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!