1. Statistics 400 - Lecture 20

2. zLast Day: Experiments zToday: Bi-Variate Data

3. Relationship Between Quantitative Variables zOften, we are concerned with describing the relationship between two quantitative (numeric) variables X and Y zObservations are paired in that they come from the same sampling unit ze.g., measure height (X) and weight (Y) of an individual zQuestions of interest: yAre the variables related? yWhat is the form of the relationship? yWhat is strength of relationship? yDoes one variable help predict another?

4. zSometimes interested in whether changes in one variable (explanatory variable) impacts the value of the other variable (response variable) zTraditionally, the explanatory variable is denoted X and the response variable is denoted, Y

5. Example zAccording to an article in The Economist (Jan. 30, 1988), there was a noticeable increase in the number of Americans dying in the summer following the Chernobyl nuclear accident zA radioactive plume arrive in the North America 11 days after the accident zMeasure of radioactivity in rain water went up 1000 fold and this was later detected in milk samples…common way to measure radioactivity

6. Example (cont.) zData: The table below gives the average radioactivity in milk samples and the percent increase in the number of deaths for 9 regions of the U.S.A.

7. Scatter-Plots zRelationship between quantitative variables can be displayed graphically using a scatter-plot zHave a sample of n observations: (x 1,y 1 ), (x 2,y 2 ),…, (x n,y n ) zPoint is plotted at position fixed by individual's X and Y values

9. Interpretation zInterpret scatter-plots in terms of: yOverall form yDirection of association yStrength of Association yDeviations from Overall Pattern

10. zPositive and Negative Associations: zLinear Association: zNo Association: zStrength of Association:

11. Strength of Linear Association zScatter diagram visualizes the nature of the relationship between X and Y zCorrelation Coefficient, r, measures strength of linear association zFor radioactivity example r =0.946

12. Properties of Correlation Coefficient zMeasure direction of linear association zMeasure strength of association zAlways between -1 and 1 zr close to 1 means zr close to -1 means zr close to 0 means

13. zr is effected by outliers zOnly useful for quantitative variables zCorrelation does not imply causation

14. Regression zWant to study relationship between an explanatory (independent, causal,...) variable, X, and a response (dependent) variable, Y zWe will be studying only linear relationships zThus first step is to do a scatter-plot to see if there relationship looks linear

15. Equations of Lines zWant to formally describe linear relationship between X and Y zHow to we write the equation of a straight line? zWhat do the components mean? zIn practice, at a given value of X, will the same Y always occur?

16. Regression Model zIf Y is the response variable and X is the explanatory variable, the linear regression model is: zwhere is the unknown error and has a distribution

17. zWe do not know the true values for the slope and intercept zWant estimate them from the data zWant best “fitting” line zWhat is meant by best fitting?

18. Least Squares Regression zThe least squares regression line is line that makes sum of squared vertical distances of data points from line as small as possible zIdea: Minimize the observed errors in prediction

20. Estimators for the Slope and Intercept zThe plot on the previous page shows the least squares regression line through the data zHow was the slope and intercept determined? zLeast Squares Estimate of the Slope: zLeast Squares Estimator of the Intercept:

21. zEstimated Regression Line: zIn our previous example, the estimated regression line is:

22. zHow would we use the line?