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	5,200
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.)

To find the regression equation for both the **number of social science doctorates** (\( x \)) and the **number of education doctorates** (\( y \)) over time, we need to set up a regression analysis for each data set based on the year.

Given the dataTo find the regression equation for both the **number of social science doctorates** (\( x \)) and the **number of education doctorates** (\( y \)) over time, we need to set up a regression analysis for each data set based on the year.

Given the 

### Social Sciences (\( x \)):
- 2010: 7,800
- 2012: 8,400
- 2014: 8,700
- 2016: 9,000
- 2018: 8,900
- 2020: 8,900

### Education (\( y \)):
- 2010: 5,200
- 2012: 4,800
- 2014: 4,800
- 2016: 5,100
- 2018: 4,800
- 2020: 4,600

For the regression analysis, we'll assign numerical values to the years where:

- \( t = 0 \) for 2010
- \( t = 2 \) for 2012
- \( t = 4 \) for 2014
- \( t = 6 \) for 2016
- \( t = 8 \) for 2018
- \( t = 10 \) for 2020

This allows us to perform a linear regression for both \( x \) (social sciences) and \( y \) (education) doctorates.

Let me calculate the regression equations.The regression equations for the number of doctorates are as follows:

### For **Social Sciences** (\( x \)):
\[
x(t) = 8095.24 + 104.29t
\]
This equation suggests that the number of social science doctorates increases by approximately 104.29 per 2-year period.

### For **Education** (\( y \)):
\[
y(t) = 5076.19 - 38.57t
\]
This equation indicates that the number of education doctorates decreases by approximately 38.57 per 2-year period.

Where \( t \) is the number of years since 2010 (i.e., \( t = 0 \) for 2010, \( t = 2 \) for 2012, etc.).

Let me know if you'd like further details or if you have any questions! 

Here are some related questions:
1. How accurate are these models in predicting future doctorate numbers?
2. What might explain the decreasing trend in education doctorates?
3. How would a quadratic model compare to the linear one?
4. Could external factors be influencing these trends?
5. How does the trend in social sciences compare to engineering?

**Tip:** When interpreting regression equations, remember that the slope indicates the rate of change over time!

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 5,200
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.)

Explore the regression analysis for PhD graduates in Social and Education Sciences from 2010 to 2020. Learn how to obtain regression equations using linear data analysis and algebraic concepts. This problem provides step-by-step solutions with detailed calculations.

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