To find the Predicted Value and Residual Value for the given data, we use the following steps:

1. Linear Regression Equation: $y = 0.5x$
2. Observed value of $y$: 3 (provided in the table for $x = 5$)

Step 1: Find the Predicted Value

The Predicted Value is calculated by plugging the value of $x$ into the given linear regression equation.

$y_{\text{predicted}} = 0.5 \times x = 0.5 \times 5 = 2.5$

Thus, the Predicted Value is $2.5$.

Step 2: Calculate the Residual Value

The Residual Value is the difference between the Observed Value and the Predicted Value. The formula is:

$\text{Residual} = y_{\text{observed}} - y_{\text{predicted}}$

Substitute the observed value ($y_{\text{observed}} = 3$) and predicted value ($y_{\text{predicted}} = 2.5$):

$\text{Residual} = 3 - 2.5 = 0.5$

Thus, the Residual Value is $0.5$.

Final Answers:

• Predicted Value: $2.5$
• Residual Value: $0.5$

Would you like further details or have any questions?

Expanding Questions:

1. What is the meaning of the residual value in regression analysis?
2. How can you interpret a positive residual versus a negative residual?
3. What role does the slope ($0.5$ here) play in the linear regression equation?
4. How does increasing the sample size affect the residuals in linear regression?
5. How would you adjust the regression model if the residuals consistently show a pattern?

Tip:

Residuals help identify how well your regression model fits the data, and patterns in residuals might suggest that a non-linear model would fit better.