Choose Ohio First Scholarship Program - Success in Mathematics Funded by the Ohio Board of Regents Regression Lines Brian Dudek Alex Miksit Melinda Toth Question Since we know the height of the goalie, will we be able to predict their save percentages? – Hypothesis: the taller the soccer goalie, the higher save percentage they will have. – We will be testing our hypothesis by looking at different regression functions. The goal of regression analysis is to create a model used to predict Y when given X. How to determine the right regression 1. Plot the data points 2. Draw different forms of regression 3. By calculating the residuals, choose the regression curve with the least error. 1. A residual is the distance between the plotted point and the function 1. the added sum of the residuals squared is the key in determining which regression curve fits best. Different forms of regression Least Square Line (LSL)- The least-squares line is a regression that has the smallest amount of space possible between each of the plotted points and itself. Thus, a line with the minimal deviation from all data points is desired. Formula for LSL y = mx + b Different forms of regression Median-Median Line- categorizes all the points into three different groups, then takes the median of all the groups, and lastly finds the slope and the y-intercept. The med-med line is more precise if outliers are present. An outlier is a statistic that lies an abnormal distance outside other values in a random sample. Formula for MedMed Line y = mx + b Different forms of regression Logarithmic Quadratic Power •y = lnx •y = ax2 + bx + c • y = xa •Ex. Reaction rate vs substrate concentration in a chemical reaction Exponential •y = ax •Ex. A graph of time vs. speed for an object affected by gravity Regressions LSL Line Med-Med Line Residuals By looking at the residuals (r2), we can see that they are no where near the preferred value of 1.00. Plot of Residuals LSL Line By looking at the graph of our residuals we see that yet again they are no where near the preferred value. Plot of Residuals Med-Med Line Regressions Quadratic Cubic Residuals By looking at the residuals (r2), we can see that they are no where near the preferred value of 1.00. Plot of Residuals Quadratic Plot of Residuals Cubic Regressions Quartic Power Residuals Plot of Residuals Quartic By looking at the residuals (r2), we can see that they are no where near the preferred value of 1.00. Plot of Residuals Power Regressions Exponential Logarithmic Exponential Plot of Residuals Exponential By looking at the residuals (r2), we can see that they are no where near the preferred value of 1.00. Plot of Residuals Logarithmic Analyze Results Do residuals represent the best possible regression model for a given set of data? YES! Question Since we know the height of the goalie, will we be able to predict their save percentages? Conclusion Looking at our information we have concluded that there is a slight correlation between the height of the player and their save percentage. This is because our residuals are nowhere close to the line of regression. Therefore, the best fit line, however horrible, is Quartic for the correlation between height and saved percentage for soccer goalies! Extension Now we are curious if there is a correlation between age and save percentage, or professional career length and save percentage. Information 07/08 Stats Name Shots on Goal Shots Saved Percent Height (m) Age Yr's playing Pat Onstad 100 76 76.00 1.93 40 21 Jon Busch 155 122 78.71 1.78 31 11 William Hesmer 130 97 74.62 1.88 26 4 Bouna Coundoul 75 54 72.00 1.88 26 2 Joe Cannon 162 124 76.54 1.88 33 10 Kevin Hartman 156 117 75.00 1.85 33 11 Nick Rimando 135 96 71.11 1.78 28 8 Dario Sala 129 92 71.32 1.93 33 14 Brad Guzan 68 48 70.59 1.93 23 4 Matt Reis 145 107 73.79 1.85 33 10 Greg Sutton 151 116 76.82 1.98 30 9 Jon Conway 140 98 70.00 1.98 30 8 Louis Crayton 62 43 69.35 1.83 30 13 Zach Wells 84 56 66.67 1.88 27 4 Preston Burpo 61 37 60.66 1.91 35 13 Steve Cronin 136 92 67.65 1.91 24 4 • • • • • • • • http://people.hofstra.edu/Stefan Waner/ calctopic1/regression.html http://www.efunda.com/math/leastsquares/leastsquares.cfm – Definition of LSL http://cnx.org/content/m17090/latest/linrgs_regeq3.png – Image of LSL http://www.marketoracle.co.uk/images/2008/dow_gold_ratio_200years_feb08.jpg – Image of Med-Med Line http://mathbits.com/Mathbits/TISection/Statistics2/logarithmic.htm – Image of logarithmic function http://calculator.maconstate.edu/quad_regression/index.html – Image of quad function http://mathbits.com/mathbits/TISection/Statistics2/power.htm – Image of power function http://mathbits.com/Mathbits/TISection/Statistics2/exponential.htm – Image of exponential function – http://www.geocities.com/dolphdamerenee/tigger32goalie.jpg -Tigger http://4.bp.blogspot.com/_Rsl_LePBeKE/SDJoUNafuKI/AAAAAAAABqM/Y3_tVD5aUY/s400/Cartoon,+Black+Guy+with+Crazy+Graph.jpg http://www.fhwa.dot.gov/construction/images/specs30.jpg Monopoly man