3'3: Least Squares Regression Line

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3.3 Least Squares Regression Line

• You have heard of this referred to as the Best
  fit line.
• However, when you eyeball a best fit line,
  there are many different ways to draw it.
• Regression Line describes how a response
  variable changes as the explanatory variable
  changes. Can be used to predict the value of y
  for a given x.

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Regression Lines

• We will use y a bx for regression lines.
• a y-intercept.
• b slope the amount the y value changes for
  every one change in x.
• Extrapolation using a regression line to predict
  values outside the range of explanatory values
  often is misleading or inaccurate.
• Prediction uses values within the range and is
  acceptable.

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Least Squares Regression Line

• Our goal is to minimize the vertical distance
  from the line and our observed values.

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• If we just add up the vertical distances, most
  likely they will cancel each other out and we
  will not be 100 sure that we have the least
  possible error. (Think of a horizontal line.)
• The correct this we will minimize the sum of the
  squares of the vertical difference.
• This is called a Least-Squares Regression Line.
  (LSRL)

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Least Squares Regression Line
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• The mathematics behind finding the LSRL is long
  and complicated luckily the calculator can do
  the work for us, or a computer printout will give
  us the important details to find

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• Every LSRL is guaranteed to pass through the
  point (x-bar, y-bar)
• So, it is possible to find the LSRL from the
  summary statistics x-bar, sx, y-bar, sy, and r.

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• Ex. A study compares the amount of fidgeting
  measured in nonexercise activity or NEA vs. the
  fat gain in young adults. They find the average
  NEA to be x-bar 324.8 cal with standard
  deviation sx 257.66 cal. The mean weight gain
  is y-bar 2.388 kg with standard deviation sy
  1.1389. The two variables have a correlation of
  r -0.7786. Find the equation of the LSRL.

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• Ex. Use the following data to answer each
  question on the number of beers consumed and BAC.
• Find the summary statistics (r 0.89)

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• Find the equation of the LSRL and graph it.
• Predict the BAC for a student who consumes 6
  beers. Compare this to the observed value.

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• Describe the direction and strength of the
  relationship between of drinks and BAC.
• Interpret the slope and y-intercept in the
  context of this problem.