https://file.aiquickdraw.com/mathbot/file/1729618091069is7o9aft.jpg

Find the predicted value and the residual for each set up data given.  Put your predicted value in the first box and the residual in the second box.  Round to 2 decimal places where necessary.

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.

Find the predicted value and the residual for each set of data given. The table provides x = 5 and y (observed) = 3. The linear regression equation is y = 0.5x. Round to 2 decimal places where necessary.

Learn how to calculate predicted and residual values using the linear regression equation y = 0.5x. For the given data with x = 5 and observed value y = 3, find the predicted value and residual step by step. Suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation