Textbook Question
Explain how to find the critical value for an F-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 10.3.5
"Finding a Critical F-Value for a Right-Tailed Test In Exercises 5–8, 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=9, d.f.D=16"
Verified step by step guidance1
Step 1: Understand the problem. We are tasked with finding the critical F-value for a right-tailed test. The level of significance (α) is 0.05, the degrees of freedom for the numerator (d.f.N) is 9, and the degrees of freedom for the denominator (d.f.D) is 16.
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 non-rejection region in the F-distribution.
Step 3: Use an F-distribution table or statistical software. Locate the row corresponding to d.f.N = 9 and the column corresponding to d.f.D = 16 in the F-distribution table for α = 0.05. This table provides the critical F-value for the specified degrees of freedom and significance level.
Step 4: If using statistical software (e.g., Excel, R, or a calculator), use the appropriate function. For example, in Excel, you can use the formula F.INV.RT(0.05, 9, 16) to find the critical F-value for a right-tailed test.
Step 5: Interpret the result. The critical F-value represents the threshold beyond which the test statistic would lead to rejecting the null hypothesis. Ensure that the value obtained matches the table or software output for accuracy.
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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 variances between two populations and is characterized by two sets of degrees of freedom: one for the numerator (d.f.N) and one for the denominator (d.f.D). The shape of the F-distribution is right-skewed, meaning it has a longer tail on the right side.
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Critical Value
A critical value is a threshold that determines the boundary for rejecting the null hypothesis in hypothesis testing. In the context of an F-test, the critical F-value is derived from the F-distribution based on the specified level of significance (α) and the degrees of freedom. If the calculated F-statistic exceeds this critical value, the null hypothesis is rejected, indicating a statistically significant difference.
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Right-Tailed Test
A right-tailed test is a type of hypothesis test where the critical region for rejecting the null hypothesis is located in the right tail of the distribution. This test is appropriate when the alternative hypothesis posits that a parameter is greater than a certain value. In the context of the F-test, a right-tailed test assesses whether the variance of one group is significantly greater than that of another.
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Related Practice
Textbook Question
Testing for Normality Using a chi-square goodness-of-fit test, you can decide, with some degree of certainty, whether a variable is normally distributed. In all chi-square tests for normality, the null and alternative hypotheses are as listed below.
H₀: The variable has a normal distribution.
Hₐ: The variable does not have a normal distribution.
To determine the expected frequencies when performing a chi-square test for normality, first estimate the mean and standard deviation of the frequency distribution. Then, use the mean and standard deviation to compute the z-score for each class boundary. Then, use the z-scores to calculate the area under the standard normal curve for each class. Multiplying the resulting class areas by the sample size yields the expected frequency for each class.In Exercises 17 and 18, (a) find the expected frequencies, (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.
In Exercises 17 and 18, (a) find the expected frequencies, (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.
Test Scores At α=0.01, test the claim that the 200 test scores shown in the frequency distribution are normally distributed.
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Textbook Question
Conditional Relative Frequencies In Exercises 37–42, use the contingency table from Exercises 33–36, and the information below.
Relative frequencies can also be calculated based on the row totals (by dividing each row entry by the row’s total) or the column totals (by dividing each column entry by the column’s total). These frequencies are conditional relative frequencies and can be used to determine whether an association exists between two categories in a contingency table.
What percent of U.S. adults ages 25 and over who are not high school graduates are unemployed?
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Textbook Question
Describe the hypotheses for a two-way ANOVA test.
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Textbook Question
"Finding a Critical F-Value for a Right-Tailed Test In Exercises 5–8, 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=2, d.f.D=11"
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Textbook Question
True or False? In Exercises 5 and 6, determine whether the statement is true or false. If it is false, rewrite it as a true statement.
When the test statistic for the chi-square independence test is large, you will, in most cases, reject the null hypothesis.
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