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	8,600	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	4,700
(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) = 
 

To find the regression equation \( y(x) \) where \( x \) is the number of social science doctorates and \( y \) is the number of education doctorates, we can use the pairs of data for Social Sciences and Education doctorates from 2010 to 2020:

\[
\text{Social Sciences } (x): 7800, 8600, 8700, 9000, 8900, 8900
\]
\[
\text{Education } (y): 5200, 4800, 4800, 5100, 4800, 4700
\]

Using these data points, I will compute the linear regression equation \( y(x) \) using least squares regression. Let me calculate that now.

The regression equation relating the number of education doctorates \( y \) to the number of social science doctorates \( x \) is:

\[
y(x) = -0.267x + 7207
\]

This equation shows that for every additional social science doctorate, the number of education doctorates decreases slightly by approximately 0.267.

Would you like more details on this or have any other questions?

Here are 5 related questions to consider:
1. How can we interpret the negative slope in the regression equation?
2. Can we predict the number of education doctorates if there are 9,500 social science doctorates?
3. What does the y-intercept (7207) represent in this context?
4. How would the regression equation change if we include more years of data?
5. What are other types of regression models we can use for this data?

**Tip:** When performing linear regression, always check the correlation to ensure that the linear model is a good fit for the data.

With x = the number of social science doctorates and y = the number of education doctorates, use technology to obtain the regression equation.

Find the linear regression equation between the number of social science doctorates and education doctorates from 2010 to 2020 using least squares regression. The problem involves key mathematical concepts like linear regression and statistical analysis. Suitable for high school and college students.

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