Understanding Best Fit Linear Regression

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Understanding Best Fit Linear Regression

• Dr. David Thomas
• Centenary College
• thomas_at_centenary.edu

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• Consider the graph pictured to the right. For
  the seven data points shown, which of the lines
  pictured is the line of best fit?
• Line A
• Line B
• Line C
• The line of best fit is not pictured.

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• The table to the right contains data on the
  amount of money spent on toys and sport supplies
  in the US from 1990 through 1995. Use this data
  to find the line of best fit.

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Results from Summer Program

• 14 students were given the preceding two
  questions at the beginning of three weeks of
  summer training and the same two questions at the
  end of the training.
• They scored as follows
• Question 1 Pretest 3 correct, Posttest 2
  correct
• Question 2 Pretest 4 correct, Posttest 10
  correct

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Regression Analysis

• Regression analysis is concerned with finding a
  mathematical model or formula that relates the
  values of one variable to those of another.
• Consider the box score data from Game 6 of the
  2006 NBA Western Conference Finals on the
  following slide.

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• Consider the third column in the table (FG and
  FGA). This represents two variables, field goals
  made and field goals attempted.
• Of the two variables, field goals made and field
  goals attempted, which is independent and which
  is dependent?
• Plot field goals attempted on the x-axis and
  field goals made on the y-axis. Plot one point
  for each player.

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We want a formula that relates field goals made
to field goals attempted. Using your data draw a
line which you thinks best fits this data.
Find the equation of your line.
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Criterion Used

• What criteria did you use to pick your line?

One common measure of best fit is to make the
sum of the squares of the differences between the
actual y-values and the predicted y-values as
small as possible. (Hence the name least squares
fit)
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Calculating the Sum of the Square of the
Differences

• Place the equation you thinks best fits the data
  in Y1.
• Enter the data for field goals attempted in L1.
• Enter the data for field goals made in L2.
• Calculate the square of the differences between
  the actual and predicted values by entering
  sum((L2 Y1(L1))2). The command sum is on the
  MATH submenu of the LIST key, which is 2nd over
  the STAT key.

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Using the Calculator to Find the Best Fit Line

• Use your calculator to find the linear regression
  for the actual data. (Option 4 off of the CALC
  submenu of the STAT key.) Enter this equation in
  Y2. Calculate the sum of the square of the
  differences as you did on the previous slide to
  find the minimum sum.
• Why do you think the difference between the
  actual and predicted values is squared?

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The sum of the square of the differences is 5.76
This is the minimum value.
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The Theory Behind This
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• Consider the graph pictured to the right. For
  the seven data points shown, which of the lines
  pictured is the line of best fit?
• Line A
• Line B
• Line C
• The line of best fit is not pictured.

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Line C
Line A
Line B
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Correlation Coefficient

• The r-value listed with the results of the linear
  regression is called the correlation coefficient.
  It is used to measure how linear the actual
  data is. If all the data points are on the same
  line the r-value is 1 (positive correlation) or
  1 (negative correlation).

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• Using the calculator enter assists in L5 and
  rebounds in L6.
• Turn off all other plots and functions. Graph
  these points with L5 as the x-values and L6 as
  the y-values.
• Is there a correlation between these variables?
  Does the number of rebounds depend on the number
  of assists?
• Use your calculator to find the linear regression
  equation and the correlation coefficient.