Math 131 Lab 1 Data Modeling 12 Jan. The purpose of lab 1 is to use graphical, numerical and analytical techniques to find a mathematical model for a given set of data and then to use that model to predict performance in the future. Refs. Sec 1.1, 1.2 of text; Chs. 6 and 16 of TI-89 Guidebook Lab 1 uses the data set for the number of US cell phone subscribers as reported in the "Statistical Abstract of the United States" at http://www.census.gov/prod/2004pubs/03statab/inforcom.pdf Year Subscribers (thousands) Year Subscribers (thousands) 1990 5283 1997 55312 1991 7557 1998 69209 1992 11033 1999 86047 1993 16009 2000 109478 1994 24134 2001 128375 1995 33786 2002 140766 1996 44043 1. Plot the data points on graph paper with year as the horizontal axis and subscribers as the vertical axis. Label the axes. Indicate the scales. Title the plot. Use your graph to determine whether a linear model (y = ax + b) or an exponential model (y = a b^x) better fits the data. Explain your answer. 2. The text explains a way to determine if a data set might be linear on p. 4 or exponential on p. 10. For the subscriber data perform some calculations to determine whether a linear model or an exponential model is a better fit. Provide a detailed explanation for your conclusion. 3. Find an exponential model for cell phone subscribers 'by hand'. Use the data points for the years 1990 and 2000 in your model. Do algebra to find the constants in y = a b^x using the 2 given data points. Show and explain your work. Example 1 on pp. 11-12 should serve as a guide for your work. 4. Enter the model in step 3 as the function y1 in Graph on TI89. Plot y1 accurately on your scattergram. You should use the Table key on your TI to obtain several points to plot on your graph of step 1. Discuss any significant differences between the data and the model. 5. Explain how to determine what your model predicts for subscribers for the years 1995 and 2002. Compare the model to the data. Explain any differences. The rest of the lab uses some of the statistical features of your graphing calculator. Some statistical instructions on the TI89 appear in a separate document. 6. Use a calculator to draw a scattergram for the data on cell phones. 7. Use your TI to find the best exponential model that uses all the data points with a calculus technique called regression. Save your model in the function y2. Write y2 on your paper. Use the Table key on your calculator to obtain points to plot y2 by hand on your scattergram. 8. Determine whether the model in step 3 or step 7 is better. Explain your answer. 9. Use your calculator to fit a quadratic regression model (y = ax^2 + bx + c) to the subscriber data. Save the model in y3. Write y3 on your paper. To avoid information overload, replat the data on another set of piece of graph paper. Then plot y3 by hand on the new scattergram. 10. Determine which of the models in steps 3, 7 or 9 most closely fits the subscriber data. Explain your answer. Use that model to predict the number of subscribers for 2004 and for 2005. Discuss the confidence you have in the predictions. Group Report Each group will submit one report. You must use computer software for your report. It contains your answers for 1 - 10. It should be clear and concise. It should follow the usual rules for grammar and style. Your English composition teacher should be able to understand your report. You submit your hand drawn graphs separately in class. Your group report should end with a short paragraph about what you learned in this lab. The report is due Wednesday 19 Jan. at 9 AM. Email your report to yanushka@cbu.edu. Feel free to see me in my office in Sci. 103g or send me email for assistance.