UNIT QUESTION: Can real world data be modeled by algebraic functions? Math II UNIT QUESTION: Can real world data be modeled by algebraic functions? Standard: MM2D1, D2 Today’s Question: How does of a graph of data with negative correlation look like? Standard: MM2D2a, b

Mathematical Modeling Mathematical modeling is the process of using mathematics to solve real- world problems. This process can be broken down into three steps: 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.) Solve the mathematical model. Interpret the solution to the mathematical model in terms of the original real-world problem.

Interpreting the Scatterplot To interpret the scatterplot, identify these following 4 things: Form: the function that best describes the relationship between the 2 variables. (Some possible forms would be linear, quadratic, cubic, exponential, or logarithmic.) Outlier(s): any values that do not follow the general pattern of the data; stray points.)

Interpreting the Scatterplot Direction: 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 Regression Linear 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

Organize Data x y 1 12 2 10 4 9 6 9 8 6 10 4 12 4 13 3 16 2 18 3 Organize the data points in ascending order according to x

Divide Data into 3 Groups 15 10 5 20 x y 1 12 2 10 4 9 6 9 8 6 10 4 12 4 13 3 16 2 18 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 12 2 10 4 9 6 9 8 6 10 4 12 4 13 3 16 2 18 3

Find the median point in each section. 15 10 5 20 x y 1 12 2 10 4 9 6 9 8 6 10 4 12 4 13 3 16 2 18 3 9 5 2 3 3

Find the equation of the line from the two outside points. 15 10 5 20 or

Plug in the middle x-coordinate and find the distance from the line. 15 10 5 20 Dist = 6.5-5 Dist = 1.5

Shift the line 1/3 down the distance you found. 15 10 5 20 Dist = 1.5 1.5/3 = 0.5 Shift Down 0.5 Final Equation

Homework Day 1: Page 250 #1, 2, 5-9 Complete the scatter plots on the handout