1. Ch 3 – Examining Relationships YMS – 3.1
Scatterplots

2. Some Vocabulary Response Variable Explanatory Variable Scatterplot
Measures an outcome of a study
AKA dependent variable
Explanatory Variable
Attempts to explain the observed outcomes
AKA independent variable
Scatterplot
Shows the relationship between two quantitative variables measured on the same individuals

4. Scatterplots Examining Drawing Categorical Variables
Look for overall pattern and any deviations
Describe pattern with form, strength, and direction
Drawing
Uniformly scale the vertical and horizontal axes
Label both axes
Adopt a scale that uses the entire available grid
Categorical Variables
Add a different color/shape to distinguish between categorical variables
Classwork p125 #3.7,
Homework: #3.16, 3.22 and 3.2 Blueprint

5. YMS – 3.2
Correlation

6. Correlation
Measures the direction and strength of the linear relationship between two quantitative variables

7. Facts About Correlation
Makes no distinction between explanatory and response variables
Requires both variables be quantitative
Does not change units when we change units of measurement
Sign of r indicates positive or negative association
r is inclusive from -1 to 1
Only measures strength of linear relationships
Is not resistant

8. Correlation Guessing Game
In Class Exercises
p146 #3.28, 3.34 and 3.37
Correlation Guessing Game
Homework
3.3 Blueprint

9. Least-Square Regression
YMS – 3.3
Least-Square Regression

11. Regression Regression Line LSRL of y on x
Describes how a response variable y changes as an explanatory variable x changes
LSRL of y on x
Makes the sum of the squares of the vertical distances of the data points from the line as small as possible
Line should be as close as possible to the points in the vertical direction
Error = Observed (Actual) – Predicted

12. LSRL
Equation of the LSRL
Slope Intercept

13. Coefficient of determination – r2
The fraction of the variation in the values of y that is explained by the least-squares regression of y on x
Measures the contribution of x in predicting y
If x is a poor predictor of y, then the sum of the squares of the deviations about the mean (SST) and the sum of the squares of deviations about the regression line (SSE) would be approximately the same.

14. Understanding r-squared: A single point simplification
Al Coons
Buckingham Browne & Nichols School
Cambridge, MA

15. y
Error w.r.t. mean model
Error eliminated by y-hat model
Proportion of error eliminated by Y-hat model
Error eliminated by y-hat model =
Error w.r.t. mean model
r2 = proportion of variability accounted for by the given model (w.r.t the mean model).

16. y ~ Error w.r.t. mean model Error eliminated by y-hat model
Proportion of error eliminated by Y-hat model
Error eliminated by y-hat model =
Error w.r.t. mean model = ~

17. Facts about Least-Squares Regression
Distinction between explanatory and response variables is essential
A change of one standard deviation in x corresponds to a change of r standard deviations in y
LSRL always passes through the point
The square of the correlation is the fraction of the variation in the values of y that is explained by the least-squares regression of y on x
Classwork: Transformations and LSRL WS
Homework: #3.39 and ABS Matching to Plots Extension Question (we’ll finish the others in class)

18. Residuals observed y – predicted y or
Positive values show that data point lies above the LSRL
The mean of residuals is always zero

19. Residual Plots
A scatterplot of the regression residuals against the explanatory variable
Helps us assess the fit of a regression line
Want a random pattern
Watch for individual points with large residuals or that are extreme in the x direction

20. Outliers vs. Influential Observations
An observation that lies outside the overall pattern of the other observations
Influential observation
Removing this point would markedly change the result of the calculation
Classwork: Residual Plots WS
Homework: p177 #3.52 and 3.61

21. Doctor’s for the Poor This will be graded for accuracy
Doctor’s for the Poor This will be graded for accuracy! Ch 3 Review p176 # , 3.56, 3.59, 3.69,