tailieunhanh - Calculus and its applications: 6.4

"Calculus and its applications: " - An application-the least squares technique have objective: find a regression line, solve applied problems involving regression lines. | An Application: The Least Squares Technique OBJECTIVE Find a regression line. Solve applied problems involving regression lines. Suppose that a car rental company that offers hybrid vehicles charts its revenue as shown below. How best could we predict the company’s revenue for the year 2011? Year, x 1991 1996 2001 2006 2011 Yearly Revenue, y (in millions of dollars) ? An Application: The Least-Squares Technique Slide 2012 Pearson Education, Inc. All rights reserved Suppose that we plot these points and try to draw a line through them that fits. Note that there are several ways in which this might be done. (See the graphs below.) Each would give a different estimate of the company’s total revenue for 2011. An Application: The Least-Squares Technique Slide 2012 Pearson Education, Inc. All rights reserved Note that the years given follow 5-yr increments. Thus, computations can be simplified if we use the data points (1, ), (2, ), (3, ), and (4, ), as in the second graph, where each horizontal line represents 5 years. To determine the equation of the line that “best” fits the data, we note that for each data point there will be a deviation, or error, between the y-value at that point and the y-value of the point on the line that is directly above or below the point. An Application: The Least-Squares Technique Slide 2012 Pearson Education, Inc. All rights reserved p. 453, formula 12, there is no “+C” Those deviations, in this case, y1 – , y2 – , y3 – , and y4 – , will be positive or negative, depending on the location of the line. An Application: The Least-Squares Technique Slide 2012 Pearson Education, Inc. All rights reserved We wish to fit these data points with a line, y = mx + b, that uses values of m and b that, somehow, minimize the deviations in order to have a good fit. One way of minimizing the deviations is based on the least-squares assumption. An . | An Application: The Least Squares Technique OBJECTIVE Find a regression line. Solve applied problems involving regression lines. Suppose that a car rental company that offers hybrid vehicles charts its revenue as shown below. How best could we predict the company’s revenue for the year 2011? Year, x 1991 1996 2001 2006 2011 Yearly Revenue, y (in millions of dollars) ? An Application: The Least-Squares Technique Slide 2012 Pearson Education, Inc. All rights reserved Suppose that we plot these points and try to draw a line through them that fits. Note that there are several ways in which this might be done. (See the graphs below.) Each would give a different estimate of the company’s total revenue for 2011. An Application: The Least-Squares Technique Slide 2012 Pearson Education, Inc. All rights reserved Note that the years given follow 5-yr increments. Thus, computations can be simplified if we use the data points (1, ), (2, ), (3, .