8.6: Formulas for Chapter 8
- Page ID
- 58296
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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 8 Formulas
Test Statistic for t-Test for the Mean
The t-test test statistic for the mean is used when testing a claim about a population mean, and the population standard deviation is unknown. It is typically applied when the sample size is small (less than 30), although it can be used for larger samples as well. The test is based on the t-distribution and uses the sample standard deviation in place of the population standard deviation. The test statistic measures how many estimated standard errors the sample mean is from the hypothesized population mean.
\( t = \dfrac{\bar{x} - \mu}{\dfrac{s}{\sqrt{n}}} \)
Where:
- \(\bar{x}\) is the sample mean
- \(\mu\) is the population mean stated in the null hypothesis
- \(s\) is the sample standard deviation (known)
- \(n\) is the sample size
- \(\dfrac{s}{\sqrt{n}}\) is the estimated standard error of the mean (based on the sample)
Test Statistic for z-Test for the Mean
The test point for the z-test for the mean is used when testing a claim about a population mean, and the population standard deviation is known. It is appropriate when the sample size is large (typically n ≥ 30) or the population is normally distributed. The test statistic measures how many standard errors the sample mean is from the hypothesized population mean.
\(z =\dfrac{\bar{x} - \mu}{\dfrac{\sigma}{\sqrt{n}}}\)
Where:
- \(\bar{x}\) is the sample mean
- \(\mu\) is the population mean stated in the null hypothesis
- \(\sigma\) is the population standard deviation (known)
- \(n\) is the sample size
- \(\dfrac{\sigma}{\sqrt{n}}\) is the standard error of the mean (measures variability in sample means)
Test Statistic for z-Test for the Proportion
The test point for the z-test for a proportion is used to test a claim about a population proportion, typically when the sample size is large enough for the normal approximation to apply (usually when both \(np\) and \(nq\) are at least 5). This test determines whether the observed sample proportion differs significantly from a hypothesized population proportion. The test statistic measures how many standard errors the sample proportion is from the claimed population proportion.
\( z = \dfrac{\hat{p} - p}{\sqrt{\dfrac{pq}{n}}} \)
Where:
- \(\hat{p}\) is the sample proportion
- \(p\) is the population proportion stated in the null hypothesis
- \(q\) is the complement of the population proportion, calculated as \(q=1−p\)
- \(n\) is the sample size
- \(\sqrt{\dfrac{pq}{n}}\) is the standard error of the sample proportion
Authors
"8.6: Formulas for Chapter 8" by Toros Berberyan, Tracy Nguyen, and Alfie Swan is licensed under CC BY 4.0