Doctorates   The following table shows the annual number of PhD graduates in a country in various fields.
Natural
Sciences	Engineering	Social
Sciences	Education
2010	5,000	7,600	7,800	5,400
2012	5,400	8,400	8,400	4,800
2014	5,900	9,600	8,700	4,800
2016	6,200	9,500	9,000	5,100
2018	6,300	10,200	8,900	4,800
2020	6,700	10,800	8,900	4,600
(a)
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.)
y(x) = 

We are tasked with finding a regression equation \( y(x) \), where \( x \) is the number of social science doctorates and \( y \) is the number of education doctorates.

The given data is:

| Year | Social Sciences (\(x\)) | Education (\(y\)) |
|------|-------------------------|-------------------|
| 2010 | 7,800                   | 5,400             |
| 2012 | 8,400                   | 4,800             |
| 2014 | 8,700                   | 4,800             |
| 2016 | 9,000                   | 5,100             |
| 2018 | 8,900                   | 4,800             |
| 2020 | 8,900                   | 4,600             |

### Steps for the linear regression:
1. Using the points \( (x, y) \), we can fit a linear regression model to the data.
2. A regression equation is of the form:
   \[
   y = mx + b
   \]
   where \( m \) is the slope and \( b \) is the intercept.
   
We can calculate the coefficients by applying least squares regression. Here is the result of the regression calculation:

### Regression equation:
\[
y(x) = -0.325x + 7,935
\]

### Explanation:
- The slope \( m = -0.325 \) indicates that for every additional social science doctorate, the number of education doctorates decreases by 0.325.
- The intercept \( b = 7,935 \) gives the estimated number of education doctorates when the number of social science doctorates is zero.

Would you like further details on the calculation or have any questions?

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Here are 5 related questions to expand on the topic:
1. How would the correlation coefficient help assess the strength of the linear relationship?
2. What might explain the negative slope in this regression?
3. Could a non-linear regression model fit the data better?
4. How can you predict future values using the regression equation?
5. What are potential limitations of using linear regression in this context?

**Tip:** Always check the residuals (differences between predicted and actual values) when evaluating the accuracy of a regression model.

Doctorates   The following table shows the annual number of PhD graduates in a country in various fields. Natural Sciences Engineering Social Sciences Education 2010 5,000 7,600 7,800 5,400 2012 5,400 8,400 8,400 4,800 2014 5,900 9,600 8,700 4,800 2016 6,200 9,500 9,000 5,100 2018 6,300 10,200 8,900 4,800 2020 6,700 10,800 8,900 4,600 (a) 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.) y(x) =

Explore how to calculate the regression equation for the relationship between social science and education PhD graduates. Learn about linear regression with real-world data, and understand key concepts such as slope, intercept, and the least squares method. Suitable for students in Grades 11-12.

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