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=0.05,d.f.N=9,d.f.D=8"
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.13
"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"
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 0.10, and the degrees of freedom for the numerator (d.f.N) and denominator (d.f.D) are 15 and 27, respectively.
Step 2: Recognize that for a two-tailed test, the significance level (α) is split equally between the two tails of the F-distribution. This means each tail will have an area of α/2 = 0.10/2 = 0.05.
Step 3: Use an F-distribution table or statistical software to find the critical F-values. For a two-tailed test, you will need to find two critical values: one for the upper tail and one for the lower tail. The degrees of freedom for the numerator (d.f.N = 15) and denominator (d.f.D = 27) are used to locate the critical values.
Step 4: For the upper critical value, look up the F-value corresponding to α/2 = 0.05 in the upper tail with d.f.N = 15 and d.f.D = 27. For the lower critical value, use the reciprocal property of the F-distribution: F_lower = 1 / F_upper.
Step 5: Verify your results by ensuring the total area under the F-distribution curve outside the critical values equals the significance level (α = 0.10). This confirms the critical values are correct for the two-tailed 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. In a two-tailed test, the critical F-value is found at both ends of the distribution, corresponding to the specified level of significance (α).
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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 the F-test, there are two types of degrees of freedom: d.f.N (numerator) and d.f.D (denominator), which correspond to the number of groups being compared and the total number of observations, respectively.
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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 0.05 or 0.10, which defines the cutoff for determining whether the observed data is statistically significant. In this case, α=0.10 indicates a 10% risk of making a Type I error.
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Related Practice
Textbook Question
In Exercises 21 and 22, (d) decide whether to reject or fail to reject the null hypothesis,
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 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?
Textbook Question
In Exercises 21 and 22, (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, the populations are normally distributed, and the population variances are equal.
[APPLET] The table shows the annual incomes (in dollars) for a sample of families from four regions of the United States. At α=0.05, can you conclude that the mean annual income of families is different in at least one of the regions? (Adapted from U.S. Census Bureau)
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 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"