One student studied for 8 years in college and her tuition bills are recorded.
Years. One semester tuition
1. 4680
2. 4960
3. 5510
4. 6060
5. 6845
6. 7600
7. 8435
8. 9250
Construct the scatter plot for the data with the scales.choose 3 points,find a quadratic equation in standard form,using a system and matrices.use quadratic regression to find the equation of the data .what is r^2? If this student remains in school for 2 more years what should she anticipate that her tuition will be?
To construct a scatter plot, we need to plot the given data points on a graph with the number of years on the x-axis and the tuition on the y-axis. Here are the points:
(1, 4680)
(2, 4960)
(3, 5510)
(4, 6060)
(5, 6845)
(6, 7600)
(7, 8435)
(8, 9250)
Plotting these points will give us a visual representation of the data.
To find the quadratic equation in standard form using a system of equations and matrices, we need to use the method of least squares regression. We want to find a quadratic equation of the form y = ax^2 + bx + c that best fits the given data points.
We set up the following system of equations using the given points:
4680 = a(1)^2 + b(1) + c
4960 = a(2)^2 + b(2) + c
5510 = a(3)^2 + b(3) + c
6060 = a(4)^2 + b(4) + c
6845 = a(5)^2 + b(5) + c
7600 = a(6)^2 + b(6) + c
8435 = a(7)^2 + b(7) + c
9250 = a(8)^2 + b(8) + c
We can rewrite this system of equations in matrix form as AX = B, where:
A = [[1, 1, 1], [4, 2, 1], [9, 3, 1], [16, 4, 1], [25, 5, 1], [36, 6, 1], [49, 7, 1], [64, 8, 1]]
X = [[a], [b], [c]]
B = [[4680], [4960], [5510], [6060], [6845], [7600], [8435], [9250]]
To find X, we need to calculate the least squares solution using matrix operations. The solution X will give us the coefficients a, b, and c for the quadratic equation.
Once we have the equation in standard form (ax^2 + bx + c), we can use the quadratic regression feature on a calculator or statistical software to find the equation of the data.
To find the value of r^2, we can use the coefficient of determination, which measures the goodness of fit for the regression model. It tells us how well the regression equation fits the data. The value of r^2 will be between 0 and 1, where 1 represents a perfect fit.
To anticipate the tuition for the next 2 years, we can substitute the values of 9 and 10 (indicating 9 years and 10 years) into the quadratic equation we obtained from the regression analysis and calculate the corresponding tuition values.