M119 Project 5 – Modeling with Logistics and Logarithmic Functions

 

Due:                                                                                                     20 points

Objective:  Use technology to fit exponential model to data

 

This first part is a repeat of the information on using the calculator that was on Project 6.  The example has been changed to a logistics example.

 

Information on Using the Regression Feature of the TI-83 calculator

 

Entering Data

            STAT – 1:Edit     

 

           If the lists L1 and L2 have values, these must be cleared out.  To empty them arrow up to the VERY top of list L1, i.e.highlight L1, and press CLEAR, similarly L2, etc.

 

          Then enter data x values into L1 and y values in L2

 

Viewing a Scatter Plot

            2nd – STAT PLOT  (the Y= key or above the 2nd key)

            1:   ENTER,   On  ENTER  ¯ENTER   ¯L1  ¯L2   -  (this defines Plot 1 as a scatter plot L1 & L2)

           Make sure that in Y=, all the functions are cleared out (or turned off)

           ZOOM-9:ZoomStat  will show a Scatter plot of the data

 

Getting the Regression Equation

            STAT      ®  CALC  ¯¯¯ to B:Logistics or 9:LnReg  ENTER  ?ENTER

            Record the equation indicated for the values of a and b, and record r

 

Graphing the Regression Equation

            In Y=, enter the regression equation, and then graph.  It should graph the function over the scatterplot points.

 

***To turn off ALL Plots ***  -  2nd – STAT PLOT -  4:PlotsOff  -   ENTER

 

Example:  U.S. Population Model

The following data obtained from the U.S. Census Bureau represent the population of the United States.  An ecologist is interested in finding a function that describes the population of the United States.

Year

1900

1910

1920

1930

1940

Population

76,212,168

92,228,496

106,021,537

123,202,624

132,164,569

 

1950

1960

1970

1980

1990

151,325,798

179,323,175

203,302,031

226,542,203

248,709,873

 

a)  Let t = 0 represent 1900.  Enter the years into L1 and population into L2.  Define, using STAT PLOT, Plot 1, as a scatter plot on L1 and L2.  Press Zoom:9 to see the  scatter plot … sketch it on paper.

b)  STAT ®CALC  ¯¯ -B:Logistic ENTER, (wait patiently …, it’s VERY slow!!) gives

f(x) = 695129657/(1+7.9105 e-0.0166223kx)    Enter this function into Y1, and graph over the points on the scatter plot.

**Note:  A shortcut to typing the function into Y1 is in the Y= screen, at the Y1= location, paste in the function by pressing the following sequence of keys:   VARS   5:Statistics  ®®EQ 1: RegEq  ?ENTER

 

b)  Use the function to predict the population of the U.S. in 2000, this years census. (I got 277,967,471).  I wonder how this year’s census will compare to this projection???

 

c)  If this model continues, when will the population of the U.S. reach 300 million?  Let Y2=300000000.  Extend the WINDOW so that Ymax is over 300,000,000, and Xmas is 120.  Graph … and find the intersection point – t = 107.9 … so approximately in 2007.

 

Now Here’s the Project!!

 

**Remember to clear out the functions in Y= between problems**

 

I.  IBM Computer Prices

 

The following data represents the price and quantity demanded in 1997 for IBM personal computers at Best Buy. 

 

Price ($/Computer)

2300

2000

1700

1500

1300

1200

1000

Quantity Demanded

152

159

164

171

176

180

189

 

a)  Sketch the scatter plot of the points.

b) Use your calculator (as described above) to find a logarithmic model (STAT ® CALC ¯¯ 9:LnReg) that fits the data.

c)  Enter the function into Y1 and graph over the scatter plot … and sketch on paper.

d)  Use the model to predict the number of IBM personal computers that would be demanded if the price were $1650.

 

II.  World Population Model

The following data obtained from the U.S. Census Bureau represent the world population.  An ecologist is interest in finding a function that describes the world population, in billions.  Let t = 0 be 1980.

Year

1981

1982

1983

1984

1985

1986

1987

1988

Population

4.533

4.614

4.695

4.775

4.856

4.941

5.029

5.117

 

1989

1990

1991

1992

1993

1994

1995

5.205

5.295

5.381

5.469

5.556

5.644

5.732

 

a)   Sketch the scatter plot of the points.

b)  Use your calculator to find a logistics model of the data.(STAT® CALC ¯¯ B:Logistic)

                               ***remember it’s very slow!!

c)  Enter the function into Y1 and graph over the scatter plot … and sketch on paper

d)  What world population would this model predict for the year 2000?

e)  When would this model predict that the world population will hit 6.500 billion?

 

II.  Walking Speed

Various studies have found a correlation between the size of a city and the average walking speed of pedestrians.  One such study obtained the following data.

Population

 5,500

 14,000

 71,000

 138,000

 342,000

Velocity (ft/sec)

  3.3

  3.7

  4.3

  4.4

  4.8

a) Use your calculator to find a logarithmic model that fits the data.

b)  Enter the function into Y1 and graph over the scatter plot … and sketch on paper.

c)  Use the model to predict the walking speed in Little Rock, Arkansas (population 176,000), New York (population 7,300,000) and Mexico City (population 20,000,000).