CHAPTER 1: FUNCTIONS, GRAPHS, AND MODELS; LINEAR FUNCTIONSSection 1.6: Fitting Lines to Data Points: Modeling Linear Functions

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SECTION 1.6: MODELING LINEAR FUNCTIONS

How do we come up with equations to model a set of points? We find a line of ‘best fit’

The table below give the number of full- and part-time employees and clinics of dentists for selected years between 1970 and 1998.

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Year197

0197

51980

1985

1990

1995

1998

Employees (in thousands)

222 331 415 480 580 644 666

SECTION 1.6: MODELING LINEAR FUNCTIONS

Draw a scatterplot of the data: x is the year since 1970 and y is the number of employees (in thousands)

Graph each of the following equations with the data y1 = -12x + 660

y2 = 13x + 220

y3 = 16x + 42

Which is the best fit? 3

SECTION 1.6: MODELING LINEAR FUNCTIONS

The line of ‘best fit’ is found using the procedure of linear regression.

Typically, the regression employs the least-squares method. reduces the some of the squares of the

distance between the data points and line

How do we get this line? Luckily, the calculator does it for us.

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SECTION 1.6: MODELING LINEAR FUNCTIONS Linear Regression on the calculator

Enter your data into the STAT List Check the scatterplot – would a line be a reasonable

fit? Find the Linear Regression

STAT CALC 4: LinReg(ax+b) To save in your Y= menu: same as above, then

VARS YVARS Y1

Graph the scatterplot and Linear Regression together ZOOM 9: ZoomStat Does the line seem to fit?

Report your equation (from Y1) in an appropriate fashion.

Try this for the employee data. What do you find?5

SECTION 1.6: MODELING LINEAR FUNCTIONS The table below shows the earnings of year-round full-

time workers by gender and educational attainment.

Create a linear model that expresses females’ annual earnings as a function of males’ earnings. Pay attention to units!! 6

Educational Attainment

Average Annual Earnings for Males ($ thousand)

Average Annual Earnings for Females ($ thousand)

< 9th grade 18.743 12.392

Some high school 18.908 12.057

High school grad. 30.414 18.092

Some college 33.614 20.241

Associate’s degree 40.047 25.079

Bachelor’s degree or more

66.810 36.755

SECTION 1.6: MODELING LINEAR FUNCTIONS

A scatter plot is a way to represent a discrete function – when there are a finite number of inputs. series of dots

When we ‘fit’ a scatter plot with a function, we are using a continuous function to describe the data continuous curve

Often we use the continuous function to determine other data that are not given. interpolate – find a value within the given domain extrapolate – find a value outside of the given domain

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SECTION 1.6: MODELING LINEAR FUNCTIONS

The average math SAT scores in Beaufort County, SC are given in the table below. Write the equation of the line that is the best fit for these data, aligning the data so that x = 0 in 1990. Compare the outputs from the equation with the original data for each of the years 1994 to 1999 and determine the year in which the model output is closest to the data value.

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Year 1994

1995

1996

1997

1998

1999

Avg. Math SAT Score

472 464 470 471 473 470

SECTION 1.6: MODELING LINEAR FUNCTIONS

According to your model, what would you predict the average Math SAT score to be in 2000.

Do you think the prediction is accurate? Why or why not?

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Year 1994

1995

1996

1997

1998

1999

Avg. Math SAT Score

472 464 470 471 473 470

SECTION 1.6: MODELING LINEAR FUNCTIONS

Homework: pp. 94-101 1-7 odd, 13-16 all, 17, 19, 21, 27, 29, 33, 37, 41,

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