1. Chapter 2 Examining Relationships

2.  Response variable measures outcome of a study (dependent variable)  Explanatory variable explains or influences changes in response variable (independent)  Practice: pg 81 #2.1, 2.2

3. 2.1 Scatterplots

4. Scatterplot  shows relationship between two quantitative variables measured on the same individuals  most common way to display relation between two variables  Example using calculator: pg 83 #2.4 powerboats vs. manatees killed

5. Examining a scatterplot  Look for overall pattern and deviations from the pattern  Describe overall shape by the form, direction, and strength of the relationship  Look for outliers

6. Clusters

7. Look for:  positively and negatively associated  strength is determined by how closely the points follow a clear form (use strong or weak)

8.  Adding categorical variables to scatterplots use different colors or symbols to plot points  Practice: pg. 89 #2.7

9. Section 2.1 practice problems: pg 91 #2.8, 9, 10, 11

10. 2.2 Correlation

11. Correlation  A linear relationship is strong if the points lie close to a straight line. Our eyes are not good judges of strength of linear relationship  measures the direction and strength of the linear relationship between two quantitative variables  equation:  to calculate on calculator: calculate with manatees

12. Correlation  r is the average product of z-scores  r is positive when there is a positive association  r requires that both variables be quantitative  r has no unit of measure its just a number

14.  r is strongly affected by outliers  r is used along with the mean and standard deviation to describe two- variable data

15. Section 2.2 practice problems pg 103 2.21, 24, 29

16. 2.3 Least-Squares Regression

17. least-squares regression line  regression line is a straight line that describes how a response variable (y) changes as an explanatory variable (x) (best fit line)  standard line that doesn’t depend on our guess as to where the line should go

18.  line that makes the sum of the squares of the vertical distances of the data points from the line as small as possible  to calculate the equation of regression line with calculator:

19. Facts about least-squares regression  the least-squares regression looks at distance from y to line (not x) so you cannot switch the x and y variables and get the same line  the least-squares regression line always passes through the point (x,y)  the square of the correlation is the proportion of data accounted for by the least-square regression line  when you report a regression, give r 2 as a measure of how successful the regression was in explaining the response  Practice problems: pg 114 #2.33, 34

20. Residuals  the difference between an observed value of the response variable and the value predicted by the regression line (y-ŷ)  to calculate residuals on calculator:  examining residuals helps assess how well the line describes the data  the mean of the residuals is always zero  residual plot is a scatterplot of the regression residuals against the explanatory variable to help assess the fit of a regression line  Make residual plot of manatees data

21. Things to look for in residual plot 1. Curved pattern – shows relationship is not linear 2. Increasing or decreasing spread – prediction of y when x is large will be less accurate 3. Individual points with large residuals – these points are outliers 4. Individual points that are extreme in the x direction

23. Consider the following data on x=height (in inches) and y=average weight (in pounds) for American females, age 30-39 X 58 59 60 61 62 63 64 65 66 67 Y 113 115 118 121 124 128 131 134 137 141 X 68 69 70 71 72 Y 145 150 153 159 164 Make scatterplot. Is the data linear? Plot residual plot.

25. Practice: pg 122 #2.36, 37 Section 2.3 practice: pg 124 # 2.42, 44, 45, 47, 48

26. 2.4 Cautions about Correlation and Regression

27. Extrapolation  use of a regression line for prediction far outside the range of values of the explanatory variable (x) that you used to obtain the line  often is not accurate

28. Using averaged data  average temp, average, salary, average age, etc.  do not apply results to individuals  correlations based on averages are usually too high when applied to individuals

29. Lurking variables  sometimes the relationship between two variables is influenced by other variables that were not measured or even thought about

30. Association is not causation  a strong association does not necessarily mean that x causes y  example: manatee deaths vs powerboat registrations

31. Section 2.4 practice: pg 136 2.57, 58, 59