Suppose a least-squares regression line is given by ŷ = 4.302x – 3.293. What is the mean value of the response variable if x = 20?

Verified step by step guidance

1. Identify the regression equation given: \(\hat{y} = 4.302x - 3.293\).
2. Understand that \(\hat{y}\) represents the predicted (mean) value of the response variable for a given value of \(x\).
3. Substitute the given value of \(x = 20\) into the regression equation to find the predicted mean response: \(\hat{y} = 4.302 \times 20 - 3.293\).
4. Perform the multiplication first: calculate \$4.302 \times 20$.
5. Then subtract \$3.293\( from the result to get the mean value of the response variable when \)x = 20$.

Key Concepts

Least-Squares Regression Line

A least-squares regression line models the relationship between an independent variable (x) and a dependent variable (y) by minimizing the sum of squared differences between observed and predicted values. It is expressed as ŷ = b₀ + b₁x, where b₀ is the intercept and b₁ is the slope.

Interpretation of the Regression Equation

The regression equation ŷ = 4.302x – 3.293 predicts the mean value of the response variable y for any given x. Here, 4.302 is the slope indicating the average change in y for a one-unit increase in x, and –3.293 is the intercept, the predicted value of y when x equals zero.

Predicting the Mean Response

To find the mean response for a specific x value, substitute x into the regression equation. This yields the predicted mean of y, which represents the expected average outcome of the response variable at that x.