# Last Time T Distribution –Confidence Intervals –Hypothesis tests Relationships Between Variables –Scatterplots (visualization) Aspects of Relations –Form.

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

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

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

Important Aspects of Relations I.Form of Relationship II.Direction of Relationship III.Strength of Relationship

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

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

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

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

Comparing Scatterplots Additional Useful Visual Tool

Comparing Scatterplots Additional Useful Visual Tool: Overlaying multiple data sets

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

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

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

Comparing Scatterplots HW HW: 2.21, 2.25

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

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

Section 2.2: Correlation Main Idea: Quantify Strength of Relationship

Section 2.2: Correlation Main Idea: Quantify Strength of Relationship Context: –A numerical summary

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

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

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

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

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)

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

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)

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

Correlation - Approach Numerical Approach

Correlation - Approach Numerical Approach: for symmetric around

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

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

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

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

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

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

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

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

Correlation – Graphical View But not all goals are satisfied

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

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

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

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

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

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

Correlation - Approach Solution to above problems

Correlation - Approach Solution to above problems: Standardize!

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

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

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

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

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

Correlation - Example Revisit above example http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg18-new.xls r is always same, and ~1, for (a), (c), (d) r < 0, and not so close to -1, for (b)

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

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

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

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)

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

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)

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

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)

Correlation – Computation From Class Scores Example: http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg16.xls

Correlation – Computation From Class Scores Example: http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg16.xls Use Excel function: CORREL

Correlation – Computation From Class Scores Example: http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg16.xls Use Excel function: CORREL

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

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

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

Correlation - Example Fun Example from Publisher’s Website: http://courses.bfwpub.com/ips6e.php

Correlation - Example Fun Example from Publisher’s Website: http://courses.bfwpub.com/ips6e.php Choose Statistical Applets

Correlation - Example Fun Example from Publisher’s Website: http://courses.bfwpub.com/ips6e.php Choose Statistical Applets Correlation and Regression

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.

Correlation - Example Correlation and Regression Applet

Correlation - Example Correlation and Regression Applet I clicked to put down 2 points

Correlation - Example Correlation and Regression Applet I clicked to put down 2 points Applet computed correlation, r

Correlation - Example Correlation and Regression Applet Applet computed correlation, r r = -1, since points on line trending down

Correlation - Example Correlation and Regression Applet Try several points close to some line

Correlation - Example Correlation and Regression Applet Try several points close to some line r ≈ -1, since points near line trending down

Correlation - Example Correlation and Regression Applet Add more points with goal of r ≈ -0.95

Correlation - Example Correlation and Regression Applet Add more points with goal of r ≈ -0.95

Correlation - Example Correlation and Regression Applet Add more points with goal of r ≈ -0.95

Correlation - Example Correlation and Regression Applet Now add a single outlier

Correlation - Example Correlation and Regression Applet Now add a single outlier Major impact on r -0.95  -0.35

Correlation - Example Correlation and Regression Applet Just 2 more outliers

Correlation - Example Correlation and Regression Applet Just 2 more outliers Leads to r > 0

Correlation - Example Correlation and Regression Applet Just 2 more outliers Leads to r > 0 (Outliers have major impact)

Correlation - Example Correlation and Regression Applet Weakness of correlation, r

Correlation - Example Correlation and Regression Applet Weakness of correlation, r Measures linear dependence

Correlation - Example Correlation and Regression Applet Weakness of correlation, r Can have r ≈ 0

Correlation - Example Correlation and Regression Applet Weakness of correlation, r Can have r ≈ 0, yet strong non-linear dependence

Correlation - HW HW: 2.31 2.33 2.39a

Correlation - Outliers Caution: Outliers can strongly affect correlation, r

Correlation - Example Correlation and Regression Applet Add more points with goal of r ≈ -0.95

Correlation - Example Correlation and Regression Applet Now add a single outlier Major impact on r -0.95  -0.35

Correlation - Example Correlation and Regression Applet Just 2 more outliers Leads to r > 0 (Outliers have major impact)

Correlation - Outliers Caution: Outliers can strongly affect correlation, r HW: 2.39b 2.45

Research Corner Relationship between more than 2 variables?

Research Corner Relationship between more than 2 variables? Each data point is (x 1, x 2, …, x d ) Called a “d-tuple”

Research Corner Relationship between more than 2 variables? Each data point is (x 1, x 2, …, x d ) Eg: d = 3 (ordered triple)

Research Corner Relationship between more than 2 variables? Each data point is (x 1, x 2, …, x d ) Eg: d = 3 (ordered triple) (height, weight, age)

Research Corner Relationship between more than 2 variables? Each data point is (x 1, x 2, …, x d ) Eg: d = 3 (ordered triple) (height, weight, age) (HW, MT1, Final)

Research Corner Visualization?

Research Corner Visualization? What is “scatterplot”?

Research Corner Visualization? What is “scatterplot”? How can we “see” structure in data?

Research Corner Visualization? Explore d = 3 (3d)

Research Corner Visualization? Explore d = 3 (3d) So can visualize “point cloud”

Research Corner Toy Example, modeling “gene expression”

Research Corner Multivariate View: Highlight one

Research Corner Multivariate View: Value of variable 1

Research Corner Multivariate View: Value of variable 2

Research Corner Multivariate View: Value of variable 3

Research Corner Multivariate View: 1-d projection, X-axis

Research Corner Multivariate View: X – Projection, 1-d View

Research Corner Multivariate View: 1-d projection, Y-axis

Research Corner Multivariate View: Y – Projection, 1-d View

Research Corner Multivariate View: 1-d projection, Z-axis

Research Corner Multivariate View: Z – Projection, 1-d View

Research Corner Multivariate View: 2-d Projection XY-plane

Research Corner Multivariate View: XY – projection, 2-d view

Research Corner Multivariate View: 2-d Projection XZ-plane

Research Corner Multivariate View: XZ – projection, 2-d view

Research Corner Multivariate View: 2-d Projection YZ-plane

Research Corner Multivariate View: YZ – projection, 2-d view

Research Corner Multivariate View: All 3 planes

Research Corner Multivariate View Now collect these views on a single page

Research Corner Multivariate View: 1-d projections on diagonal

Research Corner Multivariate View: 2-d views off diagonal

Research Corner Multivariate View: switch off color (usual view)

Research Corner Multivariate View (Useful summary of structure in data)

2 Sample Inference Main Idea: Previously studied single populations

2 Sample Inference Main Idea: Previously studied single populations Modeled as

2 Sample Inference Main Idea: Previously studied single populations Modeled as: –Measurement Error

2 Sample Inference Main Idea: Previously studied single populations Modeled as: –Measurement Error

2 Sample Inference Main Idea: Previously studied single populations Modeled as: –Measurement Error –Counts

2 Sample Inference Main Idea: Previously studied single populations Modeled as: –Measurement Error –Counts

2 Sample Inference Main Idea: Previously studied single populations Modeled as: –Measurement Error –Counts Did Inference

2 Sample Inference Main Idea: Previously studied single populations Modeled as: –Measurement Error –Counts Did Inference: –Confidence Intervals –Hypothesis Tests

2 Sample Inference Main Idea: Extend to two populations Modeled as: –Measurement Error –Counts Will Develop Inference: –Confidence Intervals –Hypothesis Tests

2 Sample Inference Location in Text

2 Sample Inference Location in Text: Measurement Error –Sec. 7.1 –Sec. 7.2

2 Sample Inference Location in Text: Measurement Error –Sec. 7.1 –Sec. 7.2 Counts –Sec. 8.2

2 Sample Measurement Error Easy Case: Paired Differences

2 Sample Measurement Error Easy Case: Paired Differences Have Treatment 1:

2 Sample Measurement Error Easy Case: Paired Differences Have Treatment 1: Treatment 2:

2 Sample Measurement Error Easy Case: Paired Differences Have Treatment 1: Treatment 2:

2 Sample Measurement Error Easy Case: Paired Differences Have Treatment 1: Treatment 2: Important: Measurements Connected

2 Sample Measurement Error Easy Case: Paired Differences Have Treatment 1: Treatment 2: Important: Measurements Connected, e.g. made on same objects

2 Sample Measurement Error Easy Case: Paired Differences Have Treatment 1: Treatment 2: Approach: Apply 1 sample methods

2 Sample Measurement Error Easy Case: Paired Differences Have Treatment 1: Treatment 2: Approach: Apply 1 sample methods to:

2 Paired Samples E.g. Old Textbook 7.32: Researchers studying Vitamin C in a product were concerned about loss of Vitamin C during shipment and storage.

2 Paired Samples E.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

2 Paired Samples E.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.

2 Paired Samples E.g. Old Textbook 7.32: The data are the two Vitamin C measurements, made for each bag.

2 Paired Samples E.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

2 Paired Samples Available in Class Example 15: http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg15.xls

2 Paired Samples Available in Class Example 15: http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg15.xls Factory, Cells B38:B64

2 Paired Samples Available 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

2 Paired Samples E.g. Old Textbook 7.32: The data are the two Vitamin C measurements, made for each bag.

2 Paired Samples E.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.

2 Paired Samples E.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.

2 Paired Samples E.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.

2 Paired Samples E.g. Old Textbook 7.32: a. Sample average difference = Computed as:

2 Paired Samples E.g. Old Textbook 7.32: a. Sample average difference = Computed as:

2 Paired Samples E.g. Old Textbook 7.32: a. Sample average difference = Computed as:

2 Paired Samples E.g. Old Textbook 7.32: a. Sample average difference = Computed as: Then average

2 Paired Samples E.g. Old Textbook 7.32: a. Sample average difference = Some evidence factory is bigger, is it strong evidence???

2 Paired Samples E.g. Old Textbook 7.32: a. Sample average difference = Some evidence factory is bigger, is it strong evidence??? Let = Difference: Haiti – Factory

2 Paired Samples E.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”

2 Paired Samples E.g. Old Textbook 7.32: b.

2 Paired Samples E.g. Old Textbook 7.32: b.

2 Paired Samples E.g. Old Textbook 7.32: b.

2 Paired Samples E.g. Old Textbook 7.32: b.

2 Paired Samples E.g. Old Textbook 7.32: b.

2 Paired Samples E.g. Old Textbook 7.32: b. But recall how TDIST works (1 – tail: upper probability only)

2 Paired Samples E.g. Old Textbook 7.32: b. But recall how TDIST works: = (symmetry)

2 Paired Samples E.g. Old Textbook 7.32: b. But recall how TDIST works: = So compute with:

2 Paired Samples E.g. Old Textbook 7.32: b.Excel Computation: Class Example 15, Part 3 http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg15.xls

2 Paired Samples E.g. Old Textbook 7.32: b.Excel Computation: Class Example 15, Part 3 http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg15.xls Standard deviation of differences, s D

2 Paired Samples E.g. Old Textbook 7.32: b.Excel Computation: Class Example 15, Part 3 http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg15.xls Standard deviation of differences, s D P-value

2 Paired Samples E.g. Old Textbook 7.32: b.Excel Computation: Class Example 15, Part 3 http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg15.xls P-value = 1.87 x 10 -5

2 Paired Samples E.g. Old Textbook 7.32: b.Excel Computation: Class Example 15, Part 3 http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg15.xls P-value = 1.87 x 10 -5 Interpretation: very strong evidence

2 Paired Samples E.g. Old Textbook 7.32: b.Excel Computation: Class Example 15, Part 3 http://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

2 Paired Samples Variations: 1.EXCEL function TTEST is useful here

2 Paired Samples Variations: 1.EXCEL function TTEST is useful here

2 Paired Samples Variations: 1.EXCEL function TTEST is useful here (same answer as above)

2 Paired Samples Variations: 1.EXCEL function TTEST is useful here Notes: a.Type is paired (discuss others later)

2 Paired Samples Variations: 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

2 Paired Samples Variations: 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

2 Paired Samples Variations: 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

2 Paired Samples Variations: 2.Can also use: Data  Data Analysis  T-test Paired

2 Paired Samples Variations: 2.Can also use: Data  Data Analysis  T-test Paired to give detailed results

2 Paired Samples Variations: 2.Can also use: Data  Data Analysis  T-test Paired to give detailed results e.g. d.f. = 26

2 Paired Samples Variations: 2.Can also use: Data  Data Analysis  T-test Paired to give detailed results e.g. d.f. = 26 P-value same

2 Paired Samples Variations: 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)

2 Paired Samples E.g. Old Textbook 7.32: c.Confidence Intervals See Class Example 15, Part 3c http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg15.xls Margin of error =

2 Paired Samples E.g. Old Textbook 7.32: c.Confidence Intervals See Class Example 15, Part 3c http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg15.xls Margin of error = (same as above, but NORMINV  TINV)

2 Paired Samples E.g. Old Textbook 7.32: c.Confidence Intervals See Class Example 15, Part 3c http://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:

Paired Sampling CIs & Tests HW: 7.33, 7.35, 7.39 Interpret P-values: (i) yes-no (ii) gray-level (suggestion: use TTEST, for hypo tests)

And now for something completely 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?

And now for something completely 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.

And now for something completely 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.

And now for something completely 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.

And now for something completely 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.

And now for something completely 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!

And now for something completely 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.

And now for something completely 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.

And now for something completely different… - And – you thought being a HORSE'S ASS wasn't important!

Carolina Course Evaluation Please give me your opinions Most highly sought: Written suggestions for improvement

Carolina Course Evaluation Please give me your opinions Most highly sought: Written suggestions for improvement Please fill out with # 2 pencil (black pen?)

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

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

STOR 155 001, Course ID: 30211211 28.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)

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