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 8,100 5,300 2012 5,400 8,400 8,700 4,800 2014 5,900 9,600 8,900 4,800 2016 6,200 9,500 9,200 5,100 2018 6,300 10,200 9,100 4,800 2020 6,700 10,800 9,100 4,700 (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 2010. (Round coefficients to three significant digits. Round your r-value to three decimal places.) y(t) = Graph the associated points and regression line. (Select Update Graph to see your response plotted on the screen. Select the Submit button to grade your response.) Update Graph Student Response Graph y 49400 Student Response Graph Description 9200 9000 8800 8600 8400 8200 8000 -1 1 2 3 |4 5 |6 |7 8 9 10 (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 8,400 per year. ○ The number of social science doctorates has been increasing at a rate of about 93 per year. The number of social science doctorates has been increasing at a rate of about 8,400 per year. ○ The number of social science doctorates has been decreasing at a rate of about 93 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-down 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-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-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-up 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. 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. ○ 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 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 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.