12.15: Formula Review
- Page ID
- 140478
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Probability Distribution Needed for Hypothesis Testing
| Type of Hypothesis Test | Population Parameter | Estimated value (point estimate) | Probability Distribution Used |
|---|---|---|---|
| Hypothesis test for the mean, when the population standard deviation is known | Population mean \(\mu\) | Sample mean \(\bar{x}\) | Normal distribution, \(\bar{X} \sim N\left(\mu_X, \dfrac{\sigma_X}{\sqrt{n}}\right)\) |
| Hypothesis test for the mean, when the population standard deviation is unknown and the distribution of the sample mean is approximately normal | Population mean \(\mu\) | Sample mean \(\bar{x}\) | Student's t-distribution, \(t_{d f}\) |
| Hypothesis test for proportions | Population proportion p | Sample proportion p ' | Normal distribution, \(P^{\prime} \sim N\left(p, \sqrt{\dfrac{p \cdot q}{n}}\right)\) |
Full Hypothesis Test Examples
Test statistic for a hypothesis test of proportions:
\(Z_c=\dfrac{\mathrm{p}^{\prime}-p_0}{\sqrt{\dfrac{p_0(1-p 0)}{n}}}\)
Comparing Two Independent Population Means
Standard error: \(S E=\sqrt{\dfrac{\left(s_1\right)^2}{n_1}+\dfrac{\left(s_2\right)^2}{n_2}}\)
Test statistic ( \(t\)-score): \(t_c=\dfrac{\left(\bar{x}_1-\bar{x}_2\right)-\delta_0}{\sqrt{\dfrac{\left(s_1\right)^2}{n_1}+\dfrac{\left(s_2\right)^2}{n_2}}}\)
Degrees of freedom:
\[d f=\dfrac{\left(\dfrac{\left(s_1\right)^2}{n_1}+\dfrac{\left(s_2\right)^2}{n_2}\right)^2}{\left(\dfrac{1}{n_1-1}\right)\left(\dfrac{\left(s_1\right)^2}{n_1}\right)^2+\left(\dfrac{1}{n_2-1}\right)\left(\dfrac{\left(s_2\right)^2}{n_2}\right)^2}\]
where:
\(s_1\) and \(s_2\) are the sample standard deviations, and \(n_1\) and \(n_2\) are the sample sizes.
\(\bar{x}_1\) and \(\bar{x}_2\) are the sample means.
Cohen's Standards for Small, Medium, and Large Effect Sizes
Cohen's \(d\) is the measure of effect size:
\[d=\dfrac{\bar{x}_1-\bar{x}_2}{s_{\text {pooled }}}\]
where \(s_{\text {pooled }}=\sqrt{\dfrac{\left(n_1-1\right) s_1^2+\left(n_2-1\right) s_2^2}{n_1+n_2-2}}\)
Test for Differences in Means: Assuming Equal Population Variances
\[t_c=\dfrac{\left(\bar{x}_1-\bar{x}_2\right)-\delta_0}{\sqrt{S_p^2\left(\dfrac{1}{n_1}+\dfrac{1}{n_2}\right)}}\]
where \(S_p^2\) is the pooled variance given by the formula:
\[S_p^2=\dfrac{\left(n_1-1\right) s_1^2+\left(n_2-1\right) s_2^2}{n_1+n_2-2}\]
Comparing Two Independent Population Proportions
Pooled Proportion: \(p_{\mathrm{C}}=\dfrac{x_A+x_B}{n_A+n_B}\)
Test Statistic (z-score): \(Z_c=\dfrac{\left(p_A^{\prime}-p_B^{\prime}\right)}{\sqrt{p_c\left(1-p_c\right)\left(\dfrac{1}{n_A}+\dfrac{1}{n_B}\right)}}\)
where
\(p_A^{\prime}\) and \(p_B^{\prime}\) are the sample proportions, \(p_A\) and \(p_B\) are the population proportions,
\(P_C\) is the pooled proportion, and \(n_A\) and \(n_B\) are the sample sizes.
Two Population Means with Known Standard Deviations
Test Statistic (z-score):
\[Z_c=\dfrac{\left(\bar{x}_1-\bar{x}_2\right)-\delta_0}{\sqrt{\dfrac{\left(\sigma_1\right)^2}{n_1}+\dfrac{\left(\sigma_2\right)^2}{n_2}}}\]
where:
\(\sigma_1\) and \(\sigma_2\) are the known population standard deviations. \(n_1\) and \(n_2\) are the sample sizes. \(\bar{x}_1\) and \(\bar{x}_2\) are the sample means.
\(\mu_1\) and \(\mu_2\) are the population means.
10.7 Matched or Paired Samples
Test Statistic ( \(t\)-score): \(t_c=\dfrac{\bar{x}_d-\mu_d}{\left(\dfrac{s_d}{\sqrt{n}}\right)}\)
where:
\(\bar{x}_d\) is the mean of the sample differences. \(\mu_d\) is the mean of the population differences. \(s_d\) is the sample standard deviation of the differences. \(n\) is the sample size.