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The image shows a table with the annual number of PhD graduates in a country across various fields (Natural Sciences, Engineering, Social Sciences, and Education) for the years 2010, 2012, 2014, 2016, 2018, and 2020.

The task asks for a **regression equation** and the **coefficient of correlation (r)** for the number of Social Science doctorates as a function of time since 2010. We are to use technology (such as a calculator or software like Excel) to compute the linear regression for this data, and round all values to three significant digits.

### Data for Social Science Doctorates:
- 2010: 8,100
- 2012: 8,700
- 2014: 8,900
- 2016: 9,200
- 2018: 9,100
- 2020: 9,100

Time \( t \) in years is taken as time since 2010, so the corresponding values for \( t \) will be:
- 2010 → \( t = 0 \)
- 2012 → \( t = 2 \)
- 2014 → \( t = 4 \)
- 2016 → \( t = 6 \)
- 2018 → \( t = 8 \)
- 2020 → \( t = 10 \)

Let's proceed to calculate the regression equation of the form:

\[
y(t) = mt + b
\]

Where:
- \( y(t) \) is the number of Social Science doctorates in year \( t \)
- \( m \) is the slope (rate of change)
- \( b \) is the y-intercept (the number of doctorates in 2010)

And also the coefficient of correlation \( r \).

I'll compute this now.The linear regression equation for the number of Social Science doctorates as a function of time \( t \) since 2010 is:

\[
y(t) = 92.9t + 8386
\]

Where:
- \( y(t) \) is the number of Social Science doctorates
- \( t \) is the number of years since 2010
- \( 92.9 \) is the slope (indicating that approximately 92.9 more doctorates are awarded per year)
- \( 8386 \) is the y-intercept (the estimated number of doctorates in 2010)

The **coefficient of correlation** \( r \) is approximately **0.850**, which suggests a strong positive correlation between the number of doctorates and time.

Would you like further details on the process, or do you have any questions?

Here are 5 related questions:
1. How do you interpret the correlation coefficient \( r \) in this context?
2. Can we use this regression model to predict future values of Social Science doctorates?
3. What does the slope of 92.9 tell us about the trend in Social Science doctorates?
4. How would you check if a quadratic or other nonlinear model fits better than this linear model?
5. Could the y-intercept of 8386 be higher or lower if the starting year was different?

**Tip**: When calculating linear regression, always check the \( r \)-value to assess how well the model fits the data.

Use technology to obtain the regression equation and the coefficient of correlation r for the number of social science doctorates as a function of time t in years since 2010. Round coefficients to three significant digits.

Learn how to calculate the regression equation and the correlation coefficient for the number of social science doctorates over time since 2010. This problem involves linear regression and the Pearson correlation coefficient, and is suitable for advanced high school or college students.

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