Chapter 7 Notes

Linear Regression Equation Sheet

Key Equations

1. Least Squares Line:[ \hat{y} = b_0 + b_1 x ]

   • ( b_1 ) (Slope): Rate of change in y with respect to x.

   • ( b_0 ) (Y-Intercept): Value of y when x = 0.

2. Residual Calculation:[ \text{Residual} = \text{Observed} - \text{Predicted} ]

   • Positive residuals indicate points above the line; negative residuals indicate points below the line.

3. Interpretation of Slope:

   • Example: For every 1g increase in protein, fat changes by ( b_1 )g.

4. Model Interpretation:

   • For Hurricane Data: For every 1 mb increase in central pressure, wind speed decreases by 0.9748 knots.

R² (Coefficient of Determination)

• Represents the fraction of the data's variation explained by the model:[ R^2 = \text{Variation explained by the model} ]

• Example for Burger King: If ( R^2 = 0.58 ), then 58% of fat variability is explained by protein variability.

Conditions for a Good Regression Model

1. Use only with quantitative variables.

2. Ensure a linear relationship exists.

3. Avoid outliers in the data set.