scatter diagrams and linear correlation chapter 1-3 single variable data examples or two variables:...
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Scatter Diagrams and Linear Correlation• Chapter 1-3 single variable data • Examples or two variables: age of person vs. time to master cell phone task ,
grade point average vs. time studying, grade point average vs. time playing video games, amount of smoking vs. rate of lung cancer
• Scatter diagram: (x,y) data plotted as individual points– x – explanatory variable (independent)– y – response variable (dependent)
• Evaluate scatterplot data– y vs x values – shows relationship between 2
quantitative variables measured on the same individual
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Scatter Diagrams and Linear Correlation• Look at overall pattern
– Any striking deviation (outliers)?
• Describe by a) form (linear or curved) b) direction - positively associated +slope negatively associated – slope c) strength - how closely do points follow form
• Examples: age of person vs. time to master cell phone task , grade point average vs. time studying, grade point average vs. time playing video games, amount of smoking vs. rate of lung cancer
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Degrees of correlation
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Scatter Diagrams and Linear Correlation• Tips for drawing
scatterplot– Scale axis: intervals for
each axis must be the same; scale can be different for each axis
– Label both axis– Adopt a scale that uses
entire grid (do not compress plot into 1 corner of grid
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Scatter Diagrams and Linear Correlation• Correlation coefficient (r)
– Assesses strength and direction of linear relationship between x and y.
– Unit less– -1≤ r ≤ 1 r = -1 or 1 perfect correlation (all
points exactly on the line)– Closer to 1or -1; better line describes relationship;
better fit of data – r > 0 positive association at x, y – r < 0 negative association a x , y – x and y are interchangeable in calculating r– r does not change if either (or both) variables have unit
changes (inches to cm, or F to C)
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Linear and non-linear correlations
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Scatter Diagrams and Linear Correlation
• r = 1 Σ( x-x . y-y_) n-1 sx sy
• Using TI-83 ex p.129 (number of police vs. muggings)• Cautions : Association does not imply causation
– Lurking variables may play rate
– r only good for linear models
– Correlation between averages higher than between individual point.
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Scatter Diagrams and Linear Correlation
• Facts– No distinction between x and y variable. The
value of r is unaffected by switching x and y – Both x and y must be quantitative– Only good for linear relationships– Not resistant to outliers
• Correlation or r is not a complete description of 2-variable data, the x and y standard deviations and means should be included
• HW: p131 2,4,6,8 a,b,c, 10 a,b,c, 12 a,b,cFor “c” use calculator to compute r
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4.2 Least Squares Regression
• Least Squares Regression– Method for finding a line (best fit) that
summarizes the relationship between 2 variables a x (explanatory) and y (response)
– Use the line to predict value of y for a given x– Must have specific response variable y and
explanatory variable x (cannot switch like r)
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4.2 Least Squares Regression
• Least Squares Regression Line (LSRL) – Minimizes square of error (y-values) – Error = observed –predicted value
Σ(y-ŷ)2 (y actual value, ŷ is predicted value) (ŷ is called y hat)
– Line of y on x that makes the sum of the squares of data points to fitted line as small as possible
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4.2 Least Squares Regression
• LSRL Equation ŷ = a + bx• ŷ predicted value of y
• Slope b = r(sy/sx)
• y – intercept a = y – bx
• x and y are means for all x and y data, respectively and are on the LSLR (x, y)
• sy sx are std. deviations of x,y data
• r correlation
• ŷ predicted value of y
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4.2 Least Squares Regression
• TI-83 – enter data into L1, L2 (x,y)– Use STAT CALC , select #8:LinReg(a+bx) to
get the best fit required
• Slope: important for interpretation of data– Rate of change of y for each increase of x
• Intercept – may not be practically important for problems.
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4.2 Least Squares Regression
• Plot LSLR: using formula ŷ = a + bx find 2 values on the line.– (x1, ŷ1) and (x2, ŷ2) make sure x1 and x2 are
near opposite ends of the data
• Influential observations and outliers– Influential – extreme in the x-direction
if we remove an influential point it will affect the LSLR significantly
– Outliers – extreme in the y-direction does not significantly change the LSLR
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Coefficient of Determination
• r2 – coefficient of determination
• r – describes the strength and direction of a straight line relationship
• r2 - fraction of variation in values of y that is explained by LSRL of y on x
• r = 1, r2 = 1 perfect correlation 100% of the variation explained by LSRL
• r = 0.7, r2 = 0.49 about 49% of y is explained by LSLR
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Residuals • Residuals – difference between observed value
and predicted value– Residual = y –ŷ– Mean of least square residuals = 0
• Residual plots – scatterplot of regression residuals against explanatory variable (x)– Useful in accessing fit of regression line i.e. do we have
a straight line?
• Linear –uniform scatter• Curved indicates relationship not linear• Increasing/ decreasing indicates predicting of y
will be less accurate for larger x
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