood of observing a test statistic as extreme as the one calculated, under the null hypothesis. In this case, α=0.01 means there is a 1% risk of concluding that a difference exists when there is none.
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Related Practice
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)"
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Textbook Question
In Exercises 1–4, (a) identify the claim and state H₀ and Hₐ, (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.
A sports website claims that the opinions of golfers about what irritates them the most on the golf course are distributed as shown in the pie chart. You randomly select 1018 golfers and ask them what irritates them the most on the golf course. The table shows the results. At α=0.05, test the sports website’s claim. (Adapted from GOLF.com)
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Textbook Question
In Exercises 5–8, (a) find the expected frequency for each cell in the contingency table, (b) identify the claim and state H0 and Ha, (c) determine the degrees of freedom, find the critical value, and identify the rejection region, (d) find the chi-square test statistic, (e) decide whether to reject or fail to reject the null hypothesis, and (f) interpret the decision in the context of the original claim.
The contingency table shows the distribution of a random sample of fatal pedestrian and bicyclist motor vehicle collisions by time of day in a recent year. At α=0.10, can you conclude that the type of crash victim and the time of day are related? (Adapted from National Highway Traffic Safety Administration)
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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=0.01,d.f.N=40,d.f.D=60
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