Regression

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Regression
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Is there a relationship between two or more
quantitative variables?

• univariate data.
• bivariate data.
• Examples of bivariate data include
• Height and Arm Length
• Movie Theater Attendance and Concession Sales
• Diastolic and systolic blood pressure
• Elevation and air temperature
• There is typically one variable that is used to
  predict the other.
• The variable used to predict is the independent
  or explanatory or predictor variable and is
  represented by x.
• The variable we want to predict is the response
  or dependent variable. It is represented by y.

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Scatter Plots

• Graphing method for understanding the
  relationship between the two variables

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Is there a relationship between the amount of
mail at the post office and the number of man
hours needed to process the mail?
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Is there a relationship between the number of
establishments in a community that sell alcohol
and the number of DUI arrests? Data from Jean
Blackwell, Diem-Trang Tran, Liane Jitchaku,
6/8/01.
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Is there a relationship between Suspended Solids
in water and Turbidity?
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Notice the outlier.
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Bivariate Data Analysis

• scatter plot
• Correlation
• Determining if there is significant Correlation
• Coefficient of Determination (r2)
• The Least Squares Regression Line
• Analyzing the Residuals
• Possibly making a prediction of y based on x.

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Correlation

• The correlation shows the strength of the linear
  relationship between x and y.
• r is the sample correlation.
• r is the population correlation.
• Correlation is the covariance of x and y divided
  by the standard deviation of x and y.

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Correlation is the covariance of x and y divided
by the standard deviation of x and y.
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Correlation

• -1 r 1,
• where r 0 means no correlation
• r 1 or 1 means perfect positive or negative
  correlation.
• Correlation does not prove cause and effect.

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Determining if there is significant Correlation
Use Excel
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Coefficient of Determination

• It is common to report the r2 value. This value
  represents the proportion of the total variation
  in the y values that can be explained by the
  linear relationship between x and y.
• 0 r2 1.

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Least Squares Regression Line

• To model the relationship between x and y, we use
  the line that is known as the least squares
  regression line.
• Performing the regression is often stated as
  regress y on x.
• The slope of the line is given by b.
• The y-intercept is given by a.
• The difference between the predicted value of y
  and the actual value is called the residual (the
  vertical deviation). It represents the error
  in the prediction. The ith residual is observed
  response minus the predicted response .