MATH 100: 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 use data to model a linear relation between two variables and then to investigate the validity of the model. Data points will be plotted, then a calculator or computer will be used to find the line that best fits the data. This line describes a mathematical model describing the relationship between the two variables. We conclude by using the model to make predictions and analyzing those predictions.

Shown below is a list of Olympic years and the best height of the gold medalist pole vaulter. 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. If we can find an equation that outputs something close to the winning height for the date input, we can use this mathematical model to predict future winning heights.  We do this by plotting the data we have, drawing a line which lies close to those data points, then finding the equation of the line.

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 or computer (in Graphmatica, View->Data Plot Editor) to plot the data presented above. Be sure to adjust the grid range (View -> Grid Range) of your display so that you can see the points plotted!

  1. The winning heights increase over time. Imagine drawing a line that describes this trend. Give the coordinates of two points (year, height) that lie close to this line.

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  2. Use the point-slope formula for the equation of a line to find the equation of a line connecting these two points.

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  3. Evaluate this equation for the year x=1960 to find out what winning height the model predicts for the year 1960.

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  4. Was the prediction accurate? Why or why not?

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It's hard to get a good mathematical model by looking at a computer screen and guessing. Luckily, your computer is programmed with some equations that help it find a good model. To find and graph a line that fits the data using Graphmatica, follow these instructions: You should find that the line of best fit has an equation like y = 0.0264x - 48.84. This model should do a better job of predicting winning pole vault heights than the previous one did.
  1. What equation did Graphmatica give you for the line of best fit?

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  2. Evaluate your equation for the year x = 1960 to find out what winning height this model predicts for the year 1960.

     

     

     

  3. Compare this prediction to the one you got from your previous equation. Does it seem more accurate?

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  4. 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?

     

     

     

  5. 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?

     

     

     

  6. Describe the domain of your mathematical model. In other words, for what years do you think your model will give an acceptable prediction of the winning pole vault height?

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The taller you are, the longer your arms tend to be. Can you use a person's forearm length to predict his or her height? We'll now try to build a mathematical model that does exactly this.
  1. In the table below, record the forearm length (distance from wrist to elbow) and height, in inches, of each member of your group. Then copy your data into the computer or onto the whiteboard at the front of the room.

    Forearm length (in)        
    Height (in)        

  2. Once the entire class has submitted their data, use Graphmatica or a calculator to find the equation of the line that best fits the class' data.  This equation describes the relationship between your classmates' forearm length and height. Write the equation below.

     

  3. Professor Burgiel's forearm is about 9.6 inches long. What does your equation predict her height will be?  Do you think this is close to her actual height?