Notation Using the weights (lb) and highway fuel consumption amounts (mi/gal) of the 48 cars listed in Data Set 35 “Car Data” of Appendix B, we get this regression equation:
y^ = 58.9 - 0.00749x, where x represents weight.
a. What does the symbol y^ represent?

Verified step by step guidance

1. The symbol y^ (read as 'y-hat') represents the predicted value of the dependent variable in a regression equation. In this case, it represents the predicted highway fuel consumption (in miles per gallon) for a car based on its weight.
2. In a regression equation, y^ is calculated using the formula y^ = b0 + b1x, where b0 is the y-intercept, b1 is the slope of the regression line, and x is the independent variable. Here, b0 = 58.9 and b1 = -0.00749.
3. The negative slope (-0.00749) indicates that as the weight of the car (x) increases, the predicted highway fuel consumption (y^) decreases. This is consistent with the idea that heavier cars tend to have lower fuel efficiency.
4. The regression equation is a model that provides an estimate of the relationship between the independent variable (weight) and the dependent variable (fuel consumption). It is not exact but gives a predicted value based on the data used to create the model.
5. To summarize, y^ is the predicted highway fuel consumption (mi/gal) for a car of a given weight (x), calculated using the regression equation provided.

Key Concepts

Regression Equation

A regression equation models the relationship between a dependent variable and one or more independent variables. In this case, the equation y^ = 58.9 - 0.00749x predicts the fuel consumption (y^) based on the weight of the cars (x). The coefficients indicate how much y is expected to change with a one-unit change in x.

Dependent and Independent Variables

In a regression analysis, the dependent variable is the outcome we are trying to predict or explain, while the independent variable is the predictor. Here, fuel consumption (y^) is the dependent variable, and the weight of the cars (x) is the independent variable, indicating that fuel consumption is influenced by the weight.

Predicted Value (y^)

The symbol y^ (y-hat) represents the predicted value of the dependent variable based on the regression model. It is an estimate of the fuel consumption for a given weight of a car, calculated using the regression equation. This predicted value helps in understanding how changes in weight affect fuel efficiency.