10: Correlation and Linear Regression
- Page ID
- 58305
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)- 10.1: Bivariate Data and Scatter Plots
- Bivariate data involves pairs of related values, typically measuring two variables for each subject. Scatter plots graph these pairs to show patterns, trends, or relationships between the variables. They are useful for identifying correlations, clusters, or outliers in the data.
- 10.2: Correlation Coefficient
- The correlation coefficient, often called r, measures the strength and direction of a linear relationship between two variables. It helps determine how closely the variables are related. The value of r ranges from -1 to 1, with values near either extreme showing stronger relationships.
- 10.3: Hypothesis Test for Correlation Using r
- Hypothesis testing for correlation using r determines whether a significant linear relationship exists between two variables in a population. It involves comparing the sample correlation to critical values based on sample size. The test helps confirm if the observed correlation is due to chance or reflects a real association.
- 10.4: Hypothesis Test for a Correlation Using t-Test
- Hypothesis testing for correlation using r with the t-value method checks if the observed correlation is statistically significant. It involves converting the correlation to a t-value and comparing it to a critical value based on degrees of freedom. This method helps determine if there’s evidence of a real linear relationship in the population
- 10.5: Linear Regression
- Linear regression models the relationship between two variables by fitting a straight line to the data. The line is defined by two coefficients: a (the y-intercept) and b (the slope), which describe where the line starts and how steep it is. This method is used to make predictions based on the trend between variables.
- 10.6: Coefficient of Determination and the Standard Error of the Estimate
- The coefficient of determination shows how well the regression line explains the variation in the dependent variable, ranging from 0 to 1. A higher value means a better fit. The standard error of the estimate measures how much the actual values differ from the predicted ones—a smaller value indicates more accurate predictions and a better model fit.
- 10.7: Formulas for Chapter 10
- In this section, the equations for Chapter 10 are displayed. These include the correlation coefficient, which measures the strength and direction of a linear relationship between two variables; the line of regression and its regression coefficients, which model the best-fitting linear relationship between the independent and dependent variables; the coefficient of determination, which indicates the proportion of variance in the dependent variable explained by the model; and the standard error.
- 10.8: Chapter 10 - Key Terms and Symbols
- This section presents the key terms and symbols for Chapter 10, which focuses on correlation and linear regression. It includes terminology for describing relationships between two variables, interpreting correlation, and making predictions using regression models. The symbols represent core concepts used to assess and model linear associations in bivariate data.