Textbook Question
Describe the hypotheses for a two-way ANOVA test.
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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 10.3.11
"Finding a Critical F-Value for a Two-Tailed Test In Exercises 9–12, 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=60, d.f.D=40"
Verified step by step guidance1
Step 1: Understand the problem. We are tasked with finding the critical F-value for a two-tailed test. The level of significance (α) is 0.05, and the degrees of freedom for the numerator (d.f.N) and denominator (d.f.D) are 60 and 40, respectively.
Step 2: Recall that for a two-tailed test, the level of significance (α) is split equally between the two tails of the F-distribution. This means each tail will have an area of α/2 = 0.05/2 = 0.025.
Step 3: Use an F-distribution table or statistical software to find the critical F-values. For the upper critical value, look up the F-value corresponding to α/2 = 0.025, d.f.N = 60, and d.f.D = 40. For the lower critical value, take the reciprocal of the upper critical value because the F-distribution is not symmetric.
Step 4: If using statistical software, input the parameters (α/2 = 0.025, d.f.N = 60, d.f.D = 40) to directly obtain the critical F-values. If using a table, locate the row for d.f.N = 60 and the column for d.f.D = 40 under the α/2 = 0.025 column.
Step 5: Interpret the results. The critical F-values define the rejection regions for the two-tailed test. If the calculated F-statistic falls outside the range of these critical values, the null hypothesis is rejected.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
F-Distribution
The F-distribution is a probability distribution that arises frequently in statistics, particularly in the context of variance analysis. It is used to compare the variances of two populations and is defined by two sets of degrees of freedom: one for the numerator and one for the denominator. The shape of the F-distribution is right-skewed, and it approaches a normal distribution as the degrees of freedom increase.
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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, d.f.N (numerator) corresponds to the number of groups minus one, while d.f.D (denominator) corresponds to the total number of observations minus the number of groups. These values are crucial for determining the critical F-value.
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Critical Value
A critical value is a threshold that determines the boundary for rejecting the null hypothesis in hypothesis testing. For an F-test, the critical F-value is derived from the F-distribution table based on the chosen significance level (α) and the degrees of freedom. In a two-tailed test, the critical values are found at both ends of the distribution, indicating the regions where the null hypothesis can be rejected.
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Related Practice
Textbook Question
"Finding a Critical F-Value for a Two-Tailed Test In Exercises 9–12, 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=27, d.f.D=19"
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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.
Carbon Monoxide Emissions An automobile manufacturer claims that the variance of the carbon monoxide emissions for a make and model of one of its vehicles is less than the variance of the carbon monoxide emissions for a top competitor’s equivalent vehicle. A sample of the carbon monoxide emissions of 19 of the manufacturer’s specified vehicles has a variance of 0.008. A sample of the carbon monoxide emissions of 21 of its competitor’s equivalent vehicles has a variance of 0.045. At α=0.10, can you support the manufacturer’s claim? (Adapted from U.S. Environmental Protection Agency)"
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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.
Use the contingency table and expected frequencies from Exercise 11. At α=0.10, test the hypothesis that the variables are independent.
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Textbook Question
Performing a Chi-Square Goodness-of-Fit Test
In Exercises 7–16, (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.
Births by Day of the Week A doctor claims that the number of births by day of the week is uniformly distributed. To test this claim, you randomly select 700 births from a recent year and record the day of the week on which each takes place. The table shows the results. At α=0.10, test the doctor’s claim. (Adapted from National Center for Health Statistics)
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Textbook Question
Contingency Tables and Relative Frequencies In Exercises 33–36, use the information below.
The frequencies in a contingency table can be written as relative frequencies by dividing each frequency by the sample size. The contingency table below shows the number of U.S. adults (in millions) ages 25 and over by employment status and educational attainment. (Adapted from U.S. Census Bureau)
What percent of U.S. adults ages 25 and over (a) are employed and are only high school graduates, (b) are not in the civilian labor force, and (c) are not high school graduates?
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