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 10.3.6
"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"
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 given parameters are the level of significance (α = 0.01), degrees of freedom for the numerator (d.f.N = 2), and degrees of freedom for the denominator (d.f.D = 11).
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 curve.
Step 3: Use an F-distribution table or statistical software. To find the critical F-value, locate the row corresponding to d.f.N = 2 and the column corresponding to d.f.D = 11 in the F-distribution table for α = 0.01. Alternatively, use statistical software or a calculator with an F-distribution function.
Step 4: Interpret the table or software output. The critical F-value is the value at which the cumulative probability in the right tail equals the level of significance (α = 0.01). This value marks the boundary of the rejection region.
Step 5: Verify the result. Double-check the table or software to ensure the correct degrees of freedom and significance level were used. This ensures the critical F-value is accurate for the given parameters.
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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 defined 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 chosen 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 used when the alternative hypothesis suggests that the parameter of interest is greater than the value specified in the null hypothesis. 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.
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=0.05, d.f.N=9, d.f.D=16"
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?
Textbook Question
Describe the hypotheses for a two-way ANOVA test.
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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?