Mathematical Modeling

Mathematical modeling is the process of using mathematics to solve real-

world problems. This process can be broken down into three steps:

1. Construct the mathematical model, a problem whose solution will provide information about the real-world problem. (We will do this on our calculator using regression analysis.)

2. Solve the mathematical model.3. Interpret the solution to the mathematical model in terms

of the original real-world problem.

Interpreting the Scatterplot

Form: the function that best describes the relationship between the 2 variables. (Some possible forms would be linear, quadratic, cubic, exponential, or logarithmic.)

To interpret the scatterplot, identify these following 4 things:

Outlier(s): any values that do not follow the general pattern of the data; stray points.)

Interpreting the ScatterplotDirection: a positive or negative direction can be found when looking at linear regression lines only. The direction is found by looking at the sign of the slope. Which of the following has a positive correlation? … negative?

Strength: how closely the points in the data are gathered around the form. Which has a strong correlation? …weak?

Some Types of RegressionLinear Regression (straight line form)- menu option 4:LinReg(ax+b)

Quadratic Regression (parabolic form)- menu option 5:QuadReg

Cubic Regression (cubic form)- menu option 6:CubicReg

Median-Median Line

x y

1 122 104 96 98 610 412 413 316 218 3

Organize the data points in ascending order according to x

Organize Data

x y

1 122 104 96 98 610 412 413 316 218 3

Divide Data into 3 Groups

x y

1 122 104 96 98 610 412 413 316 218 3

Divide Data into 3 Groups

If the data is evenly divided by 3, each group will have the same number of points.

If the data has an extra point after division, the middle section gets the extra point.

If the data falls one short of a perfect 3, the middle set will have one less than the others.

x y

1 122 104 96 98 610 412 413 316 218 3

Find the median point in each section.

9 5

233

Find the equation of the line from the two outside points.

216

103Slope

2

1

14

7Slope

10)2x(y21

11xy21

or

Plug in the middle x-coordinate and find the distance from the line.

5.6y

115.4y

11)9(y21

Dist = 6.5-5Dist = 1.5

Shift the line 1/3 down the distance you found.

Dist = 1.5

Shift Down 0.5

5.10xy21

Final Equation

1.5/3 = 0.5