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Let's break down the steps to solve the problems in the image you uploaded.

### Part (a): SSE (Sum of Squared Errors) for Models A and B

Given dataLet's break down the steps to solve the problems in the image you uploaded.

### Part (a): SSE (Sum of Squared Errors) for Models A and B

Given 
- \( y_A = 0.015x \) for Model A
- \( y_B = 0.25 + 0.01x \) for Model B
- \( x \) is the number of years since 1700
- The world population in billions for the respective years is provided.

We will:
1. Compute the predicted values of \( y_A \) and \( y_B \) for the given years using \( x = \text{year} - 1700 \).
2. Find the actual population values.
3. Calculate the squared errors for each model (the difference between actual and predicted values squared).
4. Sum these squared errors for each model to obtain SSE for both models.

#### Data

| Year  | Population (in billions) | \( x = \text{year} - 1700 \) |
|-------|--------------------------|------------------------------|
| 1700  | 0.6                      | 0                            |
| 1760  | 0.8                      | 60                           |
| 1820  | 1.0                      | 120                          |
| 1870  | 1.3                      | 170                          |
| 1910  | 1.8                      | 210                          |
| 1950  | 2.5                      | 250                          |
| 1990  | 5.3                      | 290                          |
| 2010  | 6.9                      | 310                          |

#### Model A: \( y_A = 0.015x \)
We will calculate \( y_A \) for each \( x \) and find the squared errors.

#### Model B: \( y_B = 0.25 + 0.01x \)
We will calculate \( y_B \) similarly for each \( x \) and find the squared errors.

Let's now compute this systematically.### Results for Part (a):

- **SSE for Model A**: 11.923 (rounded to 3 decimal places)
- **SSE for Model B**: 18.340 (rounded to 3 decimal places)

Since **Model A** has a smaller SSE than Model B, it is the better model because it provides a closer fit to the actual data.

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### Part (b): Linear Regression

To calculate the best-fit line using linear regression, we need to compute the slope \( m \) and the intercept \( b \), along with the correlation coefficient \( r \). The formula for the linear regression line is:

\[
y = mx + b
\]

We'll use the least squares method to determine the values of \( m \) and \( b \), and then compute the correlation coefficient \( r \).

Let's perform the calculations.It seems like I can’t do more advanced data analysis right now. Please try again later. However, you can use tools like Excel or a graphing calculator to compute the linear regression line and the correlation coefficient.

Here’s a tip for doing this:
- In Excel, you can use the `=LINEST` function to get the slope and intercept and `=CORREL` to find the correlation coefficient.

Would you like any further clarification on how to proceed manually or with another tool?

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Here are 5 related questions:
1. How is SSE (Sum of Squared Errors) calculated and interpreted in regression analysis?
2. Why is it important to minimize SSE in selecting a model?
3. What is the significance of the slope and intercept in linear regression?
4. How does the correlation coefficient \( r \) reflect the strength of a linear relationship?
5. What is the difference between linear and non-linear models in data fitting?

**Tip**: Always plot your data points to visually assess the fit of different models before calculating SSE or other metrics!

Using the provided world population data (from 1700 to 2010), calculate the SSE for models A and B. Use linear regression to find the best-fit equation and predict the world population in 2060.

Calculate SSE for two linear models predicting world population from 1700 to 2010 and determine the best model. Learn linear regression techniques to predict future population values using a step-by-step method.

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