Scatter-plot, Best-Fit Line, and Correlation Coefficient

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Scatter-plot, Best-Fit Line, andCorrelation
Coefficient
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Definitions

• Scatter Diagrams (Scatter Plots) a graph that
  shows the relationship between two quantitative
  variables.
• Explanatory Variable predictor variable
  plotted to the horizontal axis (x-axis).
• Response Variable a value explained by the
  explanatory variable plotted on the vertical
  axis (y-axis).

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Why might we want to see a Scatter Plot?

• Statisticians and quality control technicians
  gather data to determine correlations
  (relationships) between two events (variables). 
• Scatter plots will often show at a glance whether
  a relationship exists between two sets of data.
• It will be easy to predict a value based on a
  graph if there is a relationship present.

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Types of Correlations

• Strong Positive Correlation the values go up
  from left to right and are linear.
• Weak Positive Correlation - the values go up from
  left to right and appear to be linear.
• Strong Negative Correlation the values go down
  from left to right and are linear.
• Weak Negative Correlation - the values go down
  from left to right and appear to be linear.
• No Correlation no evidence of a line at all.

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Examples of each Plot
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How to create a Scatter Plot

• We will be relying on our TI 83 Graphing
  Calculator for this unit!
• 1st, get Diagnostics ON, 2nd catalog.
• Enter the data in the calculator lists.  Place
  the data in L1 and L2. STAT, 1Edit, type
  values in
• 2nd Y button StatPlot turn ON 1st type is
  scatterplot.
• Choose ZOOM 9 ZoomStat.

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Lets try one
SANDWICH Total Fat (g) Total Calories
Grilled Chicken 5 300
Hamburger 9 260
Cheeseburger 13 320
Quarter Pounder 21 420
Quarter Pounder with Cheese 30 530
Big Mac 31 560
Arch Sandwich Special 31 550
Arch Special with Bacon 34 590
Crispy Chicken 25 500
Fish Fillet 28 560
Grilled Chicken with Cheese 20 440
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The Correlation Coefficient

• The Correlation Coefficient (r) is measure of the
  strength of the linear relationship.
• The values are always between -1 and 1.
• If r /- 1 it is a perfect relationship.
• The closer r is to /- 1, the stronger the
  evidence of a relationship.

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The Correlation Coefficient

• If r is close to zero, there is little or no
  evidence of a relationship.
• If the correlation coef. is over .90, it is
  considered very strong.
• Thus all Correlation Coefficients will be
• -1lt x lt 1

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Salary with a Bachelors and Age
Age Salary (in thousands)
22 31
25 35
28 29.5
28 36
31 48
35 52
39 78
45 55.5
49 64
55 85
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Find the Equation and Correlation Coefficient

• Place data into L1 and L2
• Hit STAT
• Over to CALC.
• 4Linreg(axb)
• Is there a High or Low, Positive or Negative
  correlation?

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Movie Cost V.Gross (millions)
TITLE COST U.S. GROSS
1. Titanic (1997) 200 600.8
2. Waterworld (1995) 175 88.25
3. Armageddon (1998) 140 201.6
4. Lethal Weapon 4 (1998) 140 129.7
5. Godzilla (1998) 125 136
6. Dante's Peak (1997) 116 67.1
7. Star Wars I Phantom Menace (1999) 110 431
8. Batman and Robin (1997) 110 107
9. Speed 2 (1997) 110 48
10. Tomorrow Never Dies (1997) 110 125.3
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Finding the Line of Best Fit

• STAT ? CALC 4 LinReg(axb)
• Include the parameters L1, L2, Y1 directly after
  it.
• (Y1 comes from VARS ? YVARS, Function, Y1)
• Hit ENTER the equation of the Best Fit comes up.
  Simply hit GRAPH to see it with the scatter.

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Using the Best-Fit Line to Predict.

• Once your line of Best fit is drawn on the
  calculator, it can be used to predict other
  values.
• On the TI-83/84
• 2nd Calc
• 1Value
• x place in value

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Hypothesis Testing

• Is there evidence that there is a relationship
  between the variables?
• To test this we will do a
• TWO-TAILED t-test
• Using Table 5 for the level of Significance, and
  d.f. n 2 degrees of freedom.
• Compare the answer from the following formula to
  determine if you will REJECT a particular
  correlation.

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TI-83/84 HELP
                TI Regression ModelsRules for a ModelDiagnostics OnCorrelation CoefficientCorrelation Not CausationResiduals and Least Squares Graphing ResidualsLinear RegressionLinear Regression w/ Bio DataExponential RegressionLogarithmic RegressionPower Regression