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Last Time. T Distribution Confidence Intervals Hypothesis tests Relationships Between Variables Scatterplots (visualization) Aspects of Relations Form Direction Strength. Reading In Textbook. Approximate Reading for Today’s Material: Pages 101-105 , 447-465, 511-516

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  1. Last Time • T Distribution • Confidence Intervals • Hypothesis tests • Relationships Between Variables • Scatterplots (visualization) • Aspects of Relations • Form • Direction • Strength

  2. Reading In Textbook Approximate Reading for Today’s Material: Pages 101-105 , 447-465, 511-516 Approximate Reading for Next Class: Pages 110-135, 560-574

  3. Scatterplot E.g. Class Example 16: How does HW score predict Final Exam? xi = HW, yi = Final Exam • In top half of HW scores: Better HW  Better Final

  4. Important Aspects of Relations • Form of Relationship • Direction of Relationship • Strength of Relationship

  5. I. Form of Relationship • Linear: Data approximately follow a line Previous Class Scores Example http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg16.xls Final vs. High values of HW is “best” • Nonlinear: Data follows different pattern Nice Example: Bralower’s Fossil Data http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg17.xls

  6. Bralower’s Fossil Data http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg17.xls From T. Bralower, formerly of Geological Sci. Studies Global Climate, millions of years ago

  7. II. Direction of Relationship • Positive Association X bigger  Y bigger • Negative Association X bigger  Y smaller Note: Concept doesn’t always apply: Bralower’s Fossil Data

  8. III. Strength of Relationship Idea: How close are points to lying on a line? Revisit Class Scores Example: http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg16.xls

  9. Comparing Scatterplots Additional Useful Visual Tool

  10. Comparing Scatterplots Additional Useful Visual Tool: • Overlaying multiple data sets

  11. Comparing Scatterplots Additional Useful Visual Tool: • Overlaying multiple data sets • Allows comparison

  12. Comparing Scatterplots Additional Useful Visual Tool: • Overlaying multiple data sets • Allows comparison • Use different colors or symbols

  13. Comparing Scatterplots Additional Useful Visual Tool: • Overlaying multiple data sets • Allows comparison • Use different colors or symbols • Easy in EXCEL (colors are automatic)

  14. Comparing Scatterplots HW HW: 2.21, 2.25

  15. III. Strength of Relationship Idea: How close are points to lying on a line? Revisit Class Scores Example: http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg16.xls

  16. III. Strength of Relationship Idea: How close are points to lying on a line? Now get quantitative

  17. Section 2.2: Correlation Main Idea: Quantify Strength of Relationship

  18. Section 2.2: Correlation Main Idea: Quantify Strength of Relationship Context: • A numerical summary

  19. Section 2.2: Correlation Main Idea: Quantify Strength of Relationship Context: • A numerical summary • In spirit of mean and standard deviation

  20. Section 2.2: Correlation Main Idea: Quantify Strength of Relationship Context: • A numerical summary • In spirit of mean and standard deviation • But now applies to pairs of variables

  21. Section 2.2: Correlation Main Idea: Quantify Strength of Relationship Specific Goals

  22. Section 2.2: Correlation Main Idea: Quantify Strength of Relationship Specific Goals: • Near 1: for positive relat’ship & nearly linear

  23. Section 2.2: Correlation Main Idea: Quantify Strength of Relationship Specific Goals: • Near 1: for positive relat’ship & nearly linear • > 0: for positive relationship (slopes up)

  24. Section 2.2: Correlation Main Idea: Quantify Strength of Relationship Specific Goals: • Near 1: for positive relat’ship & nearly linear • > 0: for positive relationship (slopes up) • = 0: for no relationship

  25. Section 2.2: Correlation Main Idea: Quantify Strength of Relationship Specific Goals: • Near 1: for positive relat’ship & nearly linear • > 0: for positive relationship (slopes up) • = 0: for no relationship • < 0: for negative relationship (slopes down)

  26. Section 2.2: Correlation Main Idea: Quantify Strength of Relationship Specific Goals: • Near 1: for positive relat’ship & nearly linear • > 0: for positive relationship (slopes up) • = 0: for no relationship • < 0: for negative relationship (slopes down) • Near -1: for negative relat’ship & nearly linear

  27. Correlation - Approach Numerical Approach

  28. Correlation - Approach Numerical Approach: for symmetric around

  29. Correlation - Approach Numerical Approach: for symmetric around has similar properties

  30. Correlation - Approach Numerical Approach: for symmetric around has similar properties Worked out Example : http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg18-new.xls

  31. Correlation – Graphical View Plots (a) & (b): illustrating : • > 0 for positive relationship

  32. Correlation – Graphical View Plots (a) & (b): illustrating : • > 0 for positive relationship

  33. Correlation – Graphical View Plots (a) & (b): illustrating : • > 0 for positive relationship • < 0 for negative relationship

  34. Correlation – Graphical View Plots (a) & (b): illustrating : • > 0 for positive relationship • < 0 for negative relationship

  35. Correlation – Graphical View Plots (a) & (b): illustrating : • Bigger for data closer to line

  36. Correlation – Graphical View Plots (a) & (b): illustrating : • Bigger for data closer to line

  37. Correlation – Graphical View But not all goals are satisfied

  38. Correlation – Graphical View Problem 1: Not between -1 & 1

  39. Correlation – Graphical View Problem 2: Feels “Scale”, see plot (c) (just 10  1 vertical rescaling of)

  40. Correlation – Graphical View Problem 2: Feels “Scale”, see plot (c) (just 10  1 vertical rescaling of) ( feels factor of 1/10)

  41. Correlation – Graphical View Problem 3: Feels “Shift” even more, see (d) (even gets sign wrong!)

  42. Correlation – Graphical View Problem 3: Feels “Shift” even more, see (d) (even gets sign wrong!) • Data trend upwards

  43. Correlation – Graphical View Problem 3: Feels “Shift” even more, see (d) (even gets sign wrong!) • Data trend upwards • But < 0

  44. Correlation - Approach Solution to above problems

  45. Correlation - Approach Solution to above problems: Standardize!

  46. Correlation - Approach Solution to above problems: Standardize! Define Correlation

  47. Correlation - Approach Solution to above problems: Standardize! Define Correlation

  48. Correlation - Example Revisit above example http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg18-new.xls • r is always same, and ~1, for (a)

  49. Correlation - Example Revisit above example http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg18-new.xls • r is always same, and ~1, for (a), (c)

  50. Correlation - Example Revisit above example http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg18-new.xls • r is always same, and ~1, for (a), (c), (d)

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