finding equations of linear models section 2.2. lehmann, intermediate algebra, 3ed the average...

Finding Equations of

Linear Models

Section 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 2.2 Slide 2

Finding an Equation of a Linear ModelFinding an Equation of a Linear Model by Using Data

Example

Solution


• 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


Solution Continued

n mt b

y mx b

5.9 5.1 0.80.27

5 2 3m


• 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


Solution Continuedn mt b

0.27n t b

2,5.1 0.27n t b


• 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 2.2 Slide 5


Solution Continued0.27n t b

0.27 4.56n t Graphing Solution


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 DataFinding an Equation of a Linear Model by Using Data

Example


• View point positions in the scattergram• Use a graphing calculator• Saves time and improves

accuracy

Section 2.2 Slide 7


Solution


• Red line contains points (4, 17) and (5, 16) does not come close to the other data points

Section 2.2 Slide 8


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


• Use the points (1, 29) and (3, 22) to find the slope:

• Substitute –3.5 for m:

Section 2.2 Slide 9


Solution Continued

22 29 73.5

3 1 2m

3.5p t b


• 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


Solution Continued

3.5p t b

3.5 32.5p t


• Check correctness of equation using graphing calculator • Verify that the line contains (1, 29) and (3, 22)



Graphing Calculator


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.


Finding an Equation of a Linear LineFinding an Equation of a Linear Model by Using Data

Process


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


Finding an Equation of a Linear LineFinding an Equation of a Linear Model by Using Data

Process Continued

Graphing Calculator


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.


Finding Linear Equations of a Linear ModelFinding an Equation of a Linear Model by Using Data

Example


• Using the points (70, 37.4) and (105, 19.0) to calculate the slope:

• Equation is of the form:



Solution

37.4 19.00.53

105 70m

0.53p t b


• 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)



Solution Continued

0.53p t b

0.53 74.50p t


Comparing solution to the first example:

is close to



Solution Continued

0.53 74.50p t 0.52 73.84p t

finding equations of linear models section 2.2. lehmann, intermediate algebra, 3ed the average...

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