11: The Chi-Square Distribution
- Page ID
- 45844
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)- 11.0: Introduction to the Chi-Square Distribution
- This page introduces the chi-square distribution, highlighting its applications in analyzing frequency data like lottery numbers and preferences by age group. It covers three main tests: the goodness-of-fit test for assessing data distribution, the test of independence for evaluating event independence, and the test of a single variance for variability assessment.
- 11.1: Facts About the Chi-Square Distribution
- This page discusses the chi-square distribution (\(\chi^2\)), emphasizing its dependence on degrees of freedom (\(df\)). It notes that the mean equals \(df\) and the standard deviation is \(\sigma=\sqrt{2(df)}\). The distribution is right-skewed and non-symmetrical, with each \(df\) producing a distinct curve. As \(df\) surpasses 90, the distribution approximates a normal distribution, with the mean located slightly to the right of the peak.
- 11.2: Test of a Single Variance
- This page emphasizes the importance of understanding both the mean and variability in populations, especially in production and assessments. It covers hypothesis testing for population variance, detailing null and alternative hypotheses, test statistics, and examples of variance testing in various contexts like education and service quality.
- 11.3: Goodness-of-Fit Test
- The Goodness-of-Fit hypothesis test, you determine whether the data "fit" a particular distribution or not. For example, you may suspect your unknown data fit a binomial distribution. You use a chi-square test (meaning the distribution for the hypothesis test is chi-square) to determine if there is a fit or not. The null and the alternative hypotheses for this test may be written in sentences or may be stated as equations or inequalities.
- 11.4: Test of Independence
- This page discusses independence tests using contingency tables to analyze relationships between categorical variables. It details test statistics, hypothesis formulation, and emphasizes expected value sufficiency. A case study on volunteer hours shows dependency between variables, while another study of 400 students examines anxiety levels and academic success needs.
- 11.5: Test for Homogeneity
- This page discusses the distinction between goodness-of-fit tests and tests for homogeneity, focusing on the latter's use of χ^2 statistics to compare population distributions, requiring a minimum expected value of five. It provides a case study on male and female college students' living arrangements showing differing distributions and includes exercises about car distributions among different demographics and applications to Ivy League schools.
- 11.6: Comparison of the Chi-Square Texts
- This page discusses the χ2 test applied in three scenarios: 1) Goodness-of-Fit, assessing if a sample fits a known distribution; 2) Independence, evaluating the relationship between two qualitative variables; and 3) Homogeneity, determining if two populations share the same distribution. Each scenario has a null hypothesis positing a fit or independence, and an alternative hypothesis suggesting otherwise.
- 11.7: Key Terms
- This page defines key statistical concepts: a **Contingency Table** for evaluating two-factor probabilities; **Goodness-of-Fit** tests comparing expected versus observed values in one variable; a **Test for Homogeneity** assessing if two populations have the same distribution; and a **Test of Independence**, which also compares distributions between two populations. Each concept utilizes degrees of freedom based on specific category counts.
- 11.8: Chapter Review
- This page discusses the importance of the chi-square distribution in evaluating data fit, population distribution equality, event independence, and variability. It highlights the role of degrees of freedom in shaping the distribution, which is right-skewed and approaches normality for df > 90.
- 11.9: Formula Review
- This page provides an overview of the chi-square distribution, detailing its definition, expected mean, and standard deviation. It explains the testing of a single variance, along with its formula and the procedure that includes calculating degrees of freedom. It also covers the goodness-of-fit test, the test of independence, and the test for homogeneity, each with their formulas and degrees of freedom.
- 11.10: Practice
- This page presents statistical exercises on the chi-square distribution and hypothesis testing, including variance analysis and goodness-of-fit tests. It explores specific scenarios like archers' accuracy and waiting times in clinics, and applies these concepts to smoking levels and cigarette consumption across ethnic groups.
- 11.11: Homework
- This page discusses statistical concepts including hypothesis testing, chi-square distribution, and goodness-of-fit tests. It features exercises examining variances and means through real-world scenarios, demographic analyses of preferences and relationships, and evaluations of observed versus expected distributions. Various datasets highlight factors like age, ethnicity, and preferences, emphasizing statistical significance in testing independence and homogeneity.
- 11.12: References
- This page presents a comprehensive list of sources and references on the Chi-Square Distribution, covering variance, goodness-of-fit, independence, and homogeneity tests. It includes data from various publications such as Santa Clara County health statistics, World Bank economic data, and The Field Poll opinions on sugary beverages, detailing publication dates and access information for a wide range of studies related to obesity and consumer preferences.
- 11.13: Solutions
- This page covers key statistical concepts like mean, standard deviation, variance tests, and goodness-of-fit tests. It discusses hypothesis testing scenarios, including left-tailed and right-tailed tests, along with relevant test statistics and sample sizes. The text also includes details on degrees of freedom and specific statistical tables regarding smoking levels across ethnic groups, as well as guidance on interpreting results from tests of independence and homogeneity.
Curated and edited by Kristin Kuter | Saint Mary's College, Notre Dame, IN