Examining Relationships

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Examining Relationships

• Residuals and Residual Plots

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Residuals
Press the Button and I will use Linear Regression
to tell your Future!! (or at least something
close to it!!)
Just like our friend ZOLTAR we can make
predictions using our Line of Best Fit.
However, do we know just how good our predictions
are? Would we be willing to put a lot of CASH
MONEY down to back them up?
Luckily, we have an indicator in statistics that
can help us decide the strength of our
predictions AND tell us if a line is the Best
Fit.
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Residuals

• Unless your r value is perfect, your predictions
  wont be
• A residual is the difference between the actual
  value and your predicted value

Residual

• Each value observed value has a residual
• The sum of the residuals is always 0 (or really,
  really close)
• -roundoff error when earlier values are
  rounded, the sum may not equal exacty 0

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Residual Plot

• Scatterplot of the residuals against the
  explanatory variable (x).
• Assess the fit of the regression line
• Does your plot show the line fits?

• Individual Points w/ Large Residuals Outliers
  in y
• Individual Points extreme in x Influential
  Points

Why use Residuals?
The residual plot describes how well a LINEAR
model fits our data
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Residual Plot on Calculator

• Plot the scatterplot of the data
• Find the least squares equation (LinReg yabx)
• Put the equation into Y1 and graph it

You can do 1 Variable Stats to on L3 to find out
if the residual sum is 0.

• In L3, You need to get the residuals (quickly)
• You will take the ys and subtract the predicted
  y
• Go to the top of L3 2nd Stat - RESID
• Press enter (Your calculator finds them for
  you!! YIPPEEE!!)

Now do a scatterplot with Xlist L1 and Ylist
L3 (residuals) The line in the middle is the
least squares line.
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Residuals On Calculator (Screenshots) By hand
practice?

• Run the GESSEL program

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Influential Point vs. Outlier

• Outlier observation that is outside overall
  pattern (out of whack in the Y direction)
• Influential Point observation that IF removed
  would dramatically change the result of least
  squares line and/or predictions (way out in the X
  direction)

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Influential Point vs. Outlier
Lets Change Child 19s test score from 121 to 85
and see what happens to the EQ and Graph
Notice the minimal change in the equation and
graph This is an example of why Child 19 is
considered an outlier. An outlier in y has a
minimal effect on the equation and subsequent
predicted values.
ORIGINAL
NEW
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Influential Point vs. Outlier
Lets Change Child 18s test score from 57 to 85
and see what happens to the EQ and Graph
Notice the dramatic change in the equation and
graph This is an example of why Child 18 is
considered an influential point. A point in the
extreme x can dramatically effect the position of
the least squares line.
ORIGINAL
NEW
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Residual

• Breaking Bridges
• Anscombe Discovery
• 46