The output shown was obtained from Minitab.

a. The least-squares regression equation is y^ = 1.3962x + 12.396. What is the predicted value of y at x = 10?

Verified step by step guidance

1. Identify the regression equation given: \(\hat{y} = 12.396 + 1.3962x\).
2. Understand that to find the predicted value of \(y\) at a specific \(x\) value, you substitute that \(x\) value into the regression equation.
3. Substitute \(x = 10\) into the equation: \(\hat{y} = 12.396 + 1.3962 \times 10\).
4. Perform the multiplication first: calculate \(1.3962 \times 10\).
5. Add the result to the constant term 12.396 to get the predicted value \(\hat{y}\) at \(x=10\).

Key Concepts

Least-Squares Regression Equation

The least-squares regression equation models the relationship between a dependent variable (y) and an independent variable (x) 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 coefficient.

Prediction Using the Regression Equation

To predict the value of y for a given x, substitute the x value into the regression equation ŷ = b0 + b1x. This yields the estimated or predicted y value based on the linear relationship established by the regression model.

Interpretation of Regression Output Statistics

Regression output includes coefficients, standard errors, t-values, p-values, and R-squared. Coefficients indicate the effect size, standard errors measure estimate precision, t and p-values test significance, and R-squared shows the proportion of variance in y explained by x.