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To compute the Sum of Squares Error (SSE) for the given data points and the line \( y = x + 1 \), we need to follow these steps:

1. Calculate the predicted \( y \) values using the line equation for each \( x \) value in the table.
2. Determine the difference between the observed \( y \) values and the predicted \( y \) values.
3. Square these differences.
4. Sum all the squared differences to get the SSE.

Let's go through these steps using the provided dataTo compute the Sum of Squares Error (SSE) for the given data points and the line \( y = x + 1 \), we need to follow these steps:

1. Calculate the predicted \( y \) values using the line equation for each \( x \) value in the table.
2. Determine the difference between the observed \( y \) values and the predicted \( y \) values.
3. Square these differences.
4. Sum all the squared differences to get the SSE.

Let's go through these steps using the provided 

### Step 1: Calculate predicted \( y \) values

Using the line equation \( y = x + 1 \):
- For \( x = 1 \): \( \hat{y} = 1 + 1 = 2 \)
- For \( x = 2 \): \( \hat{y} = 2 + 1 = 3 \)
- For \( x = 3 \): \( \hat{y} = 3 + 1 = 4 \)
- For \( x = 4 \): \( \hat{y} = 4 + 1 = 5 \)
- For \( x = 6 \): \( \hat{y} = 6 + 1 = 7 \)

### Step 2: Determine the differences between observed and predicted \( y \) values
- For \( x = 1 \): \( y = 1 \), \( \hat{y} = 2 \), difference = \( 1 - 2 = -1 \)
- For \( x = 2 \): \( y = 1 \), \( \hat{y} = 3 \), difference = \( 1 - 3 = -2 \)
- For \( x = 3 \): \( y = 2 \), \( \hat{y} = 4 \), difference = \( 2 - 4 = -2 \)
- For \( x = 4 \): \( y = 5 \), \( \hat{y} = 5 \), difference = \( 5 - 5 = 0 \)
- For \( x = 6 \): \( y = 7 \), \( \hat{y} = 7 \), difference = \( 7 - 7 = 0 \)

### Step 3: Square these differences
- For \( x = 1 \): \( (-1)^2 = 1 \)
- For \( x = 2 \): \( (-2)^2 = 4 \)
- For \( x = 3 \): \( (-2)^2 = 4 \)
- For \( x = 4 \): \( 0^2 = 0 \)
- For \( x = 6 \): \( 0^2 = 0 \)

### Step 4: Sum all the squared differences to get the SSE
\[ \text{SSE} = 1 + 4 + 4 + 0 + 0 = 9 \]

Therefore, the Sum of Squares Error (SSE) is 9.

Do you want details or have any questions?

Here are 5 numbered questions you might want to ask:
1. How do we interpret the Sum of Squares Error in the context of data fitting?
2. Can you explain how to find the best-fit line for a set of data points?
3. What are the differences between SSE, MSE, and RMSE?
4. How can we use SSE to improve our model's accuracy?
5. What are some common pitfalls in calculating and interpreting SSE?

Tip: Always double-check your calculations to avoid small errors that can significantly impact the result, especially in statistical analyses.

Learn how to compute the Sum of Squares Error (SSE) for given data points using the linear regression model y = x + 1. Detailed step-by-step solution provided. Suitable for advanced high school and college students studying regression analysis and prediction modeling.

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