"Predicting y-Values In Exercises 3-6, use the multiple regression equation to predict the y-values for the values of the independent variables.
5. Black Cherry Tree Volume The volume (in cubic feet) of a black cherry tree can be modeled by the equation
y =- 52.2+0.3x_1 +4.5x_2
where x_1 is the tree's height (in feet) and x_2 is the tree's diameter (in inches). (Source: Journal of the Royal Statistical Society)
a. x_1 = 70, x_2 = 8.6
b. x_1 = 65, x_2 = 11.0
c. x_1 = 83, x_2 = 17.6
d. x_1 = 87, x_2 = 19.6"

Verified step by step guidance

1. Identify the multiple regression equation given: $y=-52.2+0.3x{1}_{}+4.5x{2}_{}$, where $x{1}_{}$ is the tree's height and $x{2}_{}$ is the tree's diameter.
2. For each set of values for $x{1}_{}$ and $x{2}_{}$, substitute the values into the regression equation. For example, for part (a), substitute $x{1}_{}=\; 70$ and $x{2}_{}=\; 8.6$.
3. Calculate the product of each coefficient and its corresponding independent variable: multiply 0.3 by $x{1}_{}$ and 4.5 by $x{2}_{}$.
4. Add the results from the multiplications to the intercept (-52.2) to find the predicted volume $y$ for each case.
5. Repeat steps 2 to 4 for parts (b), (c), and (d) using their respective values of $x{1}_{}$ and $x{2}_{}$ to find all predicted volumes.

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 predictor influences the outcome.

Interpreting Coefficients

Coefficients in a regression equation represent the expected change in the dependent variable for a one-unit increase in the corresponding independent variable, holding others constant. For example, a coefficient of 0.3 for height means each additional foot increases volume by 0.3 cubic feet, assuming diameter stays the same.

Prediction Using Regression

To predict y-values, substitute given values of independent variables into the regression equation and calculate the result. This process estimates the dependent variable based on the model, enabling practical predictions like estimating tree volume from height and diameter measurements.