"Predicting y-Values In Exercises 3-6, use the multiple regression equation to predict the y-values for the values of the independent variables.
3. Cauliflower Yield The equation used to predict the annual cauliflower yield (in pounds
per acre) is y=24,791+4.508x_1-4.723x_2
where x_1 is the number of acres planted and x_2 is the number of acres harvested.(Adapted from United States Department of Agriculture)
a. x_1 = 36,500, x_2 = 36,100
b. x_1 = 38,100, x_2 = 37,800
c. x_1 = 39,000, x_2 = 38,800
d. x_1 = 42,200, x_2 = 42,100"

Verified step by step guidance

1. Identify the multiple regression equation given: $y=\; 24791\; +\; 4.508x\_1\; -\; 4.723x\_2$, where $x\_1$ is the number of acres planted and $x\_2$ is the number of acres harvested.
2. For each set of values of $x\_1$ and $x\_2$, substitute these values into the regression equation. For example, for part (a), substitute $x\_1\; =\; 36500$ and $x\_2\; =\; 36100$.
3. Perform the multiplication for each term involving the independent variables: multiply 4.508 by $x\_1$ and multiply -4.723 by $x\_2$.
4. Add the constant term 24791 to the results of the multiplications to calculate the predicted value of $y$ (the cauliflower yield) for each case.
5. Repeat steps 2 to 4 for each set of values given in parts (b), (c), and (d) to find the predicted yields for all scenarios.

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 changes in each independent variable affect the outcome.

Interpreting Coefficients in Regression

Each coefficient in a regression equation represents the expected change in the dependent variable for a one-unit increase in the corresponding independent variable, holding other variables constant. Positive coefficients indicate a direct relationship, while negative coefficients indicate an inverse relationship.

Predicting Values Using Regression

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