9.The following table gives the number of graduates in health professions in the U.S. for various
years.
Year Number of Graduates
(in millions)
2009 |15.4
2010 |15.9
2011 |16.3
2012 |16
2013 |15.7
a. Code the years using t = 1 for 2009 and fit a quadratic function to the data with the regression program on your calculator.
equation is ________________________________
b. Predict the # of graduates in 2025. __________________
a. To fit a quadratic function to the given data, we can follow these steps using a regression program on a calculator:
1. Code the years: We need to convert the years into a new set of values starting from t = 1 for the year 2009. This will help us create a regression model with a quadratic equation. The coded years will be as follows:
Year | Coded Year (t)
---------------------
2009 | 1
2010 | 2
2011 | 3
2012 | 4
2013 | 5
2. Input the data into the regression program on your calculator: Enter the coded years (t) in the x-values and the number of graduates (in millions) in the y-values.
x-values: [1, 2, 3, 4, 5]
y-values: [15.4, 15.9, 16.3, 16, 15.7]
3. Perform the regression: Use the quadratic regression function on your calculator to get the equation. The equation in question is:
y = at^2 + bt + c, where a, b, and c are the coefficients of the quadratic equation.
The calculator will provide the values for a, b, and c, which represent the best-fit quadratic equation for the given data.
b. To predict the number of graduates in 2025, we need to use the equation from the previous step.
1. Code the year 2025: Assuming the trend of the data continues with the same coding, we can code the year 2025 as follows:
Year | Coded Year (t)
---------------------
2025 | 16
2. Plug the coded year (t = 16) into the quadratic equation obtained in step a. This equation provides the estimated number of graduates for the coded year:
y = at^2 + bt + c
By substituting t = 16 into the equation, you will get the predicted number of graduates in 2025.