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Based on the image you uploaded, the table provides the number of PhD graduates in different fields (Natural Sciences, Engineering, Social Sciences, and Education) for various years. The problem focuses on creating a regression equation using the number of Social Science doctorates as \(x\) and the number of Education doctorates as \(y\).

Here is the data extracted from the table:

| Year | Social Sciences (x) | Education (y) |
|------|---------------------|---------------|
| 2010 | 7,800               | 5,200         |
| 2012 | 8,400               | 4,800         |
| 2014 | 8,700               | 4,800         |
| 2016 | 9,000               | 5,100         |
| 2018 | 8,900               | 4,800         |
| 2020 | 8,900               | 4,700         |

We will use these points to create a linear regression equation of the form:
\[
y(x) = mx + b
\]

I will calculate the values for the slope (\(m\)) and intercept (\(b\)) using the least squares method. Let me compute that now.The regression equation based on the given data is:
\[
y(x) = -0.233x + 6911.021
\]

This means that for each increase of 1 doctorate in Social Sciences, the number of doctorates in Education decreases by about 0.233, according to this model.

Would you like more details or have any questions about the calculation?

Here are some related questions that could expand on this topic:
1. How can we interpret the negative slope in the context of this data?
2. What is the correlation coefficient (r-value) and what does it indicate about the relationship between the variables?
3. How can we use this regression equation to predict the number of Education doctorates for a given number of Social Science doctorates?
4. What would the impact be if we used a non-linear regression model?
5. What are some limitations of using this regression model for predicting future trends?

**Tip:** Linear regression assumes a straight-line relationship between the variables. Always check if the trend in your data actually supports this assumption before relying on the regression results.

With x = the number of social science doctorates and y = the number of education doctorates, use technology to obtain the regression equation. (Round coefficients to three significant digits.)

This problem focuses on using linear regression to determine the relationship between the number of Social Science doctorates (x) and Education doctorates (y) from 2010 to 2020. Learn how to calculate the regression equation using the least squares method and analyze the data for accurate predictions.

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