Chapter A, Problem 1RP

For a linear equation y = b₀ + b₁x₁ + b₂x₂ + b₃x₃, identify the

1. a. independent variables.
2. b. dependent variable.
3. c. y-intercept.
4. d. partial slopes.

To determine

Identify the independent variables in the linear equation $y=b_0+b_1{x}_1+b_2{x}_2+b_3{x}_3$.

Answer to Problem 1RP

The independent variables in the linear equation $y=b_0+b_1{x}_1+b_2{x}_2+b_3{x}_3$ are $\underset{\_}{x_1}$, $\underset{\_}{x_2}$ and $\underset{\_}{x_3}$.

Explanation of Solution

Justification:

Independent variable:

An independent variable in the multiple regression analysis is the variable which explains the variation in the response variable and which itself does not depend on any other variables in the regression. The predictor variables are the independent variables.

In the linear equation $y=b_0+b_1{x}_1+b_2{x}_2+b_3{x}_3$, the variables $x_1$, $x_2$ and $x_3$ explain the variation in y and these variables do not depend on each other or on y.

Thus, the independent variables in the linear equation $y=b_0+b_1{x}_1+b_2{x}_2+b_3{x}_3$ are $\underset{\_}{x_1}$, $\underset{\_}{x_2}$ and $\underset{\_}{x_3}$.

To determine

Identify the dependent variable in the linear equation $y=b_0+b_1{x}_1+b_2{x}_2+b_3{x}_3$.

Answer to Problem 1RP

The dependent variable in the linear equation $y=b_0+b_1{x}_1+b_2{x}_2+b_3{x}_3$ is y.

Explanation of Solution

Justification:

Dependent variable:

A dependent variable in the multiple regression analysis is the variable the variation in which is explained by the predictor variables. The response variable is the dependent variable.

In the linear equation $y=b_0+b_1{x}_1+b_2{x}_2+b_3{x}_3$, the variability in the variable y is explained by the predictors $x_1$, $x_2$ and $x_3$.

Thus, the independent variables in the linear equation $y=b_0+b_1{x}_1+b_2{x}_2+b_3{x}_3$ is y.

To determine

Identify the y-intercept in the linear equation $y=b_0+b_1{x}_1+b_2{x}_2+b_3{x}_3$.

Answer to Problem 1RP

The y-intercept in the linear equation $y=b_0+b_1{x}_1+b_2{x}_2+b_3{x}_3$ is $\underset{\_}{b_0}$.

Explanation of Solution

Justification:

The y-intercept:

The y-intercept in a multiple linear regression equation is the value taken by the response variable y, when all the predictor variables take value 0. The y-intercept gives the value of y at the point where the plane intersects the y-axis.

In a linear equation of the form, $y=b_0+b_1{x}_1+b_2{x}_2=b_3{x}_3$, the y-intercept is $b_0$.

Thus, the y-intercept in the linear equation $y=b_0+b_1{x}_1+b_2{x}_2+b_3{x}_3$ is $\underset{\_}{b_0}$.

To determine

Identify the partial slopes in the linear equation $y=b_0+b_1{x}_1+b_2{x}_2+b_3{x}_3$.

Answer to Problem 1RP

The partial slopes in the linear equation $y=b_0+b_1{x}_1+b_2{x}_2+b_3{x}_3$ are $\underset{\_}{b_1}$, $\underset{\_}{b_2}$ and $\underset{\_}{b_3}$.

Explanation of Solution

Justification:

Partial slope:

The partial slope corresponding to a predictor variable in a multiple linear regression equation is a measure of the steepness of the plane along the direction of that predictor variable. The partial slope is the change in the response variable y, when the particular predictor variable increases by one unit, keeping all other predictor variables fixed.

In a linear equation of the form, $y=b_0+b_1{x}_1+b_2{x}_2+b_3{x}_3$, the partial slope corresponding to $x_1$ is $b_1$, the partial slope corresponding to $x_2$ is $b_2$ and the partial slope corresponding to $x_3$ is $b_3$.

Thus, the partial slopes in the linear equation $y=b_0+b_1{x}_1+b_2{x}_2+b_3{x}_3$ are $\underset{\_}{b_1}$, $\underset{\_}{b_2}$ and $\underset{\_}{b_3}$.