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Finding Equations of Linear Models

Section 2.2. Finding Equations of Linear Models. Example.

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Finding Equations of Linear Models

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  1. Section 2.2 Finding Equations of Linear Models

  2. Example 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). Solution Section 2.2 Slide 2 Finding an Equation of a Linear Model Finding an Equation of a Linear Model by Using Data

  3. Solution Continued 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 2.2 Slide 3 Finding an Equation of a Linear Model Finding an Equation of a Linear Model by Using Data

  4. Solution Continued 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 2.2 Slide 4 Finding an Equation of a Linear Model Finding an Equation of a Linear Model by Using Data

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

  6. Example 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 2.2 Slide 6 Finding an Equation of a Linear Model By Using Data Finding an Equation of a Linear Model by Using Data

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

  8. Solution Continued Red line contains points (4, 17) and (5, 16) does not come close to the other data points • 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 Section 2.2 Slide 8 Finding an Equation of a Linear Model By Using Data Finding an Equation of a Linear Model by Using Data

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

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

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

  12. Process To find an equation of a linear model, given some data: Create a scattergram of the data. 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. Find an equation of the line you identified. Section 2.2 Slide 12 Finding an Equation of a Linear Line Finding an Equation of a Linear Model by Using Data

  13. Process Continued 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 Graphing Calculator Section 2.2 Slide 13 Finding an Equation of a Linear Line Finding an Equation of a Linear Model by Using Data

  14. Example 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 2.2 Slide 14 Finding Linear Equations of a Linear Model Finding an Equation of a Linear Model by Using Data

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

  16. Solution Continued 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 2.2 Slide 16 Finding Linear Equations of a Linear Model Finding an Equation of a Linear Model by Using Data

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

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