13_MMA0301_04AA_Regression Models

Mathematical Models with Applications HS Mathematics Unit: 03 Lesson: 01 Regression Models Determining representative models for data sets is not only fundamental in secondary math courses, but also in real life situations that generate data gathered through observations and experiments. A regression model is a mathematical equation used to describe the relationship between two variables. A linear or non-linear regression model can be determined for a set of data as appropriate. The model should be the one that best fits the scatterplot of the data. Linear Regression Models Concrete model: Fettuccini Fun The objective of this activity is to investigate the concept of "best fit" when determining a linear function to represent a set of data. Materials: Scatterplot of Data Points sheet, fettuccini, tape 1. Examine the Scatterplot of Data Points and visually determine a line that best seems to fit the data points. Tape a piece of fettuccine onto the scatterplot to represent your line of best fit. 2. Using additional pieces of fettuccine, break off pieces to measure the vertical distance from your line to each of the points. 3. Line up the fettuccine distances and measure them to find the total centimeters of error. Total Error ______________________ 4. Determine who in the group has the closest fit. When you have determined the "winner," replace the segments on the scatterplot and tape them down. Adapted from MTR Mathematics TEKS Refinement, Grades 9 - 12, 2006, Tab 3: Algebra 1, Spaghetti Regression, pp. 3-1 through 3-30. (C)2012, TESCCC 04/29/13 page 1 of 12 Mathematical Models with Applications HS Mathematics Unit: 03 Lesson: 01 Regression Models These distances from the line of best fit to the actual points are called residuals. The line that best fits the scatterplot will be the line with the lowest sum of the residuals. Trend Line (Parameter Changes on the Parent Function) 5. Build a table of data points. Assume the lower corner of the scatterplot is (0, 0). x y 6. Enter the actual data points into L1 and L2. Put x-values in L1 and y-values in L2. Sketch the scatterplot above. 7. Plot the scatterplot of the data. Go to y= and enter y 1= x. 8. Find a representative trend line by changing parameters the linear parent function y = x. f(x) = _____________________ Analysis of the Trend Line 9. Place the values predicted by the trend line into L3. To do this, place your cursor on L3 and enter the function, using L1 in place of the "x," such as L1 + 3. Press ENTER. The column of L3 should be populated with the predicted values. Adapted from MTR Mathematics TEKS Refinement, Grades 9 - 12, 2006, Tab 3: Algebra 1, Spaghetti Regression, pp. 3-1 through 3-30. (C)2012, TESCCC 04/29/13 page 2 of 12 Mathematical Models with Applications HS Mathematics Unit: 03 Lesson: 01 10. Calculate the residuals by subtracting the predicted values from the actual values. This can be done by placing the cursor on L4 and entering L2 - L3. Then press ENTER. The column of L4 should be populated with the residuals. Adapted from MTR Mathematics TEKS Refinement, Grades 9 - 12, 2006, Tab 3: Algebra 1, Spaghetti Regression, pp. 3-1 through 3-30. (C)2012, TESCCC 04/29/13 page 3 of 12 Mathematical Models with Applications HS Mathematics Unit: 03 Lesson: 01 Regression Models 11. If points are above the line, residuals are positive. If points are below the line, residuals are negative. Negative values cancel the effects of the positives. To fix this situation the residuals are squared. This can be done by placing the cursor on L5 and entering L4 2. Then press ENTER. The column of L5 should be populated with the squares of the residuals. 12. To find the sum of the residuals go to the home screen on the calculator. Go to 2 nd LIST MATH SUM ENTER. After the parentheses put L5) ENTER. You should get the sum of the squares of the residuals. Sum of Squares of the Residuals = _______________ Who in the class has the lowest sum of squares of residuals? ______________ What is the linear function modeled by that student? f(x) = _____________ Line of Regression The graphing calculator uses the same procedure to find the line of best fit for a set of data. When finding a regression model for a line, the line of least squares is determined. This model will have the lowest sum of the squares of differences in the predicted values and the observed values. The graphing calculator can be used to determine both linear and non-linear regression models. If the correlation is strong enough, models can then be used to predict other data. 13. Find the line of regression for the set of points using the graphing calculator. As long as the x values are in L1 and the y values are in L2, the line of regression is calculated by pressing STAT CALC LinReg(ax+b) ENTER. On the home screen values for "a" and "b" will be given so the function y=mx+b can be determined. Record the line of regression for the data on the scatterplot. y = ___________________ 14. How does the line of regression compare with the trend line you determined? 15. What would the predicted value of f(100) be when using the trend line? Line of regression? 16. Which line would be best to use for predictions? Explain your reasoning. Adapted from MTR Mathematics TEKS Refinement, Grades 9 - 12, 2006, Tab 3: Algebra 1, Spaghetti Regression, pp. 3-1 through 3-30. (C)2012, TESCCC 04/29/13 page 4 of 12 Mathematical Models with Applications HS Mathematics Unit: 03 Lesson: 01 Regression Models 17. Enter the following set of data into the graphing calculator. Sketch the scatterplot. Find the line of regression for the data set and plot it on the scatterplot. x y -10 7 -9 7 -8 6 -7 5 -6 5 -4 4 -2 3 -1 3 0 2 1 1 2 1 4 0 6 -1 7 -1 8 -2 9 -3 10 -3 Adapted from MTR Mathematics TEKS Refinement, Grades 9 - 12, 2006, Tab 3: Algebra 1, Spaghetti Regression, pp. 3-1 through 3-30. (C)2012, TESCCC 04/29/13 page 5 of 12 Mathematical Models with Applications HS Mathematics Unit: 03 Lesson: 01 18. Make a table of the points from the scatterplot. Enter the data into the graphing calculator and determine the line of regression. Plot the line of regression on the scatterplot. Adapted from MTR Mathematics TEKS Refinement, Grades 9 - 12, 2006, Tab 3: Algebra 1, Spaghetti Regression, pp. 3-1 through 3-30. (C)2012, TESCCC 04/29/13 page 6 of 12 Mathematical Models with Applications HS Mathematics Unit: 03 Lesson: 01 Regression Models Non-Linear Regression Models Just because a relationship exists between two variables, it must not be assumed the relationship is linear. In many situations the relationship between two variables is non-linear. A scatterplot must be constructed and analyzed to determine the parent function that best approximates the shape of the data. 19. For the scatterplots below use the Models of Parent Functions to determine which parent function would best approximate the data? _____________________ 20. ______________________ Sketch a scatterplot that could best be represented by a sine function. Your scatterplot must contain at least 15 points. These points will be used in a later problem. Adapted from MTR Mathematics TEKS Refinement, Grades 9 - 12, 2006, Tab 3: Algebra 1, Spaghetti Regression, pp. 3-1 through 3-30. (C)2012, TESCCC 04/29/13 page 7 of 12 Mathematical Models with Applications HS Mathematics Unit: 03 Lesson: 01 Regression Models Independent Practice The following set of data was entered into a graphing calculator. The independent variables were loaded into L1 and the dependent variables were loaded into L2. Use the data set to: o Graph a scatterplot of the data. o Select an appropriate model to mathematically represent the data. o Determine the regression model using the graphing calculator. o Describe inferences that can be made from the graph of the regression model. o Explain how the regression model would be used to make predictions. Adapted from MTR Mathematics TEKS Refinement, Grades 9 - 12, 2006, Tab 3: Algebra 1, Spaghetti Regression, pp. 3-1 through 3-30. (C)2012, TESCCC 04/29/13 page 8 of 12 Mathematical Models with Applications HS Mathematics Unit: 03 Lesson: 01 Regression Models The following set of data was entered into a graphing calculator. The independent variables were loaded into L1 and the dependent variables were loaded into L2. Use the data set to: o Set up a table to represent the data points. o Enter the data into the graphing calculator and graph the scatterplot in the graphing calculator. o Select an appropriate model to mathematically represent the data. o Determine the regression model using the graphing calculator. o Describe inferences that can be made from the graph of the regression model. o Explain how the regression model would be used to make predictions. y x Adapted from MTR Mathematics TEKS Refinement, Grades 9 - 12, 2006, Tab 3: Algebra 1, Spaghetti Regression, pp. 3-1 through 3-30. (C)2012, TESCCC 04/29/13 page 9 of 12 Mathematical Models with Applications HS Mathematics Unit: 03 Lesson: 01 Adapted from MTR Mathematics TEKS Refinement, Grades 9 - 12, 2006, Tab 3: Algebra 1, Spaghetti Regression, pp. 3-1 through 3-30. (C)2012, TESCCC 04/29/13 page 10 of 12 Mathematical Models with Applications HS Mathematics Unit: 03 Lesson: 01 Regression Models Using the points from the scatterplot you sketched for the sine function on problem #20 of the NonLinear Regression Models: o Set up a table to represent the data points. o Enter the data into the graphing calculator and graph a scatterplot of the data using the graphing calculator. o Select an appropriate model to mathematically represent the data. o Determine the regression model using the graphing calculator. o Describe inferences that can be made from the graph of the regression model. o Explain how the regression model would be used to make predictions. x y Adapted from MTR Mathematics TEKS Refinement, Grades 9 - 12, 2006, Tab 3: Algebra 1, Spaghetti Regression, pp. 3-1 through 3-30. (C)2012, TESCCC 04/29/13 page 11 of 12 Mathematical Models with Applications HS Mathematics Unit: 03 Lesson: 01 Adapted from MTR Mathematics TEKS Refinement, Grades 9 - 12, 2006, Tab 3: Algebra 1, Spaghetti Regression, pp. 3-1 through 3-30. (C)2012, TESCCC 04/29/13 page 12 of 12