11.3: Chapter 11 Formulas
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Chapter 11 Formulas
\(\chi^2\) Test Statistic for Goodness of Fit
The \(\chi^2\) test of goodness of fit is used to determine whether the observed frequencies in a categorical variable match the expected frequencies based on a specific theoretical distribution. It assesses how well the observed data fit the expected pattern. The test uses degrees of freedom given by \(df = k - 1\), where \(k\) is the number of categories.
\(\chi^2 = \sum \dfrac{(O - E)^2}{E}\)
Where:
- \( \chi^2 \) is the chi-square test statistic
- \( O \) is the observed frequency in each category
- \( E \) is the expected frequency in each category
- \( \sum \) indicates that the calculation is summed over all categories
- \( df \) is the degrees of freedom
- \( k \) is the number of distinct categories being tested
Expected Value for the \(\chi^2\) Test of Independence
The expected value in a \(\chi^2\) test for independence is the frequency that would be expected in each cell of a contingency table if the row and column variables were statistically independent. These expected values are used to compare with the observed values to compute the chi-square test statistic. This calculation is done for each cell in the table to evaluate how far the observed values deviate from what would be expected under the assumption of independence.
\( E = \dfrac{(\text{row total}) \times (\text{column total})}{\text{grand total}}\)
\(\chi^2\) Test Statistic for Independence
The chi-square test for independence is used to determine whether there is a significant association between two categorical variables in a contingency table. It compares the observed frequencies in each cell to the frequencies we would expect if the two variables were independent. The degrees of freedom for the test are calculated using the formula \(df = (r - 1)(c - 1)\), where \(r\)is the number of rows and \(c\) is the number of columns in the table.
\(\chi^2 = \sum \dfrac{(O - E)^2}{E}\)
Where:
- \( \chi^2 \) is the chi-square test statistic
- \( O \) is the observed frequency in a cell
- \( E \) is the expected frequency in the same cell
- \( \sum \) indicates the sum over all cells in the contingency table
- \( df = (r - 1)(c - 1) \) is the degrees of freedom
- \( r \) is the number of rows
- \( c \) is the number of columns
Authors
"11.3: Chapter 11 Formulas" by Toros Berberyan, Tracy Nguyen, and Alfie Swan is licensed under CC BY 4.0