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"
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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.R.9
"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=6,d.f.D=50"
Verified step by step guidance1
Step 1: Understand the problem. You are tasked with finding the critical F-value for a right-tailed test. The inputs provided are the level of significance (α = 0.05), degrees of freedom for the numerator (d.f.N = 6), and degrees of freedom for the denominator (d.f.D = 50).
Step 2: Recall the definition of the F-distribution. The F-distribution is used in hypothesis testing to compare variances. The critical F-value is the value that separates the rejection region (right tail) from the rest of the distribution, based on the given α.
Step 3: Use an F-distribution table or statistical software. Locate the row corresponding to d.f.N = 6 (numerator degrees of freedom) and the column corresponding to d.f.D = 50 (denominator degrees of freedom) in the F-distribution table for α = 0.05. If using software, input these values to obtain the critical F-value.
Step 4: Interpret the result. The critical F-value represents the threshold beyond which the null hypothesis would be rejected in a right-tailed test. Ensure that the value corresponds to the correct α and degrees of freedom.
Step 5: Verify your result. Double-check the table or software output to ensure accuracy. If using a table, confirm that you are referencing the correct α level and degrees of freedom.
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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 across different groups. In a right-tailed test, if the calculated F-statistic exceeds the critical F-value, the null hypothesis is rejected, indicating a significant difference between group variances.
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Degrees of Freedom (d.f.)
Degrees of freedom 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, there are two types of degrees of freedom: d.f.N (numerator) associated with the group variances being compared, and d.f.D (denominator) related to the error variance. These values are crucial for determining the shape of the F-distribution used to find the critical F-value.
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Level of Significance (α)
The level of significance, denoted as α, is the probability of rejecting the null hypothesis when it is actually true, also known as a Type I error. Commonly set at 0.05, it defines the threshold for determining statistical significance in hypothesis testing. A lower α value indicates a stricter criterion for significance, while a higher α allows for more leniency in rejecting the null hypothesis.
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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 each exercise,
c. find the test statistic,
In Exercises 1 and 2, use the table, which lists the distribution of educational achievement for people in the United States ages 25 and older. It also lists the results of a random survey for two additional age groups. (Adapted from U.S. Census Bureau)
Use the data for 30- to 34-year-olds and 65- to 69-year-olds to test whether age and educational attainment are related. Use α=0.01.
42
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Textbook Question
In each exercise,
d. decide whether to reject or fail to reject the null hypothesis, and
[APPLET] In Exercises 3 and 4, use the data, which list the annual wages (in thousands of dollars) for randomly selected individuals from three metropolitan areas. Assume the wages are normally distributed and that the samples are independent. (Adapted from U.S. Bureau of Economic Analysis)
Ithaca, NY: 53.0, 60.3, 34.6, 37.1, 46.6, 46.8, 41.4, 50.6, 50.8, 49.4, 35.0, 36.7, 57.1
Little Rock, AR: 50.7, 43.7, 53.4, 40.0, 45.2, 52.7, 35.2, 60.4, 40.0, 45.9, 45.7, 47.3, 46.5, 44.5, 31.5
Madison, WI: 62.4, 53.9, 67.6, 52.9, 67.7, 50.7, 62.1, 58.9, 61.1, 65.0, 60.4, 59.6, 51.3, 44.8, 66.2
Are the mean annual wages the same for all three cities? Use α=0.10. Assume that the population variances are equal.
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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)
53
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Textbook Question
In each exercise,
a. identify the claim and state H₀ and Hₐ,
In Exercises 1 and 2, use the table, which lists the distribution of educational achievement for people in the United States ages 25 and older. It also lists the results of a random survey for two additional age groups. (Adapted from U.S. Census Bureau)
Use the data for 30- to 34-year-olds and 65- to 69-year-olds to test whether age and educational attainment are related. Use α=0.01.
43
views