SECTION 9.2 Linear Regression 495 36. x 1 3 6 8 12 14 y 4 7 10 9 15 3 Transformations to Achieve Linearity When a linear model is not appropriate for representing data, other models can be used. In some cases, the values of x or y must be transformed to find an appropriate model. In a logarithmic transformation, the logarithms of the variables are used instead of the original variables when creating a scatter plot and calculating the regression line. In Exercises 37– 40, use the data shown in the table at the left, which shows the number of bacteria present after a certain number of hours. 37. Find the equation of the regression line for the data. Then construct a scatter plot of 1x, y2 and sketch the regression line with it. 38. Replace each y@value in the table with its logarithm, log y. Find the equation of the regression line for the transformed data. Then construct a scatter plot of 1x, log y2 and sketch the regression line with it. What do you notice? 39. An exponential equation is a nonlinear regression equation of the form y = abx. Use technology to find and graph the exponential equation for the original data. Include the original data in your graph. Note that you can also find this model by solving the equation log y = mx + b from Exercise 38 for y. 40. Compare your results in Exercise 39 with the equation of the regression line and its graph in Exercise 37. Which equation is a better model for the data? Explain. In Exercises 41– 44, use the data shown in the table at the left. 41. Find the equation of the regression line for the data. Then construct a scatter plot of 1x, y2 and sketch the regression line with it. 42. Replace each x@value and y@value in the table with its logarithm. Find the equation of the regression line for the transformed data. Then construct a scatter plot of 1log x, log y2 and sketch the regression line with it. What do you notice? 43. A power equation is a nonlinear regression equation of the form y = axb. Use technology to find and graph the power equation for the original data. Include a scatter plot in your graph. Note that you can also find this model by solving the equation log y = m1log x2 + b from Exercise 42 for y. 44. Compare your results in Exercise 43 with the equation of the regression line and its graph in Exercise 41. Which equation is a better model for the data? Explain. Logarithmic Equation The logarithmic equation is a nonlinear regression equation of the form y = a + b ln x. In Exercises 45– 48, use this information and technology. 45. Find and graph the logarithmic equation for the data in Exercise 25. 46. Find and graph the logarithmic equation for the data in Exercise 26. 47. Compare your results in Exercise 45 with the equation of the regression line and its graph. Which equation is a better model for the data? Explain. 48. Compare your results in Exercise 46 with the equation of the regression line and its graph. Which equation is a better model for the data? Explain. Number of hours, x Number of bacteria, y 1 165 2 280 3 468 4 780 5 1310 6 1920 7 4900 TABLE FOR EXERCISES 37–40 x y 1 695 2 410 3 256 4 110 5 80 6 75 7 68 8 74 TABLE FOR EXERCISES 41–44
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