last time t distribution –confidence intervals –hypothesis tests relationships between variables...
TRANSCRIPT
![Page 1: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/1.jpg)
Last Time
• T Distribution– Confidence Intervals– Hypothesis tests
• Relationships Between Variables– Scatterplots (visualization)
• Aspects of Relations– Form– Direction– Strength
![Page 2: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/2.jpg)
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
![Page 3: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/3.jpg)
Scatterplot E.g.Class Example 16:
How does HW score predict Final Exam?
xi = HW, yi = Final Exam
i. In top half of HW scores:Better HW Better Final
![Page 4: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/4.jpg)
Important Aspects of Relations
I. Form of Relationship
II. Direction of Relationship
III. Strength of Relationship
![Page 5: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/5.jpg)
I. Form of Relationship• Linear: Data approximately follow a line
Previous Class Scores Examplehttp://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
![Page 6: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/6.jpg)
Bralower’s Fossil Datahttp://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
![Page 7: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/7.jpg)
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
![Page 8: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/8.jpg)
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
![Page 9: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/9.jpg)
Comparing Scatterplots
Additional Useful Visual Tool
![Page 10: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/10.jpg)
Comparing Scatterplots
Additional Useful Visual Tool:
• Overlaying multiple data sets
![Page 11: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/11.jpg)
Comparing Scatterplots
Additional Useful Visual Tool:
• Overlaying multiple data sets
• Allows comparison
![Page 12: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/12.jpg)
Comparing Scatterplots
Additional Useful Visual Tool:
• Overlaying multiple data sets
• Allows comparison
• Use different colors or symbols
![Page 13: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/13.jpg)
Comparing Scatterplots
Additional Useful Visual Tool:
• Overlaying multiple data sets
• Allows comparison
• Use different colors or symbols
• Easy in EXCEL (colors are automatic)
![Page 14: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/14.jpg)
Comparing Scatterplots HW
HW:
2.21, 2.25
![Page 15: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/15.jpg)
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
![Page 16: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/16.jpg)
III. Strength of Relationship
Idea: How close are points to lying on a line?
Now get quantitative
![Page 17: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/17.jpg)
Section 2.2: Correlation
Main Idea: Quantify Strength of Relationship
![Page 18: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/18.jpg)
Section 2.2: Correlation
Main Idea: Quantify Strength of Relationship
Context:
– A numerical summary
![Page 19: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/19.jpg)
Section 2.2: Correlation
Main Idea: Quantify Strength of Relationship
Context:
– A numerical summary
– In spirit of mean and standard deviation
![Page 20: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/20.jpg)
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
![Page 21: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/21.jpg)
Section 2.2: Correlation
Main Idea: Quantify Strength of Relationship
Specific Goals
![Page 22: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/22.jpg)
Section 2.2: Correlation
Main Idea: Quantify Strength of Relationship
Specific Goals:
– Near 1: for positive relat’ship & nearly linear
![Page 23: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/23.jpg)
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)
![Page 24: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/24.jpg)
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
![Page 25: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/25.jpg)
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)
![Page 26: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/26.jpg)
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
![Page 27: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/27.jpg)
Correlation - Approach
Numerical Approach
![Page 28: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/28.jpg)
Correlation - Approach
Numerical Approach:
for symmetric around )0,0(),( ii yx
![Page 29: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/29.jpg)
Correlation - Approach
Numerical Approach:
for symmetric around
has similar properties
)0,0(),( ii yx
n
iii yx
1
![Page 30: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/30.jpg)
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
)0,0(),( ii yx
n
iii yx
1
![Page 31: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/31.jpg)
Correlation – Graphical View
Plots (a) & (b): illustrating :
• > 0 for positive relationship
n
iii yx
1
![Page 32: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/32.jpg)
Correlation – Graphical View
Plots (a) & (b): illustrating :
• > 0 for positive relationship
n
iii yx
1
![Page 33: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/33.jpg)
Correlation – Graphical View
Plots (a) & (b): illustrating :
• > 0 for positive relationship
• < 0 for negative relationship
n
iii yx
1
![Page 34: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/34.jpg)
Correlation – Graphical View
Plots (a) & (b): illustrating :
• > 0 for positive relationship
• < 0 for negative relationship
n
iii yx
1
![Page 35: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/35.jpg)
Correlation – Graphical View
Plots (a) & (b): illustrating :
• Bigger for data closer to line
n
iii yx
1
![Page 36: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/36.jpg)
Correlation – Graphical View
Plots (a) & (b): illustrating :
• Bigger for data closer to line
n
iii yx
1
![Page 37: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/37.jpg)
Correlation – Graphical View
But not all goals are satisfied
![Page 38: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/38.jpg)
Correlation – Graphical View
Problem 1: Not between -1 & 1
![Page 39: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/39.jpg)
Correlation – Graphical View
Problem 2: Feels “Scale”, see plot (c)
(just 10 1 vertical rescaling of)
![Page 40: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/40.jpg)
Correlation – Graphical View
Problem 2: Feels “Scale”, see plot (c)
(just 10 1 vertical rescaling of)
( feels factor of 1/10)
n
iii yx
1
![Page 41: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/41.jpg)
Correlation – Graphical View
Problem 3: Feels “Shift” even more, see (d)
(even gets sign wrong!)
![Page 42: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/42.jpg)
Correlation – Graphical View
Problem 3: Feels “Shift” even more, see (d)
(even gets sign wrong!)
• Data trend upwards
![Page 43: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/43.jpg)
Correlation – Graphical View
Problem 3: Feels “Shift” even more, see (d)
(even gets sign wrong!)
• Data trend upwards
• But < 0
n
iii yx
1
![Page 44: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/44.jpg)
Correlation - Approach
Solution to above problems
![Page 45: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/45.jpg)
Correlation - Approach
Solution to above problems:
Standardize!
![Page 46: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/46.jpg)
Correlation - Approach
Solution to above problems:
Standardize!
Define Correlation r
![Page 47: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/47.jpg)
Correlation - Approach
Solution to above problems:
Standardize!
Define Correlation
n
i y
i
x
i
s
yy
s
xxr
1
![Page 48: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/48.jpg)
Correlation - Example
Revisit above examplehttp://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg18-new.xls
• r is always same, and ~1, for (a)
![Page 49: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/49.jpg)
Correlation - Example
Revisit above examplehttp://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg18-new.xls
• r is always same, and ~1, for (a), (c)
![Page 50: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/50.jpg)
Correlation - Example
Revisit above examplehttp://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg18-new.xls
• r is always same, and ~1, for (a), (c), (d)
![Page 51: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/51.jpg)
Correlation - Example
Revisit above examplehttp://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg18-new.xls
• r is always same, and ~1, for (a), (c), (d)
• r < 0, and not so close to -1, for (b)
![Page 52: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/52.jpg)
Correlation - Example
Revisit Class Scores Example:http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg16.xls
Final Exam vs. HW
Correlation = r = 0.73
Strongest Dependence
![Page 53: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/53.jpg)
Correlation - Example
Revisit Class Scores Example:http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg16.xls
MT1 vs. HW
Correlation = r = 0.65
Weaker Dependence
![Page 54: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/54.jpg)
Correlation - Example
Revisit Class Scores Example:http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg16.xls
MT2 vs. MT1
Correlation = r = 0.57
Weakest Dependence
![Page 55: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/55.jpg)
Correlation - Example
Revisit Class Scores Example:http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg16.xls
• r is always > 0
(makes sense, since all trend upwards)
![Page 56: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/56.jpg)
Correlation - Example
Revisit Class Scores Example:http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg16.xls
• r is always > 0
• r is biggest for Final vs. HW
![Page 57: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/57.jpg)
Correlation - Example
Revisit Class Scores Example:http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg16.xls
• r is always > 0
• r is biggest for Final vs. HW
(visually strongest relationship)
![Page 58: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/58.jpg)
Correlation - Example
Revisit Class Scores Example:http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg16.xls
• r is always > 0
• r is biggest for Final vs. HW
(visually strongest relationship)
• r is smallest for MT2 vs. MT1
![Page 59: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/59.jpg)
Correlation - Example
Revisit Class Scores Example:http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg16.xls
• r is always > 0
• r is biggest for Final vs. HW
(visually strongest relationship)
• r is smallest for MT2 vs. MT1
(visually weakest relationship)
![Page 60: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/60.jpg)
Correlation – Computation
From Class Scores Example:http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg16.xls
![Page 61: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/61.jpg)
Correlation – Computation
From Class Scores Example:http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg16.xls
Use Excel function: CORREL
![Page 62: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/62.jpg)
Correlation – Computation
From Class Scores Example:http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg16.xls
Use Excel function: CORREL
![Page 63: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/63.jpg)
Correlation – Computation
From Class Scores Example:http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg16.xls
Use Excel function: CORREL
• Range of Xs
![Page 64: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/64.jpg)
Correlation – Computation
From Class Scores Example:http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg16.xls
Use Excel function: CORREL
• Range of Xs
• Range of Ys
![Page 65: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/65.jpg)
Correlation – Computation
From Class Scores Example:http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg16.xls
Use Excel function: CORREL
• Range of Xs
• Range of Ys
• Output is correlation, r
![Page 66: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/66.jpg)
Correlation - Example
Fun Example from Publisher’s Website:
http://courses.bfwpub.com/ips6e.php
![Page 67: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/67.jpg)
Correlation - Example
Fun Example from Publisher’s Website:
http://courses.bfwpub.com/ips6e.php
Choose
• Statistical Applets
![Page 68: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/68.jpg)
Correlation - Example
Fun Example from Publisher’s Website:
http://courses.bfwpub.com/ips6e.php
Choose
• Statistical Applets
• Correlation and Regression
![Page 69: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/69.jpg)
Correlation - Example
Fun Example from Publisher’s Website:
http://courses.bfwpub.com/ips6e.php
Choose
• Statistical Applets
• Correlation and Regression
Gives feeling for how correlation is affected by changing data.
![Page 70: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/70.jpg)
Correlation - Example
Correlation and Regression Applet
![Page 71: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/71.jpg)
Correlation - Example
Correlation and Regression Applet
I clicked to put
down 2 points
![Page 72: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/72.jpg)
Correlation - Example
Correlation and Regression Applet
I clicked to put
down 2 points
Applet computed
correlation, r
![Page 73: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/73.jpg)
Correlation - Example
Correlation and Regression Applet
Applet computed
correlation, r
r = -1, since
points on line
trending down
![Page 74: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/74.jpg)
Correlation - Example
Correlation and Regression Applet
Try several points
close to some line
![Page 75: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/75.jpg)
Correlation - Example
Correlation and Regression Applet
Try several points
close to some line
r ≈ -1, since
points near line
trending down
![Page 76: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/76.jpg)
Correlation - Example
Correlation and Regression Applet
Add more points
with goal of
r ≈ -0.95
![Page 77: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/77.jpg)
Correlation - Example
Correlation and Regression Applet
Add more points
with goal of
r ≈ -0.95
![Page 78: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/78.jpg)
Correlation - Example
Correlation and Regression Applet
Add more points
with goal of
r ≈ -0.95
![Page 79: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/79.jpg)
Correlation - Example
Correlation and Regression Applet
Now add a single
outlier
![Page 80: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/80.jpg)
Correlation - Example
Correlation and Regression Applet
Now add a single
outlier
Major impact on r
-0.95 -0.35
![Page 81: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/81.jpg)
Correlation - Example
Correlation and Regression Applet
Just 2 more
outliers
![Page 82: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/82.jpg)
Correlation - Example
Correlation and Regression Applet
Just 2 more
outliers
Leads to r > 0
![Page 83: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/83.jpg)
Correlation - Example
Correlation and Regression Applet
Just 2 more
outliers
Leads to r > 0
(Outliers have
major impact)
![Page 84: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/84.jpg)
Correlation - Example
Correlation and Regression Applet
Weakness of
correlation, r
![Page 85: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/85.jpg)
Correlation - Example
Correlation and Regression Applet
Weakness of
correlation, r
Measures linear
dependence
![Page 86: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/86.jpg)
Correlation - Example
Correlation and Regression Applet
Weakness of
correlation, r
Can have r ≈ 0
![Page 87: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/87.jpg)
Correlation - Example
Correlation and Regression Applet
Weakness of
correlation, r
Can have r ≈ 0,
yet strong
non-linear
dependence
![Page 88: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/88.jpg)
Correlation - HW
HW:
2.31
2.33
2.39a
![Page 89: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/89.jpg)
Correlation - Outliers
Caution:
Outliers can strongly affect correlation, r
![Page 90: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/90.jpg)
Correlation - Example
Correlation and Regression Applet
Add more points
with goal of
r ≈ -0.95
![Page 91: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/91.jpg)
Correlation - Example
Correlation and Regression Applet
Now add a single
outlier
Major impact on r
-0.95 -0.35
![Page 92: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/92.jpg)
Correlation - Example
Correlation and Regression Applet
Just 2 more
outliers
Leads to r > 0
(Outliers have
major impact)
![Page 93: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/93.jpg)
Correlation - Outliers
Caution:
Outliers can strongly affect correlation, r
HW:
2.39b
2.45
![Page 94: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/94.jpg)
Research Corner
Relationship between more than 2 variables?
![Page 95: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/95.jpg)
Research Corner
Relationship between more than 2 variables?
Each data point is (x1, x2, … , xd)
Called a “d-tuple”
![Page 96: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/96.jpg)
Research Corner
Relationship between more than 2 variables?
Each data point is (x1, x2, … , xd)
Eg: d = 3 (ordered triple)
![Page 97: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/97.jpg)
Research Corner
Relationship between more than 2 variables?
Each data point is (x1, x2, … , xd)
Eg: d = 3 (ordered triple)
(height, weight, age)
![Page 98: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/98.jpg)
Research Corner
Relationship between more than 2 variables?
Each data point is (x1, x2, … , xd)
Eg: d = 3 (ordered triple)
(height, weight, age)
(HW, MT1, Final)
![Page 99: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/99.jpg)
Research Corner
Visualization?
![Page 100: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/100.jpg)
Research Corner
Visualization?
What is “scatterplot”?
![Page 101: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/101.jpg)
Research Corner
Visualization?
What is “scatterplot”?
How can we “see” structure in data?
![Page 102: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/102.jpg)
Research Corner
Visualization?
Explore d = 3 (3d)
![Page 103: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/103.jpg)
Research Corner
Visualization?
Explore d = 3 (3d)
So can visualize “point cloud”
![Page 104: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/104.jpg)
Research CornerToy Example, modeling “gene expression”
![Page 105: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/105.jpg)
Research CornerMultivariate View: Highlight one
![Page 106: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/106.jpg)
Research CornerMultivariate View: Value of variable 1
![Page 107: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/107.jpg)
Research CornerMultivariate View: Value of variable 2
![Page 108: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/108.jpg)
Research CornerMultivariate View: Value of variable 3
![Page 109: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/109.jpg)
Research CornerMultivariate View: 1-d projection, X-axis
![Page 110: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/110.jpg)
Research CornerMultivariate View: X – Projection, 1-d View
![Page 111: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/111.jpg)
Research CornerMultivariate View: 1-d projection, Y-axis
![Page 112: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/112.jpg)
Research CornerMultivariate View: Y – Projection, 1-d View
![Page 113: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/113.jpg)
Research CornerMultivariate View: 1-d projection, Z-axis
![Page 114: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/114.jpg)
Research CornerMultivariate View: Z – Projection, 1-d View
![Page 115: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/115.jpg)
Research CornerMultivariate View: 2-d Projection XY-plane
![Page 116: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/116.jpg)
Research CornerMultivariate View: XY – projection, 2-d view
![Page 117: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/117.jpg)
Research CornerMultivariate View: 2-d Projection XZ-plane
![Page 118: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/118.jpg)
Research CornerMultivariate View: XZ – projection, 2-d view
![Page 119: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/119.jpg)
Research CornerMultivariate View: 2-d Projection YZ-plane
![Page 120: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/120.jpg)
Research CornerMultivariate View: YZ – projection, 2-d view
![Page 121: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/121.jpg)
Research CornerMultivariate View: All 3 planes
![Page 122: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/122.jpg)
Research CornerMultivariate View
Now collect these views
on a single page
![Page 123: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/123.jpg)
Research CornerMultivariate View: 1-d projections on diagonal
![Page 124: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/124.jpg)
Research CornerMultivariate View: 2-d views off diagonal
![Page 125: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/125.jpg)
Research CornerMultivariate View: switch off color (usual view)
![Page 126: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/126.jpg)
Research CornerMultivariate View
(Useful summary of structure in data)
![Page 127: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/127.jpg)
2 Sample InferenceMain Idea:
• Previously studied single populations
![Page 128: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/128.jpg)
2 Sample InferenceMain Idea:
• Previously studied single populations
• Modeled as
![Page 129: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/129.jpg)
2 Sample InferenceMain Idea:
• Previously studied single populations
• Modeled as:– Measurement Error
![Page 130: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/130.jpg)
2 Sample InferenceMain Idea:
• Previously studied single populations
• Modeled as:– Measurement Error
nNX
,~
![Page 131: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/131.jpg)
2 Sample InferenceMain Idea:
• Previously studied single populations
• Modeled as:– Measurement Error
– Counts
nNX
,~
![Page 132: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/132.jpg)
2 Sample InferenceMain Idea:
• Previously studied single populations
• Modeled as:– Measurement Error
– Counts
nNX
,~
n
pppNppnBiX
)1(,~ˆ),,(~
![Page 133: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/133.jpg)
2 Sample InferenceMain Idea:
• Previously studied single populations
• Modeled as:– Measurement Error
– Counts
• Did Inference
nNX
,~
n
pppNppnBiX
)1(,~ˆ),,(~
![Page 134: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/134.jpg)
2 Sample InferenceMain Idea:
• Previously studied single populations
• Modeled as:– Measurement Error
– Counts
• Did Inference:– Confidence Intervals
– Hypothesis Tests
nNX
,~
n
pppNppnBiX
)1(,~ˆ),,(~
![Page 135: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/135.jpg)
2 Sample InferenceMain Idea:
• Extend to two populations
• Modeled as:– Measurement Error
– Counts
• Will Develop Inference:– Confidence Intervals
– Hypothesis Tests
1
111 ,~n
NX
2
222 ,~n
NX
),(~ 111 pnBiX ),(~ 222 pnBiX
![Page 136: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/136.jpg)
2 Sample InferenceLocation in Text
![Page 137: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/137.jpg)
2 Sample InferenceLocation in Text:
• Measurement Error– Sec. 7.1
– Sec. 7.2
1
111 ,~n
NX
2
222 ,~n
NX
![Page 138: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/138.jpg)
2 Sample InferenceLocation in Text:
• Measurement Error– Sec. 7.1
– Sec. 7.2
• Counts– Sec. 8.2
1
111 ,~n
NX
2
222 ,~n
NX
),(~ 111 pnBiX ),(~ 222 pnBiX
![Page 139: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/139.jpg)
2 Sample Measurement Error
Easy Case: Paired Differences
![Page 140: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/140.jpg)
2 Sample Measurement Error
Easy Case: Paired Differences
Have Treatment 1: nXXX ,,, 21
![Page 141: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/141.jpg)
2 Sample Measurement Error
Easy Case: Paired Differences
Have Treatment 1:
Treatment 2:
nXXX ,,, 21
nYYY ,,, 21
![Page 142: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/142.jpg)
2 Sample Measurement Error
Easy Case: Paired Differences
Have Treatment 1:
Treatment 2:
nXXX ,,, 21
nYYY ,,, 21
![Page 143: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/143.jpg)
2 Sample Measurement Error
Easy Case: Paired Differences
Have Treatment 1:
Treatment 2:
Important: Measurements Connected
nXXX ,,, 21
nYYY ,,, 21
![Page 144: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/144.jpg)
2 Sample Measurement Error
Easy Case: Paired Differences
Have Treatment 1:
Treatment 2:
Important: Measurements Connected,
e.g. made on same objects
nXXX ,,, 21
nYYY ,,, 21
![Page 145: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/145.jpg)
2 Sample Measurement Error
Easy Case: Paired Differences
Have Treatment 1:
Treatment 2:
Approach: Apply 1 sample methods
nXXX ,,, 21
nYYY ,,, 21
![Page 146: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/146.jpg)
2 Sample Measurement Error
Easy Case: Paired Differences
Have Treatment 1:
Treatment 2:
Approach: Apply 1 sample methods to:
nXXX ,,, 21
nYYY ,,, 21
niYXD iii ,,1,
![Page 147: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/147.jpg)
2 Paired SamplesE.g. Old Textbook 7.32:
Researchers studying Vitamin C in a product were concerned about loss of Vitamin C during shipment and storage.
![Page 148: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/148.jpg)
2 Paired SamplesE.g. Old Textbook 7.32:
Researchers studying Vitamin C in a product were concerned about loss of Vitamin C during shipment and storage. They marked a collection of bags at the factory, and measured the vitamin C
![Page 149: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/149.jpg)
2 Paired SamplesE.g. Old Textbook 7.32:
Researchers studying Vitamin C in a product were concerned about loss of Vitamin C during shipment and storage. They marked a collection of bags at the factory, and measured the vitamin C. 5 months later, in Haiti, they found the same bags, and again measured the Vitamin C.
![Page 150: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/150.jpg)
2 Paired SamplesE.g. Old Textbook 7.32:
The data are the two Vitamin C measurements, made for each bag.
![Page 151: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/151.jpg)
2 Paired SamplesE.g. Old Textbook 7.32:
The data are the two Vitamin C measurements, made for each bag.
Available in Class Example 15:http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg15.xls
![Page 152: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/152.jpg)
2 Paired SamplesAvailable in Class Example 15:http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg15.xls
![Page 153: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/153.jpg)
2 Paired SamplesAvailable in Class Example 15:http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg15.xls
Factory,
Cells B38:B64
![Page 154: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/154.jpg)
2 Paired SamplesAvailable in Class Example 15:http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg15.xls
Factory,
Cells B38:B64
Haiti,
Cells C38:C64
![Page 155: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/155.jpg)
2 Paired SamplesE.g. Old Textbook 7.32:
The data are the two Vitamin C measurements, made for each bag.
![Page 156: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/156.jpg)
2 Paired SamplesE.g. Old Textbook 7.32:
The data are the two Vitamin C measurements, made for each bag.
a. Set up hypotheses to examine the question of interest.
![Page 157: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/157.jpg)
2 Paired SamplesE.g. Old Textbook 7.32:
The data are the two Vitamin C measurements, made for each bag.
a. Set up hypotheses to examine the question of interest.
b. Perform the significance test, and summarize the result.
![Page 158: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/158.jpg)
2 Paired SamplesE.g. Old Textbook 7.32:
The data are the two Vitamin C measurements, made for each bag.
a. Set up hypotheses to examine the question of interest.
b. Perform the significance test, and summarize the result.
c. Find 95% CIs for the factory mean, and the Haiti mean, and the mean change.
![Page 159: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/159.jpg)
2 Paired SamplesE.g. Old Textbook 7.32:
a. Sample average difference =
Computed as:
33.5D
![Page 160: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/160.jpg)
2 Paired SamplesE.g. Old Textbook 7.32:
a. Sample average difference =
Computed as:
33.5D
niYXD iii ,,1,
![Page 161: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/161.jpg)
2 Paired SamplesE.g. Old Textbook 7.32:
a. Sample average difference =
Computed as:
33.5D
niYXD iii ,,1,
![Page 162: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/162.jpg)
2 Paired SamplesE.g. Old Textbook 7.32:
a. Sample average difference =
Computed as:
Then average
33.5D
niYXD iii ,,1,
![Page 163: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/163.jpg)
2 Paired SamplesE.g. Old Textbook 7.32:
a. Sample average difference =
Some evidence factory is bigger,
is it strong evidence???
33.5D
![Page 164: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/164.jpg)
2 Paired SamplesE.g. Old Textbook 7.32:
a. Sample average difference =
Some evidence factory is bigger,
is it strong evidence???
Let = Difference: Haiti – Factory
33.5D
D
![Page 165: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/165.jpg)
2 Paired SamplesE.g. Old Textbook 7.32:
a. Sample average difference =
Some evidence factory is bigger,
is it strong evidence???
Let = Difference: Haiti – Factory
1-sided, from “idea of loss”
33.5D
D0:0 DH
0: DAH
![Page 166: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/166.jpg)
2 Paired SamplesE.g. Old Textbook 7.32:
b. 0|..33.5 DcmorDPvalueP
![Page 167: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/167.jpg)
2 Paired SamplesE.g. Old Textbook 7.32:
b. 0|..33.5 DcmorDPvalueP
0|33.5 DDP
![Page 168: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/168.jpg)
2 Paired SamplesE.g. Old Textbook 7.32:
b. 0|..33.5 DcmorDPvalueP
0|33.5 DDP
D
DD nsnsD
P |33.5
![Page 169: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/169.jpg)
2 Paired SamplesE.g. Old Textbook 7.32:
b. 0|..33.5 DcmorDPvalueP
0|33.5 DDP
D
DD nsnsD
P |33.5
D
Dn nstP |33.5
1
![Page 170: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/170.jpg)
2 Paired SamplesE.g. Old Textbook 7.32:b.
D
Dn nstPvalueP |33.5
1
![Page 171: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/171.jpg)
2 Paired SamplesE.g. Old Textbook 7.32:b.
But recall how TDIST works
(1 – tail: upper probability only)
D
Dn nstPvalueP |33.5
1
![Page 172: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/172.jpg)
2 Paired SamplesE.g. Old Textbook 7.32:b.
But recall how TDIST works:
=
(symmetry)
D
Dn nstPvalueP |33.5
1
![Page 173: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/173.jpg)
2 Paired SamplesE.g. Old Textbook 7.32:b.
But recall how TDIST works:
=
So compute with:
D
Dn nstPvalueP |33.5
1
DD
n nstPvalueP |33.5
1
![Page 174: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/174.jpg)
2 Paired SamplesE.g. Old Textbook 7.32:
b. Excel Computation:
Class Example 15, Part 3http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg15.xls
![Page 175: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/175.jpg)
2 Paired SamplesE.g. Old Textbook 7.32:
b. Excel Computation:
Class Example 15, Part 3http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg15.xls
Standard deviation
of differences, sD
![Page 176: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/176.jpg)
2 Paired SamplesE.g. Old Textbook 7.32:
b. Excel Computation:
Class Example 15, Part 3http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg15.xls
Standard deviation
of differences, sD
P-value
![Page 177: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/177.jpg)
2 Paired SamplesE.g. Old Textbook 7.32:
b. Excel Computation:
Class Example 15, Part 3http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg15.xls
P-value = 1.87 x 10-5
![Page 178: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/178.jpg)
2 Paired SamplesE.g. Old Textbook 7.32:
b. Excel Computation:
Class Example 15, Part 3http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg15.xls
P-value = 1.87 x 10-5
Interpretation: very strong evidence
![Page 179: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/179.jpg)
2 Paired SamplesE.g. Old Textbook 7.32:
b. Excel Computation:
Class Example 15, Part 3http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg15.xls
P-value = 1.87 x 10-5
Interpretation: very strong evidence
either yes-no or gray level
![Page 180: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/180.jpg)
2 Paired SamplesVariations:
1. EXCEL function TTEST is useful here
![Page 181: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/181.jpg)
2 Paired SamplesVariations:
1. EXCEL function TTEST is useful here
![Page 182: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/182.jpg)
2 Paired SamplesVariations:
1. EXCEL function TTEST is useful here
(same answer as above)
![Page 183: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/183.jpg)
2 Paired SamplesVariations:
1. EXCEL function TTEST is useful here
Notes:
a. Type is paired (discuss others later)
![Page 184: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/184.jpg)
2 Paired SamplesVariations:
1. EXCEL function TTEST is useful here
Notes:
a. Type is paired (discuss others later)
b. Get same answer from swapping Array 1 and Array 2
![Page 185: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/185.jpg)
2 Paired SamplesVariations:
1. EXCEL function TTEST is useful here
Notes:
a. Type is paired (discuss others later)
b. Get same answer from swapping Array 1 and Array 2
![Page 186: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/186.jpg)
2 Paired SamplesVariations:
1. EXCEL function TTEST is useful here
Notes:
a. Type is paired (discuss others later)
b. Get same answer from swapping Array 1 and Array 2
c. This is something Excel does well
![Page 187: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/187.jpg)
2 Paired SamplesVariations:
2. Can also use:
Data Data Analysis T-test Paired
![Page 188: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/188.jpg)
2 Paired SamplesVariations:
2. Can also use:
Data Data Analysis T-test Paired
to give detailed results
![Page 189: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/189.jpg)
2 Paired SamplesVariations:
2. Can also use:
Data Data Analysis T-test Paired
to give detailed results
e.g. d.f. = 26
![Page 190: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/190.jpg)
2 Paired SamplesVariations:
2. Can also use:
Data Data Analysis T-test Paired
to give detailed results
e.g. d.f. = 26
P-value same
![Page 191: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/191.jpg)
2 Paired SamplesVariations:
2. Can also use:
Data Data Analysis T-test Paired
to give detailed results
e.g. d.f. = 26
P-value same
(others we haven’t learned yet)
![Page 192: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/192.jpg)
2 Paired SamplesE.g. Old Textbook 7.32:
c. Confidence Intervals
See Class Example 15, Part 3chttp://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg15.xls
Margin of error = ns
nTINVm 1,05.0
![Page 193: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/193.jpg)
2 Paired SamplesE.g. Old Textbook 7.32:
c. Confidence Intervals
See Class Example 15, Part 3chttp://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg15.xls
Margin of error =
(same as above, but NORMINV TINV)
ns
nTINVm 1,05.0
![Page 194: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/194.jpg)
2 Paired SamplesE.g. Old Textbook 7.32:
c. Confidence Intervals
See Class Example 15, Part 3chttp://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg15.xls
Margin of error =
(same as above, but NORMINV TINV)
So CI has endpoints:
ns
nTINVm 1,05.0
mX
![Page 195: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/195.jpg)
Paired Sampling CIs & TestsHW:
7.33, 7.35, 7.39
Interpret P-values: (i) yes-no (ii) gray-level
(suggestion: use TTEST, for hypo tests)
![Page 196: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/196.jpg)
And now for somethingcompletely different…
Does the statement, “We've always done it like that” ring any bells?
The US standard railroad gauge (distance between the rails) is 4 feet, 8.5 inches.
That's an exceedingly odd number.
Why was that gauge used?
![Page 197: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/197.jpg)
And now for somethingcompletely different…
Because that's the way they built them in England, and English expatriates built the US Railroads.
Why did the English build them like that?
Because the first rail lines were built by the same people who built the pre-railroad tramways, and that's the gauge they used.
![Page 198: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/198.jpg)
And now for somethingcompletely different…
Why did "they" use that gauge then?
Because the people who built the tramways used the same jigs and tools that they used for building wagons, which used that wheel spacing.
![Page 199: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/199.jpg)
And now for somethingcompletely different…
Okay! Why did the wagons have that particular odd wheel spacing?
Well, if they tried to use any other spacing, the wagon wheels would break on some of the old, long distance roads in England , because that's the spacing of the wheel ruts.
![Page 200: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/200.jpg)
And now for somethingcompletely different…
So who built those old rutted roads?
Imperial Rome built the first long distance roads in Europe (and England ) for their legions. The roads have been used ever since.
And the ruts in the roads?
Roman war chariots formed the initial ruts, which everyone else had to match for fear of destroying their wagon wheels.
Since the chariots were made for Imperial Rome , they were all alike in the matter of wheel spacing.
![Page 201: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/201.jpg)
And now for somethingcompletely different…
The United States standard railroad gauge of 4 feet, 8.5 inches is derived from the original specifications for an Imperial Roman war chariot.
And bureaucracies live forever.
So the next time you are handed a specification and wonder what horse's ass came up with it, you may be exactly right, because the Imperial Roman army chariots were made just wide enough to accommodate the back ends of two war horses!
![Page 202: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/202.jpg)
And now for somethingcompletely different…
When you see a Space Shuttle sitting on its launch pad, there are two big booster rockets attached to the sides of the main fuel tank.
These are solid rocket boosters, or SRBs.
The SRBs are made by Thiokol at their factory at Utah.
The engineers who designed the SRBs would have preferred to make them a bit fatter, but the SRBs had to be shipped by train from the factory to the launch site.
![Page 203: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/203.jpg)
And now for somethingcompletely different…
The railroad line from the factory happens to run through a tunnel in the mountains.
The SRBs had to fit through that tunnel. The tunnel is slightly wider than the railroad track, and the railroad track, as you now know, is about as wide as two horses' behinds.
So, a major Space Shuttle design feature of what is arguably the world's most advanced transportation system was determined over two thousand years ago by the width of a horse's ass.
![Page 204: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/204.jpg)
And now for somethingcompletely different…
- And –
you thought being a HORSE'S ASS wasn't important!
![Page 205: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/205.jpg)
Carolina Course Evaluation
• Please give me your opinions
![Page 206: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/206.jpg)
Carolina Course Evaluation
• Please give me your opinions
Most highly sought:
Written suggestions for improvement
![Page 207: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/207.jpg)
Carolina Course Evaluation
• Please give me your opinions
Most highly sought:
Written suggestions for improvement
• Please fill out with # 2 pencil (black pen?)
![Page 208: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/208.jpg)
Carolina Course Evaluation
• Please give me your opinions
Most highly sought:
Written suggestions for improvement
• Please fill out with # 2 pencil (black pen?)
• Return to student volunteer
• Will turn in independently from me
![Page 209: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/209.jpg)
Carolina Course Evaluation
• Please give me your opinions
Most highly sought:
Written suggestions for improvement
• Please fill out with # 2 pencil (black pen?)
• Return to student volunteer
• Will turn in independently from me
• Dept/Course/Section: STOR 155 001
• Instructor: J. S. Marron
![Page 210: Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form](https://reader038.vdocuments.us/reader038/viewer/2022110304/551c40845503467b488b4ac2/html5/thumbnails/210.jpg)
STOR 155 001, Course ID: 3021121128. Over the course of the semester, how frequently did you review the audio/screen
recordings? (S/D) Never. I didn't know that they were available.
(D) Never. I decided not to.
(N) Seldom
(A) Sometimes
(S/A) Often
29. Did you review the recordings before taking a test or exam? (S/D) Yes / (S/A) No
30. Did you review the recordings after you missed class? (S/D) Yes / (S/A) No
31. Did you review the recordings when you didn't understand something from class? (S/D) Yes / (S/A) No
32. The recordings were helpful for me as a study aid. (S/D D N A S/A)
33. I was less likely to attend class because I knew I would have access to the lecture materials online. (S/D D N A S/A)