2.8 Least Squares Regression

• The least squares regression line minimizes the sum of squared vertical distances
• What is the formula for calculating the slope ($b_{1}$) of the least squares regression line?


  $b_{1} =$$\frac{\sum_{i = 1}^{n}(x_{i} - \bar{x})(y_{i} - \bar{y})}{\sum_{i = 1}^{n}(x_{i} - \bar{x})^{2}}$
• Match the variable with its definition in the least squares regression line:

  y ↔️ Predicted value of dependent variable

  x ↔️ Independent variable

  $b_{0}$ ↔️ y-intercept

  $b_{1}$ ↔️ Slope
• The least squares regression line always passes through the point defined by the sample means of $x$ and $y$.


  True
• Match the variables in the y-intercept formula with their definitions:

  $\bar{y}$ ↔️ Sample mean of $y$ variable

  $\bar{x}$ ↔️ Sample mean of $x$ variable

  $b_{1}$ ↔️ Slope of the regression line
• What does the least squares regression line minimize?

  Sum of squared vertical distances
• Match the concept with its explanation:

  Least Squares Regression Line ↔️ The line that minimizes the sum of squared vertical distances.

  Sum of Squares ↔️ The total of the squared vertical distances.

  Vertical Distances ↔️ The distance from each data point to the regression line.
• What is the general formula for the least squares regression line?

  y = b_{0} + b_{1}x</latex>
• The y-intercept is the mean of the $y$ variable minus the product of the slope and the mean of the $x$ variable
• The formula for the y-intercept is b_{0} = \bar{y} - b_{1}\bar{x}</latex>.


  True
• The y-intercept formula ensures that the regression line passes through the point $(\bar{x}, \bar{y})$.


  True
• Match the key concepts with their explanations:

  Least Squares Regression Line ↔️ Line that best fits two-variable data

  Minimizing Sum of Squares ↔️ Key principle to find the best-fitting line

  y-intercept ($b_{0}$) ↔️ Value of $y$ when $x =$$0$

  Slope ($b_{1}$) ↔️ Rate of change of $y$ with respect to $x$
• The sum of squares refers to the total of the squared vertical distances between observed data points and the regression line.


  True
• The slope formula calculates the ratio of the covariance between $x$ and $y$ to the variance of x
• What is the formula for calculating the y-intercept ($b_{0}$) of the least squares regression line?


  $b_{0} =$$\bar{y} - b_{1}\bar{x}$
• The y-intercept of the least squares regression line, denoted as b_{0}, represents the value of $y$ when $x =$$0$.
• The formula for the slope of the least squares regression line is $b_{1}$.
• The formula for the least squares regression line is $y =$$b_{0} +$$b_{1}x$.
• The slope of the least squares regression line, denoted as $b_{1}$.
• The formula for the y-intercept is $b_{0} =$$\bar{y} - b_{1}\bar{x}$.
• What does the y-intercept of the least squares regression line represent?

  Value of $y$ when $x =$$0$
• Match the parameter with its formula:

  Slope ↔️ $b_{1} =$$\frac{\sum_{i = 1}^{n}(x_{i} - \bar{x})(y_{i} - \bar{y})}{\sum_{i = 1}^{n}(x_{i} - \bar{x})^{2}}$

  Y-intercept ↔️ $b_{0} =$$\bar{y} - b_{1}\bar{x}$
• What does the y-intercept of the regression line represent?

  Baseline or starting point of the line
• What is the goal of the least squares regression line?

  Predict $y$ based on $x$
• The goal of the least squares regression line is to minimize the sum of the squared vertical distances between observed data points and the line.


  True
• The slope formula calculates the ratio of the covariance between $x$ and $y$ to the variance of $x$.


  True
• The y-intercept formula ensures that the regression line passes through the point $(\bar{x}, \bar{y})$.


  True
• What is the general formula for the regression line?

  $y =$$b_{0} +$$b_{1}x$
• What does the slope of the regression line represent?

  Rate of change in $y$ for a unit increase in $x$
• Given the regression line $y =$$3 +$$2x$, what is the predicted value of $y$ when $x =$$5$?


  13
• What is the formula for the slope of the regression line?

  $b_{1} =$$\frac{\sum_{i = 1}^{n}(x_{i} - \bar{x})(y_{i} - \bar{y})}{\sum_{i = 1}^{n}(x_{i} - \bar{x})^{2}}$
• What does $\bar{y}$ represent in the y-intercept formula?


  Sample mean of $y$
• The regression line always passes through the point defined by the sample means $(\bar{x}, \bar{y})$


  True
• If the mean of $x$ is 2.5 and the mean of $y$ is 6, the regression line must pass through the point (2.5, 6)


  True
• When $x$ is zero, the predicted value of $y$ is equal to the y-intercept


  True