Suppose that you are working for an automobile manufacturing company. The company president (your instructor) wants to know the following:

1. The approximate number of new passenger cars that will be sold in the United States
    in the year 2010.

2. The year in which the sale of new passenger cars will decline to 8 million.

3. The average decrease in the number of new passenger cars sold in the United States
     per year.

You will be assigned a group to complete this report. Each group will present its results to the company president and the board of directors (the class) in a 10 minute or less talk.

    The U.S. Bureau of the Census releases information about the United States in an annual publication called the Statistical Abstract of the United States. This publication is available both online and in print form and contains information gathered from a variety of sources, including the American Automobile Manufacturers Association and Ward's Communications. Using information from one of these sources, find the latest data available on new-car sales.

Part I

Complete the following exercises to help you justify your answers to the preceding questions:

1. Use data from the lastest four years. Let x = the number of years after the earliest of the
    four years and N(x) = the number of new passenger cars. (For example, if the earliest year is
    1995, then x = 0 for 1995, x = 1 for 1996, and x = 2 for 1997.) Complete the table of values
    for the independent variable x and the dependent variable N(x).

2. Graph the four coordinate pairs found in the table in exercise 1. Do the points appear to lie
    on a straight line?

3. Using the four data points two at a time, we can determine six different pairs of points.
    List the six pairs.

4. Write an equation of a line through each pair of points. Name the six equations as N₁(x),
    N₂(x), N₃(x), N₄(x), N₅(x), and N₆(x).

5. Graph N₁(x). On the same graph, plot and label the four coordinate points in exercise 1.

6. Complete the following table. Note that column three is determined by substituting values
    of x in column one into the function N₁(x).
 
 

Number of years after _____ x	Number of new automobiles in thousands N(x)	Estimated number of new automobiles	Difference between estimate and given value
		Total Difference

7. Repeat exercises 5 and 6 five times replacing N₁(x), with N₂(x), N₃(x), N₄(x), N₅(x),
    and N₆(x).

8. Compare the total differences from the six tables. The equation that results in the least total
    difference may be considered the line of best fit. Statisticians use a similar, but more
    complicated, process to determine a line of best fit. State the equation that you determined
    to be the line of best fit.

9. Use the equation from exercise 8 to predict the answer to the president's questions.

Part II

The TI-83 Plus has built-in statistical features to find the best-fitting line, called a linear regression line. In 5.4 Calculator Exercises, we used this feature to write an equation for two points. In order to use this feature for more than two points, you will need to enter the values of x in L1 and the corresponding values of N(x) in L2. Be sure that the pairs of numbers match in the two lists. Then calculate LinReg (ax + b).

1. Write an equation for N₇(x) by using this statistical feature.

2. Repeat exercise 6 in Part I replacing N₁(x) with N₇(x).

3. Would you consider N₇(x) to be a better fit to the data than the equation found in Part I?
    If so, why?

4. Use the information that you have gathered to predict the answers to the president's questions.

Part III

In this project we will use the TI-83 Plus with a Ranger program and a Claculator Based Ranger
(CBR) to collect data involving the relationship of distance walked with respect to time walked.

In order to run the Ranger program,

a. Connect the CBR to the TI-83 Plus.

b. Under programs select RANGER.

c. Under the MAIN MENU select 2: SETUP/SAMPE.

d. With the up or down arrow key select a line and press ENTER to change the settings. When the
    settings are correct, arrow to the top of the page and press ENTER to start. The screen settings
    should read as shown:

    REALTIME: YES
    TIME(S): 15
    DISPLAY:DIST
    BEGIN ON: [ENTER]
    SMOOTHING: NONE
    UNITS: FEET

e. Press ENTER to start the program.

f. Place the CBR on the table. On the floor mark a distance in front of the CBR in feet beginning
   with 3 feet for a distance of 20 feet.

g. When you are ready to collect your data, press ENTER again to start the data collection. The
    CBR will begin to make clicking noises. The calculator will collect data points for 15 seconds.

h. The calculator will display a graph having time in seconds, T, on the horizontal axis and
    distance in feet, D, on the vertical axis.

i. When you are ready to collect additional data, press ENTER and choose 5 for REPEAT
   SAMPLE. You will need one person designated as a walker and one person to control the
   calculator and CBR. For each of the given situations described below, complete parts a-e.

a. On your paper, set up a table of values to describe the situation. Use integer coordinates
    {0, 1, 2, ..., 15} for the independent variable T and determine the dependent variable D.

b. Graph the situation.

c. Preform the situation described using the CBR, the Ranger program, and your TI-83.

d. Compare your graph to the graph on your calculator.

e. Identify the slope of the line as either positive or negative and approximate its value. Determine
    the y-intercept of the graph. Write an equation for the line.

1. The walker should stand on the 3-foot line. When told to begin, walk away from the CBR at a
     constant rate of one foot per second for 15 seconds.

2. The walker should stand on the 20-foot line. When told to begin, walk toward the CBR at a
    constant rate of one foot per second for 15 seconds.

3. The walker should stand on the 5-foot line. When told to begin, the walker should remain on
     the 10-foot line for 15 seconds.

Let's see how well you can predict the graph. Sketch a graph for the following situations without
using the CBR. Then preform the situation described and check your sketch with the calculator graph.

4. The walker should stand on the 10-foot line. When told to begin, walk at a constant rate of
    one foot per second toward the CBR for 5 seconds, remain at this position for 5 seconds,
    and the walk at a constant rate of one foot per second away from the CBR.

5. The walker should stand on the 3-foot line. When told to begin, walk at a constant rate of two foot
    per second away from the CBR for 5 seconds, remain at this position for 5 seconds, and then
    continue to walk at a constant rate of one foot per second away from the CBR.