The following table shows the annual number of PhD graduates in a country in various fields. Natural Sciences Social Sciences Engineering Education 1990 70 10 70 30 1995 130 40 110 40 2000 330 130 270 130 2005 490 370 450 210 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)- Graph the associated points and regression line. 1000 1000 1000t 1000t ool 800 800 800 6o0 600 600 600 400 400 400 400 200 200 200 200 10 15 20 10 15 20 10 15 20 10 15 20 25 (b) What does the slope tell you about the number of social science doctorates? The number of social science doctorates has been decreasing at a rate of about 43 per year. The number of social science doctorates has been increasing at a rate of about 59 per year. The number of social science doctorates has been decreasing at a rate of about 59 per year. The number of social science doctorates has been increasing at a rate of about 43 per year. (c) Judging from the graph, would you say that the number of social science doctorates is increasing at a faster and faster rate, a slower and slower rate, or a more-or-less constant rate? Why? The data points suggest a concave-up curve rather than a straight line, indicating that the number of doctorates has been growing at a slower and slower rate. The data points suggest a concave-up curve rather than a straight line, indicating that the number of doctorates has been growing at a faster and faster rate. The data points suggest a straight line, indicating that the number of doctorates has been growing at a more-or-less constant rate. The data points suggest a concave-down curve rather than a straight line, indicating that the number of doctorates has been growing at a faster and faster rate. The data points suggest a concave-down curve rather than a straight line, indicating that the number of doctorates has been growing at a slower and slower rate. (d) If r had been equal to 1, could you have drawn the same conclusion as in part (c)? Explain. No, if r had been equal to 1, then the points would lie exactly on the regression line, which would indicate that the number of doctorates is growing at a faster and faster rate. Yes, if r had been equal to 1, then the points would lie exactly on the regression line, which would indicate that the number of doctorates is growing at a faster and faster rate. No, if r had been equal to 1, then the points would lie exactly on the regression line, which would indicate that the number of doctorates is growing at a slower and slower rate. o Yes, if r had been equal to 1, then the points would lie exactly on the regression line, which would indicate that the number of doctorates is growing at a constant rate. No, if r had been equal to 1, then the points would lie exactly on the regression line, which would indicate that the number of doctorates is growing at a constant rate.