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.