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

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?

xi = HW, yi = 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 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



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


http://www.geosc.psu.edu/people/faculty/personalpages/tbralower/index.html

http://www.geosci.unc.edu/

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


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


Additional Useful Visual Tool:

• Overlaying multiple data sets




• Allows comparison





• Use different colors or symbols





• Use different colors or symbols

• Easy in EXCEL (colors are automatic)

Comparing Scatterplots HW

HW:

2.21, 2.25







Now get quantitative

Section 2.2: Correlation

Main Idea: Quantify Strength of Relationship



Context:

– A numerical summary



Context:


– In spirit of mean and standard deviation



Context:


– In spirit of mean and standard deviation

– But now applies to pairs of variables



Specific Goals



Specific Goals:

– Near 1: for positive relat’ship & nearly linear



Specific Goals:


– > 0: for positive relationship (slopes up)



Specific Goals:



– = 0: for no relationship



Specific Goals:




– < 0: for negative relationship (slopes down)



Specific Goals:




– < 0: for negative relationship (slopes down)

– Near -1: for negative relat’ship & nearly linear

Correlation - Approach

Numerical Approach


Numerical Approach:

for symmetric around )0,0(),( ii yx


Numerical Approach:

for symmetric around

has similar properties

)0,0(),( ii yx

n

iii yx

1


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

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

n

iii yx

1




n

iii yx

1




• < 0 for negative relationship

n

iii yx

1




• < 0 for negative relationship

n

iii yx

1



• Bigger for data closer to line

n

iii yx

1



• Bigger for data closer to line

n

iii yx

1


But not all goals are satisfied


Problem 1: Not between -1 & 1


Problem 2: Feels “Scale”, see plot (c)

(just 10 1 vertical rescaling of)


Problem 2: Feels “Scale”, see plot (c)

(just 10 1 vertical rescaling of)

( feels factor of 1/10)

n

iii yx

1


Problem 3: Feels “Shift” even more, see (d)

(even gets sign wrong!)




• Data trend upwards




• Data trend upwards

• But < 0

n

iii yx

1


Solution to above problems


Solution to above problems:

Standardize!



Standardize!

Define Correlation r



Standardize!

Define Correlation

n

i y

i

x

i

s

yy

s

xxr

1

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)




• r is always same, and ~1, for (a), (c)




• r is always same, and ~1, for (a), (c), (d)




• r is always same, and ~1, for (a), (c), (d)

• r < 0, and not so close to -1, for (b)




Final Exam vs. HW

Correlation = r = 0.73

Strongest Dependence




MT1 vs. HW


Weaker Dependence




MT2 vs. MT1


Weakest Dependence




• r is always > 0

(makes sense, since all trend upwards)




• r is always > 0

• r is biggest for Final vs. HW



• r is always > 0


(visually strongest relationship)



• r is always > 0



• r is smallest for MT2 vs. MT1



• r is always > 0



• 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



Use Excel function: CORREL







• Range of Xs




• Range of Xs

• Range of Ys




• Range of Xs

• Range of Ys

• Output is correlation, r


Fun Example from Publisher’s Website:

http://courses.bfwpub.com/ips6e.php




Choose

• Statistical Applets




Choose


• Correlation and Regression




Choose


• Correlation and Regression

Gives feeling for how correlation is affected by changing data.


Correlation and Regression Applet



I clicked to put

down 2 points



I clicked to put

down 2 points

Applet computed

correlation, r



Applet computed

correlation, r

r = -1, since

points on line

trending down



Try several points

close to some line



Try several points

close to some line

r ≈ -1, since

points near line

trending down



Add more points

with goal of

r ≈ -0.95



Add more points

with goal of

r ≈ -0.95



Add more points

with goal of

r ≈ -0.95



Now add a single

outlier



Now add a single

outlier

Major impact on r

-0.95 -0.35



Just 2 more

outliers



Just 2 more

outliers

Leads to r > 0



Just 2 more

outliers

Leads to r > 0

(Outliers have

major impact)



Weakness of

correlation, r



Weakness of

correlation, r

Measures linear

dependence



Weakness of

correlation, r

Can have r ≈ 0



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



Add more points

with goal of

r ≈ -0.95



Now add a single

outlier

Major impact on r

-0.95 -0.35



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


Each data point is (x1, x2, … , xd)

Called a “d-tuple”

Research Corner



Eg: d = 3 (ordered triple)

Research Corner




(height, weight, age)

Research Corner




(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 CornerToy Example, modeling “gene expression”

Research CornerMultivariate View: Highlight one

Research CornerMultivariate View: Value of variable 1



Research CornerMultivariate View: 1-d projection, X-axis

Research CornerMultivariate View: X – Projection, 1-d View

Research CornerMultivariate View: 1-d projection, Y-axis

Research CornerMultivariate View: Y – Projection, 1-d View

Research CornerMultivariate View: 1-d projection, Z-axis

Research CornerMultivariate View: Z – Projection, 1-d View

Research CornerMultivariate View: 2-d Projection XY-plane

Research CornerMultivariate View: XY – projection, 2-d view

Research CornerMultivariate View: 2-d Projection XZ-plane

Research CornerMultivariate View: XZ – projection, 2-d view

Research CornerMultivariate View: 2-d Projection YZ-plane

Research CornerMultivariate View: YZ – projection, 2-d view

Research CornerMultivariate View: All 3 planes

Research CornerMultivariate View

Now collect these views

on a single page

Research CornerMultivariate View: 1-d projections on diagonal

Research CornerMultivariate View: 2-d views off diagonal

Research CornerMultivariate View: switch off color (usual view)

Research CornerMultivariate View

(Useful summary of structure in data)

2 Sample InferenceMain Idea:

• Previously studied single populations



• Modeled as



• Modeled as:– Measurement Error




nNX

,~




– Counts

nNX

,~




– Counts

nNX

,~

n

pppNppnBiX

)1(,~ˆ),,(~




– Counts

• Did Inference

nNX

,~

n

pppNppnBiX

)1(,~ˆ),,(~




– Counts

• Did Inference:– Confidence Intervals

– Hypothesis Tests

nNX

,~

n

pppNppnBiX

)1(,~ˆ),,(~


• Extend to two populations


– Counts

• Will Develop Inference:– Confidence Intervals

– Hypothesis Tests

1

111 ,~n

NX

2

222 ,~n

NX

),(~ 111 pnBiX ),(~ 222 pnBiX

2 Sample InferenceLocation in Text

2 Sample InferenceLocation in Text:

• Measurement Error– Sec. 7.1

– Sec. 7.2

1

111 ,~n

NX

2

222 ,~n

NX

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

2 Sample Measurement Error

Easy Case: Paired Differences



Have Treatment 1: nXXX ,,, 21



Have Treatment 1:

Treatment 2:

nXXX ,,, 21

nYYY ,,, 21



Have Treatment 1:

Treatment 2:

nXXX ,,, 21

nYYY ,,, 21



Have Treatment 1:

Treatment 2:

Important: Measurements Connected

nXXX ,,, 21

nYYY ,,, 21



Have Treatment 1:

Treatment 2:

Important: Measurements Connected,

e.g. made on same objects

nXXX ,,, 21

nYYY ,,, 21



Have Treatment 1:

Treatment 2:

Approach: Apply 1 sample methods

nXXX ,,, 21

nYYY ,,, 21



Have Treatment 1:

Treatment 2:

Approach: Apply 1 sample methods to:

nXXX ,,, 21

nYYY ,,, 21

niYXD iii ,,1,

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.


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


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.


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


Factory,

Cells B38:B64


Factory,

Cells B38:B64

Haiti,

Cells C38:C64





a. Set up hypotheses to examine the question of interest.




b. Perform the significance test, and summarize the result.




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.


a. Sample average difference =

Computed as:

33.5D



Computed as:

33.5D

niYXD iii ,,1,



Computed as:

33.5D

niYXD iii ,,1,



Computed as:

Then average

33.5D

niYXD iii ,,1,



Some evidence factory is bigger,

is it strong evidence???

33.5D





Let = Difference: Haiti – Factory

33.5D

D





Let = Difference: Haiti – Factory

1-sided, from “idea of loss”

33.5D

D0:0 DH

0: DAH


b. 0|..33.5 DcmorDPvalueP



0|33.5 DDP



0|33.5 DDP

D

DD nsnsD

P |33.5



0|33.5 DDP

D

DD nsnsD

P |33.5

D

Dn nstP |33.5

1

2 Paired SamplesE.g. Old Textbook 7.32:b.

D

Dn nstPvalueP |33.5

1


But recall how TDIST works

(1 – tail: upper probability only)

D

Dn nstPvalueP |33.5

1


But recall how TDIST works:

=

(symmetry)

D

Dn nstPvalueP |33.5

1


But recall how TDIST works:

=

So compute with:

D

Dn nstPvalueP |33.5

1

DD

n nstPvalueP |33.5

1


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




Standard deviation

of differences, sD

P-value




P-value = 1.87 x 10-5




P-value = 1.87 x 10-5

Interpretation: very strong evidence




P-value = 1.87 x 10-5

Interpretation: very strong evidence

either yes-no or gray level

2 Paired SamplesVariations:

1. EXCEL function TTEST is useful here





(same answer as above)



Notes:

a. Type is paired (discuss others later)



Notes:


b. Get same answer from swapping Array 1 and Array 2



Notes:





Notes:



c. This is something Excel does well


2. Can also use:

Data Data Analysis T-test Paired


2. Can also use:


to give detailed results


2. Can also use:



e.g. d.f. = 26


2. Can also use:



e.g. d.f. = 26

P-value same


2. Can also use:



e.g. d.f. = 26

P-value same

(others we haven’t learned yet)


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




Margin of error =

(same as above, but NORMINV TINV)

ns

nTINVm 1,05.0




Margin of error =

(same as above, but NORMINV TINV)

So CI has endpoints:

ns

nTINVm 1,05.0

mX

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)

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?


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.


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.


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.


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.


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!


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.


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 –

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



Most highly sought:


• Please fill out with # 2 pencil (black pen?)



Most highly sought:



• Return to student volunteer

• Will turn in independently from me



Most highly sought:



• 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: 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)

Documents

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