Chapter 12, Problem 12.134LM

Suppose you have developed a regression model to explain the relationship between y and x₁, x₂, and x₃. The ranges of the variables you observed were as follows: 10 ≤ y ≤ 100, 5 ≤ x₁ ≤ 55, .5 ≤ x₂ ≤ 1, and 1,000 ≤ x₃ ≤ 2,000. Will the error of prediction be smaller when you use the least squares equation to predict y when x₁ = 30, x₂ = .6, and x₃ = 1,300, or when x₁ = 60, x₂ = .4, and x₃ = 900? Why?

To determine

For which set of values the error of prediction would be small for predicting y using the least squares regression equation.

Answer to Problem 12.134LM

The error of prediction would be small for predicting y using the least squares regression equation for the values $x_1=30,x_2=0.6,x_3=1,300$.

Explanation of Solution

Given info: The independent variables in the study are $x_1,x_2\text{and}{x}_3$. The ranges for each of the independent variables are $5\le x_1\le 55$, $0.5\le x_2\le 1$ and $1,000\le x_3\le 2,000$. The response variable in the regression model is y, the range of y is $10\le y\le 100$. The two set of values mentioned are $x_1=30,x_2=0.6\text{and}{x}_3=1,300$, $x_1=60,x_2=0.4\text{and}{x}_3=900$

Justification:

When the values of the independent variables in the study are closer or equal to the values of sample means then the error of prediction would be small and if the values of the independent variables in the study are further to the values of sample means then the error of prediction would be large.

The independent variables are $x_1,x_2\text{and}{x}_3$. Consider the values $x_1=60,x_2=0.4\text{and}{x}_3=900$, it can be observed that all the values of the independent variables $x_1,x_2\text{and}{x}_3$ are not within the range of the variables; this implies that the values $x_1=60,x_2=0.4\text{and}{x}_3=900$ would be far away from the value of sample means. Hence, the error of prediction would be large.

Consider the values $x_1=30,x_2=0.6\text{and}{x}_3=1,300$, it can be observed that all the values of the independent variables $x_1,x_2\text{and}{x}_3$ are within the range of the variables this implies that the values $x_1=30,x_2=0.6\text{and}{x}_3=1,300$ would be closer from the value of sample means. Hence, the error of prediction would be less.

Thus, The error of prediction would be small for predicting y using the least squares regression equation for the values $x_1=30,x_2=0.6,x_3=1,300$.