"Predicting y-Values In Exercises 3-6, use the multiple regression equation to predict the y-values for the values of the independent variables.
4. Sorghum Yield The equation used to predict the annual sorghum yield (in bushels per
acre) is y = 80.1-20.2x_1 +21.2x_2
where x_1 is the number of acres planted (in millions) and x_2 is the number of acres harvested (in millions). (Adapted from United States Department of Agriculture)
a. x_1 = 5.5, x_2 = 3.9
b. x_1 = 8.3, x_2 = 7.3
c. x_1 = 6.5, x_2 = 5.7
d. x_1 = 9.4, x_2= 7.8"

Verified step by step guidance

1. Identify the multiple regression equation given: $y=80.1-20.2x{1}_{}+21.2x{2}_{}$, where $x{1}_{}$ is the number of acres planted (in millions) and $x{2}_{}$ is the number of acres harvested (in millions).
2. For each pair of values of $x{1}_{}$ and $x{2}_{}$, substitute these values into the regression equation. For example, for part (a), substitute $x{1}_{}=\; 5.5$ and $x{2}_{}=\; 3.9$.
3. Perform the multiplication for each term involving the independent variables: multiply $-20.2$ by $x\_1$ and $21.2$ by $x\_2$.
4. Add the results of these multiplications to the constant term 80.1, following the signs in the equation (subtract the first product and add the second product). This will give the predicted value of $y$ for the given $x\_1$ and $x\_2$.
5. Repeat steps 2 to 4 for each set of values given in parts (b), (c), and (d) to find the predicted sorghum yield for each scenario.

Key Concepts

Multiple Regression Equation

A multiple regression equation models the relationship between one dependent variable and two or more independent variables. It predicts the dependent variable (y) by combining the independent variables (x₁, x₂, etc.) multiplied by their coefficients, plus a constant term. This allows for understanding how each independent variable influences the outcome while holding others constant.

Interpretation of Coefficients

In a multiple regression equation, each coefficient represents the expected change in the dependent variable for a one-unit increase in the corresponding independent variable, assuming other variables remain constant. For example, a coefficient of -20.2 for x₁ means that increasing x₁ by one unit decreases y by 20.2 units, holding x₂ fixed.

Prediction Using Regression Equation

To predict y-values, substitute the given values of independent variables into the regression equation and perform the arithmetic operations. This process yields estimated values of the dependent variable based on the model, enabling practical forecasting or decision-making using the provided data.