Activity 2.4 Line of Best Fit
Overview:
Two related discovery activities, one short and one more extensive, are given for finding line of
best fit to data.
Estimated Time Required:
Activity 2.4a requires 20 minutes of class time.
Activity 2.4b requires 30 minutes of class time and 15 minutes of out-of-class time if the Dow
Jones model is assigned.
Technology: A calculator that will compute linear regression equations
Prerequisite Concepts: Linear functions and their characteristics
Define Now:
 Define the “line of best fit” as the model that best predicts of the outputs for given inputs.
The line may contain some of the data points but it is not required to contain any of the
points.

With perfect correlation, the predicted output values are the same as the actual output
values. The correlation coefficient is a measure of how linear the data are.
Define Later:
 Interpolation of table values and extrapolation
Group Activity:
Have each student group complete the worksheet then present the following to the class:

a line of best fit for the data illustrated on a scatter plot

a justification of why they choose that line

a discussion the “goodness of fit” of their line
Class Discussion: Discuss how the models differ and how they are the same. (Aside: How
many models are possible is each model contains two points of the data? Answer 6. List them.)
Class Activity: After students explain their models, demonstrate to the class finding the linear
regression model on their calculators and have them enter the data and create the equation on
their calculators.
Lecture: Present interpolation of values for a table by finding interpolating and extrapolation to
find that when x = 7, y = 9.
Activity 2.4b Assignment: Assign Dow Jones data as an out-of-class activity
Activity 2.4a Fitting Linear Functions to Data
1. Plot the points (3, 1), (4, 3), (6, 6), and (8, 12) on the graph below.
2. Next, draw the line that “best fits” the points. Also, give an approximate equation of your line.
3. How did you decide what makes a line the “best fit” to a set of data points? Explain.
4. What is the equation of the linear regression line for these data found in
class?________________________________
5. How close was your line to the line in Question 4?
6. Using the linear regression equation, find approximates output value when the input is 4 and
when the input is 15.
7. Explain the difference between interpolation and extrapolation.
y
x
Activity 2.4b Line of best Fit
Consider the set of data point given below.
Table 1
x
y
3
1
4
3
6
6
8
12
1. Create the scatter plot of the data.
The following will be presented to the class by a group representative:
2. Create a linear function that you think will best predict output values for the given inputs by
using two points. Sketch the graph of your model on the scatter plot. Justify your choice of
linear model and be able to explain to the class why you decided on that model
3. Discuss how good a predictor your model is by finding the predicted values and comparing
them to the given values by finding each error. The difference y – y’ is called the error.
Table 2
x
y
Predicted y’
Error
y – y’
3
1
4
3
6
6
8
12
Predicting The Dow Jones Average
The table below represents an approximate average closing value of the Dow Jones for the
given years.
Year
‘80
‘83
‘85
‘86
‘88
‘90
DJ Close
750
1250
1500
1800
2100
2700
1. Create a scatter plot of the data. Let x = 0 represent the number of years since 1980.
2. Find the line of best fit for the data. Discuss what m and b represent.
3. State the value of the correlation coefficient r and discuss what it measures and signifies.
3. Predict the average Dow Jones closing value for 1984 by using interpolation of table values
and by using the regression model. Compare the two values.
Predict 1996 and 2002 by using the regression model. Can you use interpolation to find either
value? Why or why not?
Compare each predicted value with the actual values of 1250, 5800, and 8900 respectively,
and state the errors.
Year t
‘80
‘83
Actual DJ Close y
750
1250
‘84
‘85
‘86
‘88
‘90
1500
1800
2100
2700
96
‘02
Predicted y’
Difference y – y’
4. Discuss how the predicted values for the years 1984, 1996, and 2002 are different from
each other. Which methods can be used in each case? List in order the predicted values in
which you have the most confidence.
5. Interpret how the actual closing value of 10,800 for the year 2000 relates to the predicted
value.