1. Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute with TDIST & TINV (Recall different organization) (relative to NORMDIST & NORMINV)

2. Reading In Textbook Approximate Reading for Today’s Material: Pages 420-427, 86-94 Approximate Reading for Next Class: Pages 101-105, 447-465, 511-516

3. Deeper look at Inference Recall: “inference” = CIs and Hypo Tests Main Issue: In sampling distribution Usually σ is unknown, so replace with an estimate, s. For n large, should be “OK”, but what about: n small? How large is n “large”?

4. Unknown SD Then So can write: Replace by, then has a distribution named: “t-distribution with n-1 degrees of freedom”

5. t - Distribution Notes: 4.Calculate t probs (e.g. areas & cutoffs), using TDIST & TINV Caution: these are set up differently from NORMDIST & NORMINV

6. EXCEL Functions Summary: Normal: plug in: get out: NORMDIST: cutoff area NORMINV: area cutoff (but TDIST is set up really differently)

7. EXCEL Functions t distribution: Area 2 tail: plug in: get out: TDIST: cutoff area TINV: area cutoff (EXCEL note: this one has the inverse)

8. t - Distribution Application 1: Confidence Intervals

9. t - Distribution Application 1: Confidence Intervals Recall:

10. t - Distribution Application 1: Confidence Intervals Recall: margin of error

11. t - Distribution Application 1: Confidence Intervals Recall: margin of error from NORMINV

12. t - Distribution Application 1: Confidence Intervals Recall: margin of error from NORMINV or CONFIDENCE

13. t - Distribution Application 1: Confidence Intervals Recall: margin of error from NORMINV or CONFIDENCE Using TINV?

14. t - Distribution Application 1: Confidence Intervals Recall: margin of error from NORMINV or CONFIDENCE Using TINV? Careful need to standardize

15. t - Distribution Using TINV? Careful need to standardize

16. t - Distribution Using TINV? Careful need to standardize

17. t - Distribution Using TINV? Careful need to standardize

18. t - Distribution Using TINV? Careful need to standardize # spaces on number line

19. t - Distribution Using TINV? Careful need to standardize # spaces on number line Need to work in to use TINV

20. t - Distribution Using TINV? Careful need to standardize # spaces on number line Need to work in to use TINV

21. t - Distribution

22. distribution

23. t - Distribution distribution

24. t - Distribution distribution So want:

25. t - Distribution distribution So want: i.e. want:

26. t - Distribution Class Example 15, Part I http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg15.xls Old text book problem 7.24

27. t - Distribution Class Example 15, Part I http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg15.xls Old text book problem 7.24: In a study of DDT poisoning, researchers fed several rats a measured amount. They measured the “absolutely refractory period” required for a nerve to recover after a stimulus. Measurements on 4 rats gave:

28. t - Distribution Class Example 15, Part I http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg15.xls Old text book problem 7.24: Measurements on 4 rats gave: 1.6 1.7 1.8 1.9 a)Find the mean refractory period, and the standard error of the mean b)Give a 95% CI for the mean “absolutely refractory period” for all rats of this strain

29. t - Distribution Class Example 15, Part I Data in cells B9:E9

30. t - Distribution Class Example 15, Part I Data in cells B9:E9 Note: small sample size (n = 4)

31. t - Distribution Class Example 15, Part I Data in cells B9:E9 Note: small sample size (n = 4), population sd, σ, unknown

32. t - Distribution Class Example 15, Part I Data in cells B9:E9 Note: small sample size (n = 4), population sd, σ, unknown, so use sample sd, s

33. t - Distribution Class Example 15, Part I Data in cells B9:E9 Note: small sample size (n = 4), population sd, σ, unknown, so use sample sd, s, and t distribution

34. t - Distribution Class Example 15, Part I Data in cells B9:E9 Center CI at Sample Mean

35. t - Distribution Class Example 15, Part I Data in cells B9:E9 Center CI at Sample Mean Measure Sample Spread by S. D.

36. t - Distribution Class Example 15, Part I Data in cells B9:E9 Center CI at Sample Mean Measure Sample Spread by S. D. Divide by to get Standard Error

37. t - Distribution Class Example 15, Part I Data in cells B9:E9 Center CI at Sample Mean Measure Sample Spread by S. D. Divide by to get Standard Error Which answers (a)

38. t - Distribution Class Example 15, Part I (b) 95% CI for μ Data in cells B9:E9

39. t - Distribution Class Example 15, Part I (b) 95% CI for μ Data in cells B9:E9 CI Radius = Margin of Error

40. t - Distribution Class Example 15, Part I (b) 95% CI for μ Data in cells B9:E9 CI Radius = Margin of Error Compute using TINV

41. t - Distribution Class Example 15, Part I (b) 95% CI for μ Data in cells B9:E9 CI Radius = Margin of Error Recall: d.f. = n – 1 = 4 – 1

42. t - Distribution Class Example 15, Part I (b) 95% CI for μ Data in cells B9:E9 CI Radius = Margin of Error Compare to old Normal CIs

43. t - Distribution Class Example 15, Part I (b) 95% CI for μ Data in cells B9:E9 CI Radius = Margin of Error Compare to old Normal CIs Compute using CONFIDENCE

44. t - Distribution Class Example 15, Part I (b) 95% CI for μ Data in cells B9:E9 CI Radius = Margin of Error Compare to old Normal CIs T CIs are wider

45. t - Distribution Class Example 15, Part I (b) 95% CI for μ Data in cells B9:E9 CI Radius = Margin of Error Compare to old Normal CIs T CIs are wider (as expected)

46. t - Distribution Class Example 15, Part I (b) 95% CI for μ Data in cells B9:E9 CI Radius = Margin of Error Compare to old Normal CIs Left End

47. t - Distribution Class Example 15, Part I (b) 95% CI for μ Data in cells B9:E9 CI Radius = Margin of Error Compare to old Normal CIs Left End Right End

48. t - Distribution Confidence Interval HW: 7.24 (a. Q-Q roughly linear, so OK, b. 43.17, 4.41, 0.987 c. [41.1, 45.2]) 7.25

49. And now for something completely different An extreme “sport” video:

50. t - Distribution Application 2: Hypothesis Tests

51. t - Distribution Application 2: Hypothesis Tests Idea: Calculate P-values using TDIST

52. t - Distribution Application 2: Hypothesis Tests Idea: Calculate P-values using TDIST

53. t – Distribution Hypo Testing E.g. Old Textbook Example 7.26 For the above DDT poisoning example

54. t – Distribution Hypo Testing E.g. Old Textbook Example 7.26 For the above DDT poisoning example Recall Data in cells B9:E9

55. t – Distribution Hypo Testing E.g. Old Textbook Example 7.26 For the above DDT poisoning example Recall Data in cells B9:E9 As above: t – distribution appropriate

56. t – Distribution Hypo Testing E.g. Old Textbook Example 7.26 For the above DDT poisoning example Recall Data in cells B9:E9 As above: t – distribution appropriate (small sample, and using s ≈ σ)

57. t – Distribution Hypo Testing E.g. Old Textbook Example 7.26 For the above DDT poisoning example, Suppose that the mean “absolutely refractory period” is known to be 1.3

58. t – Distribution Hypo Testing E.g. Old Textbook Example 7.26 For the above DDT poisoning example, Suppose that the mean “absolutely refractory period” is known to be 1.3 (recall observed in data)

59. t – Distribution Hypo Testing E.g. Old Textbook Example 7.26 For the above DDT poisoning example, Suppose that the mean “absolutely refractory period” is known to be 1.3. DDT poisoning should slow nerve recovery, and so increase this period.

60. t – Distribution Hypo Testing E.g. Old Textbook Example 7.26 For the above DDT poisoning example, Suppose that the mean “absolutely refractory period” is known to be 1.3. DDT poisoning should slow nerve recovery, and so increase this period. Do the data give good evidence for this supposition?

61. t – Distribution Hypo Testing E.g. Old Textbook Example 7.26 Let = population mean absolutely refractory period for poisoned rats.

62. t – Distribution Hypo Testing E.g. Old Textbook Example 7.26 Let = population mean absolutely refractory period for poisoned rats. (checking strong evidence for this)

63. t – Distribution Hypo Testing E.g. Old Textbook Example 7.26 Let = population mean absolutely refractory period for poisoned rats. (from before)

64. t – Distribution Hypo Testing E.g. Old Textbook Example 7.26 P-value = P{what saw or more conclusive | H 0 – H A Bdry}

65. t – Distribution Hypo Testing E.g. Old Textbook Example 7.26 P-value = P{what saw or more conclusive | H 0 – H A Bdry}

66. t – Distribution Hypo Testing E.g. Old Textbook Example 7.26 P-value = P{what saw or more conclusive | H 0 – H A Bdry}

67. t – Distribution Hypo Testing E.g. Old Textbook Example 7.26 P-value = P{what saw or more conclusive | H 0 – H A Bdry}

68. t – Distribution Hypo Testing E.g. Old Textbook Example 7.26 P-value = P{what saw or more conclusive | H 0 – H A Bdry}

69. t – Distribution Hypo Testing E.g. Old Textbook Example 7.26 P-value = P{what saw or more conclusive | H 0 – H A Bdry} Now use TDIST

70. t – Distribution Hypo Testing E.g. Old Textbook Example 7.26 P-value = P{what saw or more conclusive | H 0 – H A Bdry} Now use TDIST

71. t – Distribution Hypo Testing E.g. Old Textbook Example 7.26 P-value = P{what saw or more conclusive | H 0 – H A Bdry} Now use TDIST

72. t – Distribution Hypo Testing E.g. Old Textbook Example 7.26 P-value = P{what saw or more conclusive | H 0 – H A Bdry} Now use TDIST

73. t – Distribution Hypo Testing E.g. Old Textbook Example 7.26 P-value = P{what saw or more conclusive | H 0 – H A Bdry} Now use TDIST Degrees of Freedom = n – 1 = 4 - 1

74. t – Distribution Hypo Testing E.g. Old Textbook Example 7.26 P-value = P{what saw or more conclusive | H 0 – H A Bdry} Now use TDIST Tails

75. t – Distribution Hypo Testing E.g. Old Textbook Example 7.26 From Class Example 27, part 2: http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg15.xls P - value = 0.003

76. t – Distribution Hypo Testing E.g. Old Textbook Example 7.26 From Class Example 27, part 2: http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg15.xls P - value = 0.003 Interpretation: very strong evidence, for either yes-no or gray-level

77. t – Distribution Hypo Testing Variations: For “opposite direction” hypotheses:

78. t – Distribution Hypo Testing Variations: For “opposite direction” hypotheses: P-value =

79. t – Distribution Hypo Testing Variations: For “opposite direction” hypotheses: P-value =

80. t – Distribution Hypo Testing Variations: For “opposite direction” hypotheses: P-value = [wrong way for TDIST(…,1)]

81. t – Distribution Hypo Testing Variations: For “opposite direction” hypotheses: P-value = Then use symmetry

82. t – Distribution Hypo Testing Variations: For “opposite direction” hypotheses: P-value = Then use symmetry, i.e. put - into TDIST.

83. t – Distribution Hypo Testing Variations: For 2-sided hypotheses

84. t – Distribution Hypo Testing Variations: For 2-sided hypotheses: H 0 : μ = H 1 : μ ≠

85. t – Distribution Hypo Testing Variations: For 2-sided hypotheses: H 0 : μ = H 1 : μ ≠

86. t – Distribution Hypo Testing Variations: For 2-sided hypotheses: H 0 : μ = H 1 : μ ≠ Use 2-tailed version of TDIST

87. t – Distribution Hypo Testing Variations: For 2-sided hypotheses: H 0 : μ = H 1 : μ ≠ Use 2-tailed version of TDIST, i.e. TDIST(…,2)

88. t – Distribution Hypo Testing HW: Interpret P-values: (i)yes-no (ii)gray-level 7.21e ((i)significant, (ii) significant, but not very strongly so) 7.22e (0.0619, (i) not significant (ii) not sig., but nearly significant)

89. Research Corner Another SiZer analysis: Mollusk Extinction Data

90. Research Corner Another SiZer analysis: Mollusk Extinction Data From Matthew Campbell UNC Master’s Student In Geological Sciences

91. Research Corner Another SiZer analysis: Mollusk Extinction Data Data points: Fossilized shells

92. Research Corner Another SiZer analysis: Mollusk Extinction Data Data points: Fossilized shells (fossil beds up and down Eastern Seaboard)

93. Research Corner Another SiZer analysis: Mollusk Extinction Data Data points: Fossilized shells Dated (by fossil bed)

94. Research Corner Another SiZer analysis: Mollusk Extinction Data Data points: Fossilized shells Dated (by fossil bed) Biologically categorized

95. Research Corner Another SiZer analysis: Mollusk Extinction Data Data points: Fossilized shells Dated (by fossil bed) Biologically categorized (family – genus – species, etc.)

96. Research Corner Another SiZer analysis: Mollusk Extinction Data Data points: Fossilized shells Dated (by fossil bed) Biologically categorized Goal: study extinctions over long periods

97. Research Corner Another SiZer analysis: Mollusk Extinction Data Data points: Fossilized shells Dated (by fossil bed) Biologically categorized Goal: study extinctions over long periods (via last time saw each)

98. Research Corner Another SiZer analysis: Mollusk Extinction Data Oversmoothed: nothing interesting

99. Research Corner Another SiZer analysis: Mollusk Extinction Data Undersmoothed: many bumps appear

100. Research Corner Another SiZer analysis: Mollusk Extinction Data Undersmoothed: many bumps appear but not statistically significant

101. Research Corner Another SiZer analysis: Mollusk Extinction Data Intermediate Smoothing two bumps appear

102. Research Corner Another SiZer analysis: Mollusk Extinction Data Intermediate Smoothing two bumps appear SiZer result:not statistically significant

103. Research Corner Another SiZer analysis: Mollusk Extinction Data Matthew’s Comment: Whoah, those are times of mass extinctions

104. Research Corner Another SiZer analysis: Mollusk Extinction Data Matthew’s Comment: Whoah, those are times of mass extinctions Any way to show these are “really there”?

105. Research Corner Another SiZer analysis: Mollusk Extinction Data Any way to show these are “really there”?

106. Research Corner Another SiZer analysis: Mollusk Extinction Data Any way to show these are “really there”? Standard Answer: Get more data

107. Research Corner Another SiZer analysis: Mollusk Extinction Data Any way to show these are “really there”? Challenge: Took 100s of year to get these!

108. Research Corner Another SiZer analysis: Mollusk Extinction Data Any way to show these are “really there”? Alternate Approach: Refined from Genus level to Species

109. Research Corner Another SiZer analysis: Mollusk Extinction Data Species level result

110. Research Corner Another SiZer analysis: Mollusk Extinction Data Species level result: Now both bumps are significant

111. Research Corner Another SiZer analysis: Mollusk Extinction Data Species level result: Now both bumps are significant Consistent with Global Climactic Events

112. Variable Relationships Chapter 2 in Text

113. Variable Relationships Chapter 2 in Text Idea: Look beyond single quantities

114. Variable Relationships Chapter 2 in Text Idea: Look beyond single quantities, to how quantities relate to each other.

115. Variable Relationships Chapter 2 in Text Idea: Look beyond single quantities, to how quantities relate to each other. E.g. How do HW scores “relate” to Exam scores?

116. Variable Relationships Chapter 2 in Text Idea: Look beyond single quantities, to how quantities relate to each other. E.g. How do HW scores “relate” to Exam scores? Section 2.1: Useful graphical device: Scatterplot

117. Plotting Bivariate Data Toy Example:Ordered pairs (1,2) (3,1) (-1,0) (2,-1)

118. Plotting Bivariate Data Toy Example:Ordered pairs Captures relationship between X & Y (1,2)as (X,Y) (3,1) (-1,0) (2,-1)

119. Plotting Bivariate Data Toy Example:Ordered pairs Captures relationship between X & Y (1,2)as (X,Y) (3,1)e.g. (height, weight) (-1,0) (2,-1)

120. Plotting Bivariate Data Toy Example:Ordered pairs Captures relationship between X & Y (1,2)as (X,Y) (3,1)e.g. (height, weight) (-1,0)e.g. (MT Score, Final Exam Score) (2,-1)

121. Plotting Bivariate Data Toy Example: (1,2)Think in terms of: (3,1) (-1,0) X coordinates (2,-1)

122. Plotting Bivariate Data Toy Example: (1,2)Think in terms of: (3,1) (-1,0) X coordinates (2,-1)Y coordinates

123. Plotting Bivariate Data Toy Example: (1,2)Think in terms of: (3,1) (-1,0) X coordinates (2,-1)Y coordinates And plot in x,y plane, to see relationship

124. Plotting Bivariate Data Toy Example: (1,2) (3,1) (-1,0) (2,-1)

125. Plotting Bivariate Data Toy Example: (1,2) (3,1) (-1,0) (2,-1)

126. Plotting Bivariate Data Toy Example: (1,2) (3,1) (-1,0) (2,-1)

127. Plotting Bivariate Data Toy Example: (1,2) (3,1) (-1,0) (2,-1)

128. Plotting Bivariate Data Sometimes: Can see more insightful patterns by connecting points

129. Plotting Bivariate Data Sometimes: Useful to switch off points, and only look at lines/curves

130. Plotting Bivariate Data Common Name: “Scatterplot”

131. Plotting Bivariate Data Common Name: “Scatterplot” A look under the hood in Excel

132. Plotting Bivariate Data Common Name: “Scatterplot” A look under the hood in Excel:  Insert Tab

133. Plotting Bivariate Data Common Name: “Scatterplot” A look under the hood in Excel:  Insert Tab  Charts

134. Plotting Bivariate Data Common Name: “Scatterplot” A look under the hood in Excel:  Insert Tab  Charts  Scatter Button

135. Plotting Bivariate Data Common Name: “Scatterplot” A look under the hood in Excel:  Insert Tab  Charts  Scatter Button  Choose Dots

136. Plotting Bivariate Data Common Name: “Scatterplot” A look under the hood in Excel:  Insert Tab  Charts  Scatter Button  Choose Dots (but note other options)

137. Plotting Bivariate Data Common Name: “Scatterplot” A look under the hood in Excel:  Insert Tab  Charts  Scatter Button  Choose Dots  Manipulate plot as done before for bar plots

138. Plotting Bivariate Data Common Name: “Scatterplot” A look under the hood in Excel:  Insert Tab  Charts  Scatter Button  Choose Dots  Manipulate plot as done before for bar plots (e.g. titles, labels, colors, styles, …)

139. Scatterplot E.g. Class Example 16: http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg16.xls Data from related Intro. Statistics Class

140. Scatterplot E.g. Class Example 16: http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg16.xls Data from related Intro. Statistics Class (actual scores)

141. Scatterplot E.g. Class Example 16: http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg16.xls Data from related Intro. Statistics Class (actual scores) A.How does HW score predict Final Exam?

142. Scatterplot E.g. Class Example 16: http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg16.xls Data from related Intro. Statistics Class (actual scores) A.How does HW score predict Final Exam? x i = HW, y i = Final Exam

143. Scatterplot E.g. Class Example 16: http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg16.xls Data from related Intro. Statistics Class (actual scores) A.How does HW score predict Final Exam? x i = HW, y i = Final Exam (Study Relationship using scatterplot)

144. Scatterplot E.g. Class Example 16: How does HW score predict Final Exam? x i = HW, y i = Final Exam

145. Scatterplot E.g. Class Example 16: How does HW score predict Final Exam? x i = HW, y i = Final Exam (Scatterplot View)

146. Scatterplot E.g. Class Example 16: How does HW score predict Final Exam? x i = HW, y i = Final Exam i.In top half of HW scores

147. Scatterplot E.g. Class Example 16: How does HW score predict Final Exam? x i = HW, y i = Final Exam i.In top half of HW scores: Better HW  Better Final

148. Scatterplot E.g. Class Example 16: How does HW score predict Final Exam? x i = HW, y i = Final Exam i.In top half of HW scores: Better HW  Better Final ii.For lower HW

149. Scatterplot E.g. Class Example 16: How does HW score predict Final Exam? x i = HW, y i = Final Exam i.In top half of HW scores: Better HW  Better Final ii.For lower HW: Final is more “random”

150. Scatterplots Common Terminology: When thinking about “X causes Y”,

151. Scatterplots Common Terminology: When thinking about “X causes Y”, Call X the “Explanatory Var.”

152. Scatterplots Common Terminology: When thinking about “X causes Y”, Call X the “Explanatory Var.” or “Indep. Var.”

153. Scatterplots Common Terminology: When thinking about “X causes Y”, Call X the “Explanatory Var.” or “Indep. Var.” Call Y the “Response Var.”

154. Scatterplots Common Terminology: When thinking about “X causes Y”, Call X the “Explanatory Var.” or “Indep. Var.” Call Y the “Response Var.” or “Dep. Var.”

155. Scatterplots Common Terminology: When thinking about “X causes Y”, Call X the “Explanatory Var.” or “Indep. Var.” Call Y the “Response Var.” or “Dep. Var.” (think of “Y as function of X”)

156. Scatterplots Common Terminology: When thinking about “X causes Y”, Call X the “Explanatory Var.” or “Indep. Var.” Call Y the “Response Var.” or “Dep. Var.” (think of “Y as function of X”) (although not always sensible)

157. Scatterplots Note: Sometimes think about causation

158. Scatterplots Note: Sometimes think about causation, Other times: “Explore Relationship”

159. Scatterplots Note: Sometimes think about causation, Other times: “Explore Relationship” HW: 2.9

160. Class Scores Scatterplots http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg16.xls B.How does HW predict Midterm 1? x i = HW, y i = MT1

161. Class Scores Scatterplots http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg16.xls B.How does HW predict Midterm 1? x i = HW, y i = MT1 (Replace Final above with 1 st Midterm)

162. Class Scores Scatterplots http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg16.xls B.How does HW predict Midterm 1? x i = HW, y i = MT1

163. Class Scores Scatterplots http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg16.xls B.How does HW predict Midterm 1? x i = HW, y i = MT1 i.Better HW  better Exam (general upwards tendency still the same)

164. Class Scores Scatterplots http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg16.xls B.How does HW predict Midterm 1? x i = HW, y i = MT1 i.Better HW  better Exam ii.Wider range MT1 scores (for each range of HW scores) (relative to final scores)

165. Class Scores Scatterplots http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg16.xls B.How does HW predict Midterm 1? x i = HW, y i = MT1 i.Better HW  better Exam ii.Wider range MT1 scores iii.HW doesn’t predict MT1 (as well as HW predicted the final)

166. Class Scores Scatterplots http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg16.xls B.How does HW predict Midterm 1? x i = HW, y i = MT1 i.Better HW  better Exam ii.Wider range MT1 scores iii.HW doesn’t predict MT1 iv.“Outliers” in scatterplot

167. Class Scores Scatterplots http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg16.xls B.How does HW predict Midterm 1? x i = HW, y i = MT1 i.Better HW  better Exam ii.Wider range MT1 scores iii.HW doesn’t predict MT1 iv.“Outliers” in scatterplot e.g. HW = 72, MT1 = 94

168. Class Scores Scatterplots http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg16.xls B.How does HW predict Midterm 1? x i = HW, y i = MT1 i.Better HW  better Exam ii.Wider range MT1 scores iii.HW doesn’t predict MT1 iv.“Outliers” in scatterplot may not be outliers in either individual variable e.g. HW = 72, MT1 = 94 (bad HW, but good MT1?, fluke???)

169. Class Scores Scatterplots http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg16.xls C.How does MT1 predict MT2? x i = MT1, y i = MT2

170. Class Scores Scatterplots http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg16.xls C.How does MT1 predict MT2? x i = MT1, y i = MT2 (Different choice of x and y, since studying different relationship)

171. Class Scores Scatterplots http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg16.xls C.How does MT1 predict MT2? x i = MT1, y i = MT2 (Study Relationship using tool of scatterplot)

172. Class Scores Scatterplots http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg16.xls C.How does MT1 predict MT2? x i = MT1, y i = MT2 i.Idea: less “causation”, more “exploration” (don’t expect better MT1 to lead to better MT2)

173. Class Scores Scatterplots http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg16.xls C.How does MT1 predict MT2? x i = MT1, y i = MT2 i.Idea: less “causation”, more “exploration” ii.High MT1  High MT2 (again clear overall upwards trend)

174. Class Scores Scatterplots http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg16.xls C.How does MT1 predict MT2? x i = MT1, y i = MT2 i.Idea: less “causation”, more “exploration” ii.High MT1  High MT2 iii.Wider range of MT2 (for each range of MT1)

175. Class Scores Scatterplots http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg16.xls C.How does MT1 predict MT2? x i = MT1, y i = MT2 i.Idea: less “causation”, more “exploration” ii.High MT1  High MT2 iii.Wider range of MT2 i.e. “not good predictor” (MT1)(of MT2)

176. Class Scores Scatterplots http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg16.xls C.How does MT1 predict MT2? x i = MT1, y i = MT2 i.Idea: less “causation”, more “exploration” ii.High MT1  High MT2 iii.Wider range of MT2 i.e. “not good predictor” iv.Interesting Outliers

177. Class Scores Scatterplots http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg16.xls C.How does MT1 predict MT2? x i = MT1, y i = MT2 i.Idea: less “causation”, more “exploration” ii.High MT1  High MT2 iii.Wider range of MT2 i.e. “not good predictor” iv.Interesting Outliers: MT1 = 100, MT2 = 56 (oops!)

178. Class Scores Scatterplots http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg16.xls C.How does MT1 predict MT2? x i = MT1, y i = MT2 i.Idea: less “causation”, more “exploration” ii.High MT1  High MT2 iii.Wider range of MT2 i.e. “not good predictor” iv.Interesting Outliers: MT1 = 100, MT2 = 56 MT1 = 23, MT2 = 74 (woke up!)

179. Class Scores Scatterplots http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg16.xls C.How does MT1 predict MT2? x i = MT1, y i = MT2 i.Idea: less “causation”, more “exploration” ii.High MT1  High MT2 iii.Wider range of MT2 i.e. “not good predictor” iv.Interesting Outliers: MT1 = 100, MT2 = 56 MT1 = 23, MT2 = 74 MT1 70s, MT2 90s (moved up!)

180. And now for something completely different A thought provoking movie clip: http://www.aclu.org/pizza/

181. Important Aspects of Relations I.Form of Relationship (Linear or not?)

182. Important Aspects of Relations I.Form of Relationship II.Direction of Relationship (trending up or down?)

183. Important Aspects of Relations I.Form of Relationship II.Direction of Relationship III.Strength of Relationship (how much of data “explained”?)

184. I.Form of Relationship Linear: Data approximately follow a line

185. 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

186. 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

187. 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”

188. 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” But saw others with “rough linear trend”

189. 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” But saw others with “rough linear trend”

190. 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” But saw others with “rough linear trend” Interesting question: Measure strength of linear trend

191. 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 (non-linear)

192. 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

193. 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.T. BralowerGeological Sci.

194. 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.T. BralowerGeological Sci.

195. 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.T. BralowerGeological Sci. Studies Global Climate

196. 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.T. BralowerGeological Sci. Studies Global Climate, millions of years ago

197. 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.T. BralowerGeological Sci. Studies Global Climate, millions of years ago: Small shells from ocean floor cores

198. 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.T. BralowerGeological Sci. Studies Global Climate, millions of years ago: Small shells from ocean floor cores Ratios of Isotopes of Strontium

199. 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.T. BralowerGeological Sci. Studies Global Climate, millions of years ago: Small shells from ocean floor cores Ratios of Isotopes of Strontium Reflects Ice Ages, via Sea Level

200. 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.T. BralowerGeological Sci. Studies Global Climate, millions of years ago: Small shells from ocean floor cores Ratios of Isotopes of Strontium Reflects Ice Ages, via Sea Level (50 meter difference!)

201. 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.T. BralowerGeological Sci. Studies Global Climate, millions of years ago: Small shells from ocean floor cores Ratios of Isotopes of Strontium Reflects Ice Ages, via Sea Level (50 meter difference!) As function of time

202. 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.T. BralowerGeological Sci. Studies Global Climate, millions of years ago: Small shells from ocean floor cores Ratios of Isotopes of Strontium Reflects Ice Ages, via Sea Level (50 meter difference!) As function of time Clearly nonlinear relationship

203. II. Direction of Relationship Positive Association

204. II. Direction of Relationship Positive Association X bigger  Y bigger

205. II. Direction of Relationship Positive Association X bigger  Y bigger E.g. Class Scores Data above

206. II. Direction of Relationship Positive Association X bigger  Y bigger Negative Association

207. II. Direction of Relationship Positive Association X bigger  Y bigger Negative Association X bigger  Y smaller

208. II. Direction of Relationship Positive Association X bigger  Y bigger Negative Association X bigger  Y smaller E.g. X = alcohol consumption, Y = Driving Ability Clear negative association

209. II. Direction of Relationship Positive Association X bigger  Y bigger Negative Association X bigger  Y smaller Note: Concept doesn’t always apply:

210. 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

211. 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

212. 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 Final Exam is “closely related to HW”

213. 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 Final Exam is “closely related to HW” Midterm 1 less closely related to HW

214. 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 Final Exam is “closely related to HW” Midterm 1 less closely related to HW Midterm 2 even less related to Midterm 1

215. 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 Final Exam is “closely related to HW” Midterm 1 less closely related to HW Midterm 2 even less related to Midterm 1 Interesting Issue: Measure this strength

216. Linear Relationship HW HW: 2.11, 2.13, 2.15, 2.17