Questions: Suppose a sample regression equation is given by Yhat = 3 + 0.40° X. Suppose when X is 10, Y is observed to be 8. What is the residual of the model prediction and does the model under or overpredict the value of Y ? = -1, the model overpredicts y = -1, the model underpredicts y = 1, the model overpredicts y = 1, the model underpredicts y

Suppose a sample regression equation is given by Yhat = 3 + 0.40° X.
Suppose when X is 10, Y is observed to be 8.
What is the residual of the model prediction and does the model under or overpredict the value of Y ?
= -1, the model overpredicts y
= -1, the model underpredicts y
= 1, the model overpredicts y
= 1, the model underpredicts y

Transcript text: Suppose a sample regression equation is given by Yhat $=3+0.40^{\circ} \mathrm{X}$. Suppose when X is $10, \mathrm{Y}$ is observed to be 8. What is the residual of the model prediction and does the model under or overpredict the value of $Y$ ? $=-1$, the model overpredicts $y$ $=-1$, the model underpredicts $y$ $=1$, the model overpredicts $y$ $=1$, the model underpredicts $y$

Solution

Solution Steps

To find the residual of the model prediction, we first need to calculate the predicted value of $ Y $ using the given regression equation when $ X = 10 $. The residual is then the difference between the observed value of $ Y $ and the predicted value. If the residual is negative, the model overpredicts; if positive, it underpredicts.

Step 1: Calculate the Predicted Value of $ Y $

Given the regression equation:

\[ \hat{Y} = 3 + 0.40 \times X \]

Substitute $ X = 10 $ into the equation:

\[ \hat{Y} = 3 + 0.40 \times 10 = 3 + 4 = 7 \]

Step 2: Calculate the Residual

The residual is the difference between the observed value of $ Y $ and the predicted value $ \hat{Y} $:

\[ \text{Residual} = Y_{\text{observed}} - \hat{Y} = 8 - 7 = 1 \]

Step 3: Determine if the Model Overpredicts or Underpredicts

Since the residual is positive ($ 1 $), the model underpredicts the value of $ Y $.