MATH 105: Fitting Lines to Data
Name:
Names of people you worked with:
Work in groups of four to fill in answers to the questions
below. Each group member should contribute equally to the
analysis. This will count as one quiz grade.
The purpose of this exercise is to analyze data describing a linear
relation between two variables. A calculator or computer will be used
to find the line that best fits the data. This describes a
mathematical model for the relationship between the two variables.
Mathematical models are used to make predictions or guesses about
values that aren't known for sure.
Below is a list of Olympic years and the best
height of the gold medalist pole vaulter (Source: Precalculus:
Mathematics for Calculus by Stewart, Redlin and Watson).
To model the way winning
heights change over time, we use the date as a
dependent variable, or x-coordinate, and the height as our
independent variable, or y-coordinate. Use your calculator or
computer to plot the points below. Note that you need to adjust the veiwing
window to accomodate the domain and range of the data -- otherwise your graph will appear blank!
| Year |
1900 |
1912 |
1924 |
1936 |
1948 |
1960 |
1972 |
1984 |
1996 |
| Height (m) |
3.30 |
3.95 |
3.95 |
4.35 |
4.30 |
4.70 |
5.64 |
5.75 |
5.92 |
Use your calculator to find and graph a line that fits the data or do
this using Graphmatica:
- Choose "Options" from the Data Plot pannel.
- Select Equation Type: Polynomial.
- Set "Maximum order of polynomial" to 1 (to fit a line rather than a curve to the data).
- Click OK to close the Global Settings window.
- Choose "Curve Fit" in the Data Plot pannel.
- Read the equation of the line shown below the graph in Graphmatica.
You should find that the line of best fit has an equation like y =
0.0226x - 39.44. Your computer or calculator has used a program to
determine that this line is the closest it can find to the points
you've plotted.
Why do we care what line lies closest to the data? The line
forms the basis of a model describing the relationship between
year and winning pole vault height. We can use this model to predict
winning pole vault heights for other years.
- Evaluate the expression 0.0226x - 39.44 for the year x = 2004 to
find out what winning height our model predicts for the year 2004.
- The actual winning height for 2004 was 5.95 meters.
Was the prediction accurate?
- What winning height does the model predict for the year 1972? Is
this prediction close to the actual height of 5.64 meters?
- What winning height does the model predict for the year 2008? Do
you think this will be close to the actual height? Why or why not?
- What winning height does the model predict for the year 3000? Do
you think this will be close to the actual height? Why or why not?
We now compare the length of our
classmates' forearms to their heights; this experiment is described in Section 3.1 of your text.
- In the table below, record the forearm length (distance from
wrist to elbow) and height, in inches, of each member of your group.
Then type your data into the computer at the front of the room.
| Forearm length (in) |
|
|
|
|
| Height (in) |
|
|
|
|
- Once the entire class has entered their data, use Graphmatica or a calculator to find the equation of the line
that best fits the data entered. This equation describes the relationship between your classmates' forearm
length and height. Write the equation below.
- Professor Burgiel's forearm is about 9.6 inches long. Use your
equation to guess her height.
- You may have wondered why you had to wait for the entire class to enter their data before
computing a linear model. To find out, use the data shown below to predict Professor Burgiel's height.
Why does this data (collected by real students!) yield such an inaccurate prediction?
| Forearm length (in) |
11 |
10.4 |
| Height (in) |
62 |
71 |
- In most cases, the more data you use when computing a linear model, the better your model.
Can you think of any situations in which you could improve your model by leaving out some of your data?