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

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Reading In Textbook Approximate Reading for Today’s Material: Pages , , Approximate Reading for Next Class: Pages ,

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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

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

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I.Form of Relationship Linear: Data approximately follow a line Previous Class Scores Example Final vs. High values of HW is “best” Nonlinear: Data follows different pattern Nice Example: Bralower’s Fossil Data

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Bralower’s Fossil Data From T. Bralower, formerly of Geological Sci.T. BralowerGeological Sci. Studies Global Climate, millions of years ago

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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

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III. Strength of Relationship Idea: How close are points to lying on a line? Revisit Class Scores Example:

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Comparing Scatterplots Additional Useful Visual Tool

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Comparing Scatterplots Additional Useful Visual Tool: Overlaying multiple data sets

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

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

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

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Comparing Scatterplots HW HW: 2.21, 2.25

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III. Strength of Relationship Idea: How close are points to lying on a line? Revisit Class Scores Example:

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

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Section 2.2: Correlation Main Idea: Quantify Strength of Relationship

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

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

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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

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

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

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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)

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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

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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)

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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

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Correlation - Approach Numerical Approach

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Correlation - Approach Numerical Approach: for symmetric around

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

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Correlation - Approach Numerical Approach: for symmetric around has similar properties Worked out Example :

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

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

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

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

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

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

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Correlation – Graphical View But not all goals are satisfied

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

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

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

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

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

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

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Correlation - Approach Solution to above problems

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Correlation - Approach Solution to above problems: Standardize!

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

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

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Correlation - Example Revisit above example r is always same, and ~1, for (a)

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Correlation - Example Revisit above example r is always same, and ~1, for (a), (c)

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Correlation - Example Revisit above example r is always same, and ~1, for (a), (c), (d)

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Correlation - Example Revisit above example r is always same, and ~1, for (a), (c), (d) r < 0, and not so close to -1, for (b)

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Correlation - Example Revisit Class Scores Example: Final Exam vs. HW Correlation = r = 0.73 Strongest Dependence

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Correlation - Example Revisit Class Scores Example: MT1 vs. HW Correlation = r = 0.65 Weaker Dependence

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Correlation - Example Revisit Class Scores Example: MT2 vs. MT1 Correlation = r = 0.57 Weakest Dependence

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Correlation - Example Revisit Class Scores Example: r is always > 0 (makes sense, since all trend upwards)

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Correlation - Example Revisit Class Scores Example: r is always > 0 r is biggest for Final vs. HW

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Correlation - Example Revisit Class Scores Example: r is always > 0 r is biggest for Final vs. HW (visually strongest relationship)

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Correlation - Example Revisit Class Scores Example: r is always > 0 r is biggest for Final vs. HW (visually strongest relationship) r is smallest for MT2 vs. MT1

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Correlation - Example Revisit Class Scores Example: r is always > 0 r is biggest for Final vs. HW (visually strongest relationship) r is smallest for MT2 vs. MT1 (visually weakest relationship)

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Correlation – Computation From Class Scores Example:

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Correlation – Computation From Class Scores Example: Use Excel function: CORREL

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Correlation – Computation From Class Scores Example: Use Excel function: CORREL

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Correlation – Computation From Class Scores Example: Use Excel function: CORREL Range of Xs

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Correlation – Computation From Class Scores Example: Use Excel function: CORREL Range of Xs Range of Ys

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Correlation – Computation From Class Scores Example: Use Excel function: CORREL Range of Xs Range of Ys Output is correlation, r

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Correlation - Example Fun Example from Publisher’s Website:

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Correlation - Example Fun Example from Publisher’s Website: Choose Statistical Applets

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Correlation - Example Fun Example from Publisher’s Website: Choose Statistical Applets Correlation and Regression

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Correlation - Example Fun Example from Publisher’s Website: Choose Statistical Applets Correlation and Regression Gives feeling for how correlation is affected by changing data.

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Correlation - Example Correlation and Regression Applet

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

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

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

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

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

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

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

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

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

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

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Correlation - Example Correlation and Regression Applet Just 2 more outliers

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

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

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

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

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

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

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Correlation - HW HW: a

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

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

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

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

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

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Research Corner Relationship between more than 2 variables?

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

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

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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)

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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)

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Research Corner Visualization?

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Research Corner Visualization? What is “scatterplot”?

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

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

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

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Research Corner Toy Example, modeling “gene expression”

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Research Corner Multivariate View: Highlight one

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Research Corner Multivariate View: Value of variable 1

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Research Corner Multivariate View: Value of variable 2

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Research Corner Multivariate View: Value of variable 3

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

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

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

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

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

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

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

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

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

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

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

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

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Research Corner Multivariate View: All 3 planes

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

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Research Corner Multivariate View: 1-d projections on diagonal

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Research Corner Multivariate View: 2-d views off diagonal

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

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

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2 Sample Inference Main Idea: Previously studied single populations

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

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

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

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

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

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

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

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

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2 Sample Inference Location in Text

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

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

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2 Sample Measurement Error Easy Case: Paired Differences

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

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

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

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

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

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

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

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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.

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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

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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.

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

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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:

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2 Paired Samples Available in Class Example 15:

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2 Paired Samples Available in Class Example 15: Factory, Cells B38:B64

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2 Paired Samples Available in Class Example 15: Factory, Cells B38:B64 Haiti, Cells C38:C64

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

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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.

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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.

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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.

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

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

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

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

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

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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

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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”

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

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

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

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

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

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

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

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

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

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2 Paired Samples E.g. Old Textbook 7.32: b.Excel Computation: Class Example 15, Part 3 Standard deviation of differences, s D

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2 Paired Samples E.g. Old Textbook 7.32: b.Excel Computation: Class Example 15, Part 3 Standard deviation of differences, s D P-value

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2 Paired Samples E.g. Old Textbook 7.32: b.Excel Computation: Class Example 15, Part 3 P-value = 1.87 x 10 -5

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2 Paired Samples E.g. Old Textbook 7.32: b.Excel Computation: Class Example 15, Part 3 P-value = 1.87 x Interpretation: very strong evidence

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2 Paired Samples E.g. Old Textbook 7.32: b.Excel Computation: Class Example 15, Part 3 P-value = 1.87 x Interpretation: very strong evidence either yes-no or gray level

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

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

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

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

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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

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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

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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

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

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

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

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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

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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)

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2 Paired Samples E.g. Old Textbook 7.32: c.Confidence Intervals See Class Example 15, Part 3c Margin of error =

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2 Paired Samples E.g. Old Textbook 7.32: c.Confidence Intervals See Class Example 15, Part 3c Margin of error = (same as above, but NORMINV TINV)

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2 Paired Samples E.g. Old Textbook 7.32: c.Confidence Intervals See Class Example 15, Part 3c Margin of error = (same as above, but NORMINV TINV) So CI has endpoints:

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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)

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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?

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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.

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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.

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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.

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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.

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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!

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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.

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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.

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

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Carolina Course Evaluation Please give me your opinions

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

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

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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

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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 Instructor: J. S. Marron

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STOR , Course ID: 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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