Algebra Review

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Algebra Review

• The equation of a straight line
• y mx b
• m is the slope the change in y over the change
  in x or rise over run.
• b is the y-intercept the value where the line
  cuts the y axis.

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Review

• y 3x 2
• x 0 y 2 (y-intercept)
• x 3 y 11
• Change in y (9) divided by the change in x (3)
  gives the slope, 3.

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Linear Regression

• Example Tar (mg) and CO (mg) in cigarettes.
• y, Response CO (mg).
• x, Explanatory Tar (mg).
• Cases 25 brands of cigarettes.

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Correlation Coefficient

• Tar and nicotine
• r 0.9575

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Linear Regression

• There is a strong positive linear association
  between tar and nicotine.
• What is the equation of the line that models the
  relationship between tar and nicotine?

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Linear Model

• The linear model is the equation of a straight
  line through the data.
• A point on the straight line through the data
  gives a predicted value of y, denoted .

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Residual

• The difference between the observed value of y
  and the predicted value of y, , is called the
  residual.
• Residual

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Residual
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Line of Best Fit

• There are lots of straight lines that go through
  the data.
• The line of best fit is the line for which the
  sum of squared residuals is the smallest the
  least squares line.

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Line of Best Fit
Least squares slope intercept
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Summary of the Data
Tar, x
CO, y
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Least Squares Estimates
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Interpretations

• Slope for every 1 mg increase in tar, the CO
  content increases, on average, 0.801 mg.
• Intercept there is not a reasonable
  interpretation of the intercept in this context
  because one wouldnt see a cigarette with 0 mg of
  tar.

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Predicted CO 2.743 0.801Tar
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Prediction

• Least squares line

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Residual

• Tar, x 16.0 mg
• CO, y 16.6 mg
• Predicted, 15.56 mg
• Residual, 16.615.56
• 1.04 mg

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Residuals

• Residuals help us see if the linear model makes
  sense.
• Plot residuals versus the explanatory variable.
• If the plot is a random scatter of points, then
  the linear model is the best we can do.

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Interpretation of the Plot

• The residuals appear to have a pattern. For
  values of Tar between 0 and 20 the residuals tend
  to increase. The brand with Tar 30, appears to
  have a large residual.

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(r)2 or R2

• The square of the correlation coefficient gives
  the amount of variation in y, that is accounted
  for or explained by the linear relationship with
  x.

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Tar and Nicotine

• r 0.9575
• (r)2 (0.9575)2 0.917 or 91.7
• 91.7 of the variation in CO content can be
  explained by the linear relationship with Tar
  content.

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Regression Conditions

• Quantitative variables both variables should be
  quantitative.
• Linear model does the scatter plot show a
  reasonably straight line?
• Outliers watch out for outliers as they can be
  very influential.

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Regression Cautions

• Beware of extraordinary points.
• Dont extrapolate beyond the data.
• Dont infer x causes y just because there is a
  good linear model relating the two variables.
• Dont choose a model based on R2 alone.