The following table shows the annual number of PhD graduates in a country in various fields.
Natural
Sciences	Engineering	Social
Sciences	Education
1990	70	10	70	30
1995	130	40	110	40
2000	330	130	290	130
2005	490	370	460	210
2010	590	550	830	520
2012	690	590	1,000	900
(a)
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 1990. (Round coefficients to three significant digits. Round your r-value to three decimal places.)
y(t)	=	
 
r	=	
Graph the associated points and regression line. 
What does the slope tell you about the number of social science doctorates?

### (a) Regression Equation and Correlation Coefficient for Social Science Doctorates

Let’s start by interpreting the problem and setting up the data for regression analysis.

We are given the following data for the number of **Social Science PhD graduates** across different years.

| Year  | Social Sciences (y) |
|-------|---------------------|
| 1990  | 70                  |
| 1995  | 110                 |
| 2000  | 290                 |
| 2005  | 460                 |
| 2010  | 830                 |
| 2012  | 1,000               |

We define **t** as the number of years since 1990. So for each year:

- \( t = \text{Year} - 1990 \)

This transforms the data table into:

| t (Years since 1990) | Social Sciences (y) |
|----------------------|---------------------|
| 0                    | 70                  |
| 5                    | 110                 |
| 10                   | 290                 |
| 15                   | 460                 |
| 20                   | 830                 |
| 22                   | 1,000               |

Next, we will perform **linear regression** on this data to obtain the regression equation \( y(t) \), where \( y(t) = mt + b \), and the **correlation coefficient** \( r \).

Using a calculator or software, the linear regression equation is obtained by minimizing the sum of squared differences between the actual and predicted values of \( y \). I'll perform this now.

The regression equation for the number of social science doctorates as a function of time \( t \) (years since 1990) is:

\[
y(t) = 42.784t - 53.405
\]

The correlation coefficient \( r \) is:

\[
r = 0.962
\]

### Interpretation of the Slope:
The slope of the regression equation, \( 42.784 \), represents the rate of increase in the number of social science PhD graduates per year. This means that, on average, approximately 42.784 more PhDs are awarded in social sciences each year since 1990.

Next, I'll graph the data points and the regression line.

Would you like to proceed with that or ask any questions?

---

**Questions to Explore Further:**
1. How does the correlation coefficient \( r \) indicate the strength of the relationship?
2. What is the significance of the negative intercept in this context?
3. Could a quadratic or other non-linear model fit this data better?
4. How can we use the regression model to predict future PhD graduates?
5. How do the trends in other fields compare with social sciences?

**Tip:** The closer the correlation coefficient \( r \) is to 1 or -1, the stronger the linear relationship between the variables.

The following table shows the annual number of PhD graduates in a country in various fields. Natural Sciences Engineering Social Sciences Education 1990 70 10 70 30 1995 130 40 110 40 2000 330 130 290 130 2005 490 370 460 210 2010 590 550 830 520 2012 690 590 1,000 900 (a) 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 1990. (Round coefficients to three significant digits. Round your r-value to three decimal places.) y(t) = r = Graph the associated points and regression line. What does the slope tell you about the number of social science doctorates?

Analyze the annual growth of social science PhD graduates from 1990 to 2012 using linear regression. Learn how to compute the regression equation y(t) = 42.784t - 53.405 and the correlation coefficient r = 0.962, along with the interpretation of the slope and the strength of the relationship.

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