1. 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

3. 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

4. Degrees of correlation

5. 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

6. Scatter Diagrams and Linear Correlation
Correlation coefficient (r)
Assesses strength and direction of linear relationship between x and y.
Unit less
-1≤ r ≤ 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)

7. Linear and non-linear correlations

8. Scatter Diagrams and Linear Correlation
r = 1 Σ( x-x y-y_) n 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.

9. 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,c For “c” use calculator to compute r

10. 4.2 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)

11. 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

12. 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

13. 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.

14. 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

15. 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 = about 49% of y is explained by LSLR

16. 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