1. Finding Equations of Linear Models Section 2.2

2. Lehmann, Intermediate Algebra, 3ed The average number of credit card offers a household receives in one month is increased approximately linearly from 5.1 offers in 2002 to 5.9 offers in 2005 (Source: Synovate). Let n be the average number of credit card offers a household receives in one month at t years since 2000. Find the model. The known values are shown (right). Section 2.2Slide 2 Finding an Equation of a Linear Model Finding an Equation of a Linear Model by Using Data Example Solution

3. Lehmann, Intermediate Algebra, 3ed Linear function can be put into the form Here y depends on x t and n are approximately linear So, t depends on n Thus, our model equation is Now we find the slope: Section 2.2Slide 3 Finding an Equation of a Linear Model Finding an Equation of a Linear Model by Using Data Solution Continued

4. Lehmann, Intermediate Algebra, 3ed Substitute 0.27 for m in the equation : Now we need to find b Substitute one of the coordinates and solve for b. Substituting into : Section 2.2Slide 4 Finding an Equation of a Linear Model Finding an Equation of a Linear Model by Using Data Solution Continued

5. Lehmann, Intermediate Algebra, 3ed Substitute 4.56 for b in the equation : Now we need to find Verify using TRACE checking (2, 5.1) and (5, 5.9) Section 2.2Slide 5 Finding an Equation of a Linear Model Finding an Equation of a Linear Model by Using Data Solution Continued Graphing Solution

6. Lehmann, Intermediate Algebra, 3ed During the 1900s there was great consumer demand for food products claiming to be “low fat” or “no fat.” Since then, this demand has declined greatly. The table (on slide 7) shows the percentage of new food products claiming to be “low fat” or “no fat” from 1996 to 2001. Let p be the percentage of new food products claiming to be “low fat” or “no fat” at t years since 1995. Find an equation of a line that comes close to the points in the scattergram of data. Section 2.2Slide 6 Finding an Equation of a Linear Model By Using Data Finding an Equation of a Linear Model by Using Data Example

7. Lehmann, Intermediate Algebra, 3ed View point positions in the scattergram Use a graphing calculator Saves time and improves accuracy Section 2.2Slide 7 Finding an Equation of a Linear Model By Using Data Finding an Equation of a Linear Model by Using Data Solution

8. Lehmann, Intermediate Algebra, 3ed Red line contains points (4, 17) and (5, 16) does not come close to the other data points Section 2.2Slide 8 Finding an Equation of a Linear Model By Using Data Finding an Equation of a Linear Model by Using Data Solution Continued Green line contains the points (1, 29) and (3, 22) and appears to come close to the rest of the points So, we must find the equation of the green line

9. Lehmann, Intermediate Algebra, 3ed Use the points (1, 29) and (3, 22) to find the slope: Substitute –3.5 for m: Section 2.2Slide 9 Finding an Equation of a Linear Model By Using Data Finding an Equation of a Linear Model by Using Data Solution Continued

10. Lehmann, Intermediate Algebra, 3ed To find b substitute the point (1, 29) into the equation and then solve for b. Substituting 32.5 for b: Section 2.2Slide 10 Finding an Equation of a Linear Model By Using Data Finding an Equation of a Linear Model by Using Data Solution Continued

11. Lehmann, Intermediate Algebra, 3ed Check correctness of equation using graphing calculator Verify that the line contains (1, 29) and (3, 22) Section 2.2Slide 11 Finding an Equation of a Linear Model By Using Data Finding an Equation of a Linear Model by Using Data Graphing Calculator

12. Lehmann, Intermediate Algebra, 3ed To find an equation of a linear model, given some data: 1.Create a scattergram of the data. 2.Determine whether there is a line that comes close to the data points. If so, choose two points (not necessarily data points) that you can use to find the equation of a linear function. 3.Find an equation of the line you identified. Section 2.2Slide 12 Finding an Equation of a Linear Line Finding an Equation of a Linear Model by Using Data Process

13. Lehmann, Intermediate Algebra, 3ed 4.Use a graphing calculator to verify that the graph of your equation comes close to the point of the scattergram. Linear equation found by linear regression are called linear regression equations/functions Most graphing calculators have regression features Section 2.2Slide 13 Finding an Equation of a Linear Line Finding an Equation of a Linear Model by Using Data Process Continued Graphing Calculator

14. Lehmann, Intermediate Algebra, 3ed Cigarette smoking has been on the decline for the past several decades. Let p be the percentage of Americans who smoke at t years since 1900. 1.Use two well-chosen points to find an equation of a model that describes the relationship between t and p. 2.Find the linear regression equation and line by using a graphing calculator. Compare this model with the one your found in the part 1. Section 2.2Slide 14 Finding Linear Equations of a Linear Model Finding an Equation of a Linear Model by Using Data Example

15. Lehmann, Intermediate Algebra, 3ed Using the points (70, 37.4) and (105, 19.0) to calculate the slope: Equation is of the form: Section 2.2Slide 15 Finding Linear Equations of a Linear Model Finding an Equation of a Linear Model by Using Data Solution

16. Lehmann, Intermediate Algebra, 3ed To find b, substitute the point (70, 37.4) into the equation : Equation is Use graphing calculator to verify that the linear model contains the points (70, 37.4) and (105, 19.0) Section 2.2Slide 16 Finding Linear Equations of a Linear Model Finding an Equation of a Linear Model by Using Data Solution Continued

17. Lehmann, Intermediate Algebra, 3ed Comparing solution to the first example: is close to Section 2.2Slide 17 Finding Linear Equations of a Linear Model Finding an Equation of a Linear Model by Using Data Solution Continued