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Warm-Up. = “change”. In the 2 scenarios below, find the change in x and the change in y. What conclusions can you draw? What are the similarities & differences? How would you model this data?. Quadratic Regression. Statistics, Fall 2010. Linear Regression.
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Warm-Up = “change” • In the 2 scenarios below, find the change in x and the change in y. • What conclusions can you draw? What are the similarities & differences? How would you model this data?
Quadratic Regression Statistics, Fall 2010
Linear Regression • So far, the data we have worked with has a linear relationship • We have discussed 3 forms of linear modeling: • Best Fit line • Least Squares Regression Line • Median-Median line
Linear Change is Constant • The data we have used has a linear relationship. This means that the rate of change (the slope) is constant. • For linear data, every increase in the independent variable (X) has a constant increase in the dependent variable (Y). y Slope = = constant x
Not All Data Is Linear • Data does not always follow a linear model. • Data may increase sharply, reach a maximum, then decline. Or, it may decrease sharply, reach a minimum, then increase again. • The data in the graph at the right is shaped like aparabola – so it would followa quadratic model instead of a linear model.
Quadratic Regression • Recall that parabolas are graphs of quadratic equations. They follow the model of y = ax2 + bx +c. • If a > 0, the parabola opens up (smiling) • If a < 0, the parabola opens down (frowning) • Real-life examples of data that follows a quadratic pattern include: • Stock market (Peaks and Valleys) • Disease outbreaks (Black Plague, Polio, AIDS) • Particle motion (Ball trajectory, Draining water)
Exponential Regression • Data can also follow an exponential model. • Exponential data either • Increases exponentially, where the change in y continues to increase for each change in x OR • Decreases exponentially, where the change in y continues to decrease for each change in x.
Exponential Regression • Examples of Real-Life data that follows an exponential model include: • Population growth (increasing) • National Debt (increasing) • Radioactive Decay (decreasing) • Exponential equationsfollow the modely = a(b)x (where a and b are constants)
Which Model Fits Best? • To determine if data is linear, quadratic or exponential • Create a Scatterplot of the data and look for the overall pattern • Evaluate the change in Y for each change in X y LINEAR = Constant = Increasing, 0, then Decreasing or Decreasing, 0, then Increasing = Increasing exponentially (e.g., doubles every time) or Decreases exponentially (e.g., halves every time) x QUADRATIC IF EXPONENTIAL
Which model fits best?(Hint: Look at the change in Y for each change in X)
Quadratic Regression Example • The table below lists the total estimated numbers of AIDS cases, by year of diagnosis from 1999 to 2003 in the United States (Source: US Dept. of Health and Human Services, Centers for Disease Control and Prevention, HIV/AIDS Surveillance, 2003.) • Notice the data peaksin 2001, then drops off. • This is a good indicatorthat Quadratic Regressionwill provide the mostaccurate model of the data
Find the Quadratic Regression Equation (Model) • 1). Plot the data, letting x = 0 correspond to the year 1998. • 2). Find a quadratic function that models the data. • Using your calculator, enter the Year as L1 and #of Cases as L2 • Use the QuadReg function on your calculator to calculate the regression equation • 3). Plot the function on the graph with the data and determine how well the graph fits the data, • 4). Use the model (equation) to predict the cumulative number of AIDS cases for the year 2006.
