Mathematical Models

A model is a mathematical representation of the relationship between two real-world quantities.

After 2 hours of driving, Freddy finds that 13 gallons of gas are left in his car’s fuel tank, and after 3 hours of driving 10.5 gallons are left in the tank.

a) Construct a linear model in which the number of gallons, g, left in the tank is a function of hours, h, of driving.

b) At what rate is the car consuming gas?c) How many gallons of gas were in the tank before the car was

driven?a) Find a linear function of the form g(h) = ah + b that contains (2, 13) and (3, 10.5).

The slope of the line is10.5 1

23

3

2.5 2. 5h bg 2.5( )13 2 b

18b 2.5 18g h

b) Since the slope is -2.5, the rate the car is consuming gas is 2.5 gallons per hour

c) If h = 0, then g = -2.5(0) + 18

18g

The car started with 18 gallons of gas.

Scatterplots The accompanying table shows the number of applications for admissions that a college received for certain years. Create a scatterplot that models this data.

Year# of

Applications

1991 297

1993 331

1995 409

1996 482

1999 647

2000 615

Let x = 1 represent 1991, x = 3 represent 1993 and so on so that x = 10 represents 2000.Store the data

Press STAT ENTEREnter x values in L1 and corresponding y values in L2.Set up the Scatterplot

Press 2nd STAT PLOT ENTER ENTER

Display the Scatterplot

Press ZOOM 9 Or press GRAPH

That was easy

Calculating a Regression LineA Regression Line is a linear model that best approximates a

set of data points. The Regression Line is also called a Line of Best Fit.

Let’s use the data from the previous table to calculate the line of regression.

Year# of

Applications

1991 297

1993 331

1995 409

1996 482

1999 647

2000 615

Press STAT > (CALC) 4 (LinReg(ax+b))

It’s helpful to store the equation as Y1VARS > (Y-VARS) 1 (Function) 1 (Y1) ENTER

The equation of the regression line is

41.140 230.371xy

The r-value is the Correlation Coefficient.

This value will be between -1 and +1. The closer the absolute value is to 1, the more closely the regression line fits the data.

Comparing Correlation Coefficients

0r

As x increases, y increases. The closer r is to 1, the better a line fits the data and the stronger the linear relationship is.

0 1r

0r

There is no significant linear relationship between x and y. The closer r is to 0, the weaker the linear relationship is.

0r

0r

As x increases, y decreases. The closer r is to -1, the better a line fits the data and the stronger the linear relationship is.

1 0r

Making Predictions

Interpolation:Estimating within the range of observed data.

Extrapolation:

Those are some pretty funny-looking words. I

wonder what they mean.

Estimating outside the range of observed data.

Let’s use the data from the previous table to do some interpolating and some extrapolating.

Year# of

Applications

1991 297

1993 331

1995 409

1996 482

1999 647

2000 615

Interpolation: Estimate the number of applicants the school had in 1997.Plug the x-value

into the equation. ( 41.140 23 ) 0.371y 7

518.351y The school had about 518 applicants in 1997.

Extrapolation: Estimate the number of applicants the school will have in 2009.Plug the x-value

into the equation. ( 41.140 23 ) 0.371y 19

1,012.031y The school will have about 1,012 applicants in 2009.

Calculating Exponential Regression

AUG06 31

ExpReg y = abx

a = 379.92 b = 1.04 r2 = 1.00 r = 1.00

379.92 1.( )04 xy

10(379.92 1.04)y 562.37y $ 562y

This is Sam Ting as linear regression, only you push different buttons.Asi De Facil

DefCon 3

Calculating Power RegressionJAN07 30 DefCon 3

PwrReg y = axb

a = 451.431 b = -.243 r2 = .956 r = .978

.243451.431y x

.243451.431( )8y 272.358y

272y

This is Sam Ting as linear regression and exponential regression, only you push different buttons.

Asi De Facil

Multiple Choice QuestionsWhich scatter diagram shows the

strongest positive correlation?Which graph represents data used in a linear regression that produces a correlation coefficient closest to -1?

That was easy