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## Mathematical Models

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**Mathematical Models**A model is a mathematical representation of the relationship between two real-world quantities. After 2 hours of driving, Freddy finds that 13 gallons of gas are left in his car’s fuel tank, and after 3 hours of driving 10.5 gallons are left in the tank. • Construct a linear model in which the number of gallons, g, left in the tank is a function of hours, h, of driving. • At what rate is the car consuming gas? • How many gallons of gas were in the tank before the car was driven? a) Find a linear function of the form g(h) = ah + b that contains (2, 13) and (3, 10.5). The slope of the line is b) Since the slope is -2.5, the rate the car is consuming gas is 2.5 gallons per hour c) If h = 0, then g = -2.5(0) + 18 The car started with 18 gallons of gas.**Scatterplots**The accompanying table shows the number of applications for admissions that a college received for certain years. Create a scatterplot that models this data. Let x = 1 represent 1991, x = 3 represent 1993 and so on so that x = 10 represents 2000. Store the data Press STAT ENTER Enter x values in L1 and corresponding y values in L2. Set up the Scatterplot Press 2nd STAT PLOT ENTER ENTER Display the Scatterplot Or press GRAPH Press ZOOM 9 That was easy**Calculating a Regression Line**A Regression Line is a linear model that best approximates a set of data points. The Regression Line is also called a Line of Best Fit. Let’s use the data from the previous table to calculate the line of regression. Press STAT > (CALC) 4 (LinReg(ax+b)) It’s helpful to store the equation as Y1 > VARS (Y-VARS) 1 (Function) 1 (Y1) ENTER The r-value is the Correlation Coefficient. This value will be between -1 and +1. The closer the absolute value is to 1, the more closely the regression line fits the data. The equation of the regression line is**Comparing Correlation Coefficients**As x increases, y increases. The closer r is to 1, the better a line fits the data and the stronger the linear relationship is. As x increases, y decreases. The closer r is to -1, the better a line fits the data and the stronger the linear relationship is. There is no significant linear relationship between x and y. The closer r is to 0, the weaker the linear relationship is.**Those are some pretty funny-looking words. I wonder what**they mean. Making Predictions Interpolation: Estimating within the range of observed data. Extrapolation: Estimating outside the range of observed data. Let’s use the data from the previous table to do some interpolating and some extrapolating. Interpolation: Estimate the number of applicants the school had in 1997. Plug the x-value into the equation. 7 The school had about 518 applicants in 1997. Extrapolation: Estimate the number of applicants the school will have in 2009. Plug the x-value into the equation. 19 The school will have about 1,012 applicants in 2009.**Calculating Exponential Regression**AUG06 31 DefCon 3 ExpReg y = abx a = 379.92 b = 1.04 r2 = 1.00 r = 1.00 This is Sam Ting as linear regression, only you push different buttons. Asi De Facil**Calculating Power Regression**JAN07 30 DefCon 3 PwrReg y = axb a = 451.431 b = -.243 r2 = .956 r = .978 This is Sam Ting as linear regression and exponential regression, only you push different buttons. Asi De Facil**Multiple Choice Questions**Which scatter diagram shows the strongest positive correlation? Which graph represents data used in a linear regression that produces a correlation coefficient closest to -1? That was easy