12: F Distribution and One-Way ANOVA
- Page ID
- 46008
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)- 12.0: Introduction to the F-Distribtuion
- This page covers the F probability distribution and its applications in hypothesis testing, particularly in one-way ANOVA and variance testing. It highlights ANOVA's relevance across various fields like psychology and business, introducing single factor ANOVA. The chapter provides an initial overview, with plans for more in-depth exploration in future courses, and is produced by OpenStax College for free access.
- 12.1: Test of Two Variances
- This page discusses the F distribution, crucial for comparing variances in contexts like ANOVA. It covers the F test for variance equality, highlighting the need for normality and independence, with the F statistic as a ratio of sample variances compared to critical values.
- 12.2: One-Way ANOVA
- This page explains the one-way ANOVA test, which evaluates significant differences among group means based on variance. It requires five assumptions: normality of populations, independence of samples, equal variances, a categorical factor, and a numerical response. The null hypothesis posits that all group means are equal, while the alternative indicates at least one differs.
- 12.3: The F Distribution and the F-Ratio
- This page covers the \(F\) distribution and one-way ANOVA in hypothesis testing, showing how to calculate the \(F\) ratio using variances between and within samples. It details the process of conducting one-way ANOVA, including sums of squares calculations and provides an example involving tomato production experiments at Marist College.
- 12.4: Facts About the F Distribution
- This page covers the F distribution's properties and its application in ANOVA for variance analysis. It highlights the right-skewed nature of the F distribution, its relation to degrees of freedom, and the significance of the F statistic in testing null hypotheses. Several examples demonstrate one-way ANOVA, including analyses of tomato yields, sorority GPAs, and bean plant heights, where null hypotheses were not rejected due to high p-values.
- 12.5: Key Terms
- This page discusses key statistical concepts, focusing on Analysis of Variance (ANOVA), which tests whether the means of three or more populations are equal under specific conditions including normal distribution and equal standard deviations. It highlights the One-Way ANOVA as a specific application, defines variance with its formula involving sample size, and notes that the F-ratio is used as the test statistic for both types of ANOVA.
- 12.6: Chapter Review
- This page discusses the F test for comparing two variances, emphasizing normal distribution assumptions. It includes one-way ANOVA, which compares multiple groups under similar conditions. The F statistic, the ratio of variation between group means to within-group variation, is introduced. The positively skewed F distribution is explained, along with the relationship between the null hypothesis and the F statistic.
- 12.7: Formula Review
- This page explains hypothesis testing for two variances, focusing on the null hypothesis that the ratio of variances equals a specified value (δ0) and the alternative hypothesis that it does not. It discusses the F statistic as the ratio of sample variances and covers the F distribution, including calculations for sums of squares, degrees of freedom, and mean squares, culminating in the F-ratio employed in analysis of variance.
- 12.8: Practice
- This page discusses statistical exercises on variances focusing on F tests and one-way ANOVA. It presents various scenarios involving comparisons among coworkers, students, cyclists, and teams regarding commute times, test scores, sports performance, and ages for obtaining driver licenses.
- 12.9: Homework
- This page presents a variety of statistical experiments analyzing differences in means across multiple groups. Key analyses include weight gain in rats, commuting distances among social classes, and various measurements related to magazine lengths and skiing conditions. Each study employs methods like One-Way ANOVA, forming hypotheses and testing significance levels (5%-10%) to assess variances.
- 12.10: References
- This page encompasses diverse statistical references, including historical MLB standings, unpublished student research on tomatoes, and educational insights on the F distribution. It notably cites the Handbook of Small Datasets on fruit fly reproduction and critiques body temperature standards from a 1992 medical journal. Overall, it showcases a blend of academic research and statistical evaluations from varied sources.
- 12.11: Solutions
- This page covers statistical tests, notably ANOVA and hypothesis testing, focusing on variances and means among populations. It highlights the assumptions of normal distributions and equal variances, noting instances where null hypotheses are not rejected, indicating no significant differences in certain datasets. However, some cases confirm significant differences, particularly in fruit flies' egg-laying behaviors.
Curated and edited by Kristin Kuter | Saint Mary's College, Notre Dame, IN