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"BEST FIT" LINES

Being able to make predictions about the future based on evidence (data) from the past is an important skill in the worlds of business and science. Companies can plan how much they will budget for new equipment or emphoyees if they can see a steady growth in sales. Scientists can predict the results of experiments based on patterns they see in data. Linear relationships, data that seems to fall near a line, occur frequently in the real world. In this lesson, each student will graph the "best fit" line for a set of data and determine its equation.

MATERIALS

graph paper
ruler
TI-83 calculator

PROCEDURE

  1. Label the axes of your graph with the variables represented in the data and choose an appropriate scale based on the range of the data.
  2. Graph the coordinates of corresponding ordered pairs of data. This is a SCATTERPLOT.
  3. Using the TI-83, choose Stat, Edit, then enter the data for one variable in List 1 and the other variable in List 2. If you need to delete old information from the lists, highlight, at the top, the list you want to empty, press clear and enter.
  4. Choose List, Math, #3 mean ( , L1 (above #1 on the keypad), enter to find the average of your x-coordinates. Repeat using L2 to find the average of the y- coordinates. This point will be on your "best fit" line.
  5. Place your ruler on this point and position the ruler to locate a line that fits the slope of the data. Draw the line.
  6. Find the coordinates of any other point on your line.
  7. Now you are ready to write the equation for your line. Using the two points you have identified, follow the procedure in the previous lesson.

Example: The total amount (in millions of dollars) spent by the government on mathematics research from 1985 through 1995 is shown in the table. Find the "best fit" line and write its equation. Predict the amount to be spent in 2000. To center your graph, let x = 0 correspond with 1985.

Year 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995
Amount

$91

$118

$128

$130

$151

$184

$185

$205

$212

$240

$245

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The point for the average of the coordinates is (5, 171.72)

Another point on the line is (10,245)

slope = (245 - 171.72) / (10 - 5) = 14.7

equation is:

y - 245 = 14.7(x - 10)
y - 245 = 14.7x - 147
y = 14.7x + 98

Now you try these:

  1. A store decided to monitor its sales of VCRs as related to their price. They randomly chose 15 weeks from the past year. The data is shown in the table below. Make a scatterplot, draw the "best fit" line and write its equation. How many VCRs should the store expect to sell if the price dropped to $250?

Week

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

Price

350

360

365

325

395

400

330

335

365

390

375

350

390

400

425

#sold

45

35

30

54

25

18

35

50

45

20

45

50

19

20

17
  1. The data in the table shows the age (in years), and the corresponding height (in inches), for a young man from age 2 to age 19. Find the "best fit" line and write the equation.
Age

2

3

6

8

10

12

14

15

17

18

19
Height

28

33

40

46

52

55

61

64

70

72

72
  1. Use the data found in the Average Velocity Lab in Physics class to graph and write the equation for the "best fit" line.
  2. Do you think you could get a "best fit" line by using any 2 consecutive data points? What if you used the first and last data points? Write a few sentences explaining why or why not? Why do you think we used the average of the data points for one point on our line?

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