11.9: Formula Review
- Page ID
- 46054
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)11.2 Facts About the Chi-Square Distribution
\(X^2=\left(Z_1\right)^2+\left(Z_2\right)^2+\ldots\left(Z_{d f}\right)^2\) chi-square distribution random variable
\(\mu_{X^2}=d f\) chi-square distribution population mean
\(\sigma_{\chi^2}=\sqrt{2(d f)}\) Chi-Square distribution population standard deviation
11.3 Test of a Single Variance
\(\chi^2=\dfrac{(n-1) s^2}{\sigma_0^2}\) Test of a single variance statistic where:
\(n:\) sample size
s: sample standard deviation
\(\sigma_0\) : hypothesized value of the population standard deviation
\(d f=n-1\) Degrees of freedom
Test of a Single Variance
- Use the test to determine variation.
- The degrees of freedom is the number of samples -1 .
- The test statistic is \(\dfrac{(n-1) s^2}{\sigma_0^2}\), where \(n=\) sample size, \(s^2=\) sample variance, and \(\sigma^2=\) population variance.
- The test may be left-, right-, or two-tailed.
11.4 Goodness-of-Fit Test
\(\sum_k \dfrac{(O-E)^2}{E}\) goodness-of-fit test statistic where:
O: observed values
\(E\) : expected values
\(k\) : number of different data cells or categories
\(d f=k-1\) degrees of freedom
11.5 Test of Independence
Test of Independence
- The number of degrees of freedom is equal to (number of columns - 1)(number of rows - 1).
- The test statistic is \(\sum_{i \cdot j} \dfrac{(O-E)^2}{E}\) where \(O=\) observed values, \(E=\) expected values, \(i=\) the number of rows in the table, and \
- (j=\) the number of columns in the table.
- If the null hypothesis is true, the expected number \(E=\dfrac{\text { (row total)(column total) }}{\text { total surveyed }}\).
11.6 Test for Homogeneity.
\(\sum_{i \cdot j} \dfrac{(O-E)^2}{E}\) Homogeneity test statistic where: \(O=\) observed values
\(E=\) expected values
\(i=\) number of rows in data contingency table
\(j=\) number of columns in data contingency table
\(d f=(i-1)(j-1)\) Degrees of freedom