Study Notes on the Least-Squares Regression Line

Objective 1: Compute the least-squares regression line.
Objective 2: Compute the correlation coefficient.
Objective 3: Use the least-squares regression line to make predictions.
Objective 4: Interpret predicted values, the slope, and the y-intercept of the least-squares regression line.

A sample of house data is provided: size in square feet and selling price in thousands of dollars.
Previous analysis revealed a strong positive linear association between the size and sales price of houses.
Data Table:
- Size (Square Feet) and Selling Price ($1000s):
- 2521 sq ft: 400
- 2555 sq ft: 426
- 2735 sq ft: 428
- 2846 sq ft: 435
- 3028 sq ft: 469
- 3049 sq ft: 475
- 3198 sq ft: 488
- 3198 sq ft: 455

The least-squares regression line minimizes the sum of the squared vertical distances between the data points and the line itself.
Best-Fitting Line:
- The optimal fitting line is the one for which these squared vertical distances are smallest.
Equation of the Least-Squares Regression Line:
- The general form for predicting y from x is given as:
- $y = a + bx$
  - where:
  - $a$ = y-intercept
  - $b$ = slope.
Variable Definitions:
- The variable we want to predict (selling price) is known as the outcome variable or dependent variable.
- The variable we are given (size) is known as the explanatory variable or independent variable.

A one-time setting on the calculator is required to display the correlation coefficient.
TI-84 Plus Calculator Steps:
- To configure: Press 2nd, 0, and select DiagnosticOn.
- After setting, compute the least-squares regression line and the correlation coefficient for the house data.

House Size:
- 2521 sq ft: 400
- 2555 sq ft: 426
- 2735 sq ft: 428
- 2846 sq ft: 435
- 3028 sq ft: 469
- 3049 sq ft: 475
- 3198 sq ft: 488
- 3198 sq ft: 455

Predicted Value: Predictions can be made by substituting the explanatory variable's value into the regression equation.
Example Calculation:
- Given the equation $y = 160.1939 + 0.0992x$ for selling price based on size:
- Predicting selling price for house size of 2800 sq ft:
- Calculation: $y = 160.1939 + 0.0992 imes 2800$ .
Point of Averages:
- Average size of houses: $x = 2891.25$ sq ft.
- Average selling price: $y = 447.0$ thousand dollars.
- Substituting average size back into the regression gives expected average selling price, confirming the linear relationship.

Predicted Values: Estimates of the average outcome for given values of the explanatory variable.
- Example: With the equation $y = 160.1939 + 0.0992x$ , to estimate average price for 3000 sq ft:
- Substitute: $y = 160.1939 + 0.0992 imes 3000$ .
Interpreting the y-Intercept (a):
- The y-intercept is where the line crosses the y-axis:
- Interpreted only if data includes both positive and negative x-values.
  - If the range of x-values contains only positive or negative numbers, the intercept value does not hold significant practical interpretation.

The slope represents how much the predicted value (outcome variable) changes with a unit change in x (explanatory variable):
- If the values of the explanatory variable differ by 1, the predicted values change by the amount of the slope, $b$ .
- If the change in the explanatory variable is by a factor $d$ , the predicted values change by the amount $b imes d$ .
Example Comparison:
- Considering two houses: One at 1900 sq ft and another at 1750 sq ft, predict their price difference.

At final exams, students reported hours studied, leading to the regression line for predicting scores:
- $y = 50 + 5x$ .
- Predict Antoine's score studying for 6 hours:
- Substitute: $y = 50 + 5 imes 6$ .
- Effect of studying more hours: If Emma studied 3 hours longer than Jeremy, predict the score difference based on the slope of 5.

Definitions of outcome (response) and explanatory (predictor) variables.
Calculation of least-squares regression line.
Application of regression line for predictions.
Interpretation of predicted values, y-intercepts, and slopes in regression analysis.