1. Interpret the meaning of the coefficient -8.2 in the multiple regression equation y=112.1+0.43x_1-8.2x_2+29.5x_3.

Verified step by step guidance

1. Identify the multiple regression equation given: $y=112.1+0.43x{1}_{}-8.2x{2}_{}+29.5x{3}_{}$.
2. Recall that in a multiple regression equation, each coefficient represents the expected change in the dependent variable $y$ for a one-unit increase in the corresponding independent variable, holding all other variables constant.
3. Focus on the coefficient of $x\_2$, which is $-8.2$. This means that for every one-unit increase in $x\_2$, the predicted value of $y$ decreases by 8.2 units, assuming $x\_1$ and $x\_3$ remain unchanged.
4. Interpret the negative sign of the coefficient as indicating an inverse relationship between $x\_2$ and $y$: as $x\_2$ increases, $y$ tends to decrease.
5. Summarize the interpretation: the coefficient $-8.2$ quantifies the strength and direction of the effect of $x\_2$ on $y$ within the context of the other variables in the model.

Key Concepts

Multiple Regression Equation

A multiple regression equation models the relationship between one dependent variable and two or more independent variables. Each coefficient represents the expected change in the dependent variable for a one-unit change in the corresponding independent variable, holding other variables constant.

Interpretation of Regression Coefficients

A regression coefficient quantifies the effect of an independent variable on the dependent variable. A negative coefficient, like -8.2, indicates that as the independent variable increases by one unit, the dependent variable decreases by 8.2 units, assuming other variables remain fixed.

Holding Other Variables Constant

In multiple regression, the effect of each independent variable is interpreted while keeping other variables constant. This means the coefficient isolates the unique contribution of that variable to changes in the dependent variable, controlling for the influence of other predictors.