Chapter 12 – Linear Regression and Correlation

=## Linear Equations

Two-variable linear regressions take on the form:

$$y=a+bx$$

where $x$ is the independent variable and $y$ is the dependent variable.

Scatterplots

Scatterplots show the direction of a bivariate relationship.

The Regression Equation

The line of best fit is also called a least-squares line in the form $\hat{y}=a+bx$. $\hat{y}$ is an estimate.

• The least-squares line always passes through the point of the $x$ and $y$ sample means $(\bar{x},\bar{y})$
  • $a=\bar{y}-b\bar{x}$
  • $b=r\frac{s_y}{s_x}$

Error/Residuals

Residuals are calculated using the formula $y_0-{\hat{y}}_0={\epsilon }_0$.

The sum of squared errors (SSE) is calculated as $\Sigma (\epsilon {)}^2$. The minima of the SSE returns the line of best fit.

A residual plot should be random/show no pattern when a linear relation is present. A dataset with a strong linear relation has residuals which produce no pattern.

Correlation Coefficient

The correlation coefficient $(r)$ measures the strength of the relationship between $x$ and $y$.

$$r=\frac{n\Sigma (xy)-(\Sigma x)(\Sigma y)}{\sqrt{[n\Sigma (x^2)-(\Sigma y{)}^2][n\Sigma (y^2)-(\Sigma x{)}^2]}}$$

• $-1\le r\le 1$
• Values of $r$ close to -1 or 1 indicate a strong linear relationship
• $r=0$ can be a horizontal linear correlation or no linear correlation
• $r=1$ or $-1$ indicate a perfect correlation

Coefficient of Determination

The coefficient of determination $(r^2)$ is typically stated as a percentage.

• $r^2$ as a percentage represents the percent variation in $y$ that can be explained by $x$
• $1-r^2$ represents the percent variation not explained by $x$