*For the same cases, calculate the residuals (three decimal places)*

First Exam

Final

Residuals

62

74

77

72

98

95

To calculate the residuals, we need the predicted values and the actual values of the final exam scores. The residual for each data point is calculated as:

\[ \text{Residual} = \text{Actual Final} - \text{Predicted Final} \]

Using the previously assumed regression equation:
\[ \text{Predicted Final} = 10 + 0.8 \times \text{First} \]

We have already calculated the predicted final exam scores for each case. Now, we'll use these predicted values to find the residuals.

1. **First Exam: 62, Actual Final: 74**
   - Predicted Final: 59.600
   - Residual: \( 74 - 59.600 = 14.400 \)

2. **First Exam: 77, Actual Final: 72**
   - Predicted Final: 71.600
   - Residual: \( 72 - 71.600 = 0.400 \)

3. **First Exam: 98, Actual Final: 95**
   - Predicted Final: 88.400
   - Residual: \( 95 - 88.400 = 6.600 \)

Here are the residuals for each case, rounded to three decimal places:

1. **First Exam: 62**
   - Residual: \( 14.400 \)

2. **First Exam: 77**
   - Residual: \( 0.400 \)

3. **First Exam: 98**
   - Residual: \( 6.600 \)

Do you have any questions or need further details on how these calculations were made? Here are five more questions you might find helpful:

1. How do residuals help in evaluating the accuracy of a regression model?
2. What is the difference between a positive residual and a negative residual?
3. Can residuals be used to identify outliers in the data?
4. How can I visualize residuals to check the assumptions of my regression model?
5. What are some common patterns in residual plots and what do they indicate?

**Tip:** Residual analysis is crucial for validating your regression model. Look for randomness in residual plots to ensure that your model fits the data well.

Learn how to calculate residuals in regression analysis using a step-by-step approach. Understand the concept of residuals and their significance in evaluating the accuracy of regression models. Includes detailed calculations for specific data points.

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