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

Last Time

• Interpretation of Confidence Intervals

• Handling unknown μ and σ

• T Distribution

• Compute with TDIST & TINV

(Recall different organization)

(relative to NORMDIST & NORMINV)

Reading In Textbook

Approximate Reading for Today’s Material:

Pages 420-427, 86-94

Approximate Reading for Next Class:

Pages 101-105 , 447-465, 511-516

Deeper look at Inference

Recall: “inference” = CIs and Hypo Tests

Main Issue: In sampling distribution

Usually σ is unknown, so replace with an estimate, s.

For n large, should be “OK”, but what about:

• n small?

• How large is n “large”?

nNX /,0~

Unknown SD

Then

So can write:

Replace by , then

has a distribution named:

“t-distribution with n-1 degrees of freedom”

nNX /,~

1,0~ N

n

X

sn

sX

t - Distribution

Notes:

4. Calculate t probs (e.g. areas & cutoffs),

using TDIST & TINV

Caution: these are set up differently from NORMDIST & NORMINV

EXCEL Functions

Summary:

Normal:

plug in: get out:

NORMDIST: cutoff area

NORMINV: area cutoff

(but TDIST is set up really differently)

EXCEL Functions

t distribution:

Area

2 tail:

plug in: get out:

TDIST: cutoff area

TINV: area cutoff

(EXCEL note: this one has the inverse)

t - Distribution

Application 1: Confidence Intervals

t - Distribution


Recall: mX

t - Distribution


Recall:

margin of error

mX

t - Distribution


Recall:

margin of error

from NORMINV

mX

t - Distribution


Recall:

margin of error

from NORMINV

or CONFIDENCE

mX

t - Distribution


Recall:

margin of error

from NORMINV

or CONFIDENCE

Using TINV?

mX

t - Distribution


Recall:

margin of error

from NORMINV

or CONFIDENCE

Using TINV? Careful need to standardize

mX

t - DistributionUsing TINV? Careful need to standardize


mXmXbyveredcoP ,95.0



mXmXP


# spaces on number line


mXmXP

mXP



Need to work into use TINV


mXmXP

mXP ns



Need to work into use TINV


mXmXP

mXP

ns

mns

XP

ns

t - Distribution

ns

mns

XP

95.0

t - Distribution

distribution

ns

mns

XP

95.0

nsX

t - Distribution

distribution

ns

mns

XP

95.0

nsm

nsX

t - Distribution

distribution

So want:

ns

mns

XP

95.0

nsm

nTINV )1,05.0( nsm

nsX

t - Distribution

distribution

So want:

i.e. want:

ns

mns

XP

95.0

nsm

nTINV )1,05.0(

ns

nTINVm )1,05.0(

nsm

nsX

t - Distribution

Class Example 15, Part Ihttp://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg15.xls

Old text book problem 7.24

http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg15.xls

t - Distribution


Old text book problem 7.24:

In a study of DDT poisoning, researchers fed several rats a measured amount. They measured the “absolutely refractory period” required for a nerve to recover after a stimulus. Measurements on 4 rats gave:


t - Distribution


Old text book problem 7.24:

Measurements on 4 rats gave:

1.6 1.7 1.8 1.9

a) Find the mean refractory period, and the standard error of the mean

b) Give a 95% CI for the mean “absolutely refractory period” for all rats of this strain


t - Distribution

Class Example 15, Part I

Data in cells B9:E9

t - Distribution


Data in cells B9:E9

Note: small sample size (n = 4)

t - Distribution


Data in cells B9:E9

Note: small sample size (n = 4),

population sd, σ, unknown

t - Distribution


Data in cells B9:E9


population sd, σ, unknown,

so use sample sd, s

t - Distribution


Data in cells B9:E9


population sd, σ, unknown,

so use sample sd, s,

and t distribution

t - Distribution


Data in cells B9:E9

Center CI at Sample Mean

t - Distribution


Data in cells B9:E9


Measure Sample Spread by S. D.

t - Distribution


Data in cells B9:E9



Divide by to get Standard Errorn

t - Distribution


Data in cells B9:E9



Divide by to get Standard Error

Which

answers (a)

n

t - Distribution

Class Example 15, Part I (b) 95% CI for μ

Data in cells B9:E9

t - Distribution


Data in cells B9:E9

CI Radius = Margin of Error

t - Distribution


Data in cells B9:E9


Compute using TINV

t - Distribution


Data in cells B9:E9


Recall:

d.f. = n – 1

= 4 – 1

t - Distribution


Data in cells B9:E9


Compare to old Normal CIs

t - Distribution


Data in cells B9:E9



Compute using

CONFIDENCE

t - Distribution


Data in cells B9:E9



T CIs are wider

t - Distribution


Data in cells B9:E9



T CIs are wider

(as expected)

t - Distribution


Data in cells B9:E9



Left End

t - Distribution


Data in cells B9:E9



Left End

Right End

t - Distribution

Confidence Interval HW:

7.24 (a. Q-Q roughly linear, so OK, b. 43.17, 4.41, 0.987 c. [41.1, 45.2])

7.25

And now for something completely different

An extreme “sport” video:

t - Distribution

Application 2: Hypothesis Tests

t - Distribution


Idea: Calculate P-values using TDIST

t - Distribution


Idea: Calculate P-values using TDIST

t – Distribution Hypo Testing

E.g. Old Textbook Example 7.26

For the above DDT poisoning example




Recall Data in cells B9:E9





As above: t – distribution appropriate





As above: t – distribution appropriate

(small sample, and using s ≈ σ)



For the above DDT poisoning example, Suppose that the mean “absolutely refractory period” is known to be 1.3



For the above DDT poisoning example, Suppose that the mean “absolutely refractory period” is known to be 1.3

(recall observed in data)



For the above DDT poisoning example, Suppose that the mean “absolutely refractory period” is known to be 1.3. DDT poisoning should slow nerve recovery, and so increase this period.



For the above DDT poisoning example, Suppose that the mean “absolutely refractory period” is known to be 1.3. DDT poisoning should slow nerve recovery, and so increase this period. Do the data give good evidence for this supposition?



Let = population mean absolutely

refractory period for poisoned rats.





(checking strong evidence for

this)

3.1:0 H

3.1: AH





(from before)

3.1:0 H

3.1: AH

75.1X


E.g. Old Textbook Example 7.26 P-value = P{what saw or more conclusive | H0 – HA Bdry}



3.1|75.1 XP



3.1|75.1 XP

3.1|

3.175.1 nsns

XP



3.1|75.1 XP

3.1|

3.175.1 nsns

XP

ns

tP3.175.1

3



ns

tP3.175.1

3



Now use TDIST

ns

tP3.175.1

3



Now use TDIST

ns

tP3.175.1

3



Now use TDIST

ns

tP3.175.1

3



Now use TDIST

ns

tP3.175.1

3



Now use TDIST

Degrees of Freedom = n – 1 = 4 - 1

ns

tP3.175.1

3



Now use TDIST

Tails

ns

tP3.175.1

3

t – Distribution Hypo TestingE.g. Old Textbook Example 7.26

From Class Example 27, part 2:http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg15.xls

P - value = 0.003


t – Distribution Hypo TestingE.g. Old Textbook Example 7.26

From Class Example 27, part 2:http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg15.xls

P - value = 0.003

Interpretation: very strong evidence, for either yes-no or gray-level


t – Distribution Hypo TestingVariations:

• For “opposite direction” hypotheses:

:AH



P-value =

:AH

tP



P-value =

:AH

tP



P-value =

[wrong way for TDIST(…,1)]

:AH

tP



P-value =

Then use symmetry

:AH

tP



P-value =

Then use symmetry, i.e. put - into TDIST.

:AH

tP


• For 2-sided hypotheses


• For 2-sided hypotheses:

H0: μ =

H1: μ ≠



H0: μ =

H1: μ ≠



H0: μ =

H1: μ ≠

Use 2-tailed version of TDIST



H0: μ =

H1: μ ≠

Use 2-tailed version of TDIST,

i.e. TDIST(…,2)


HW: Interpret P-values:

(i) yes-no

(ii) gray-level

7.21e ((i)significant, (ii) significant, but not

very strongly so)

7.22e (0.0619, (i) not significant (ii) not sig.,

but nearly significant)

Research Corner

Another SiZer analysis:

Mollusk Extinction Data

Research Corner



From Matthew Campbell

UNC Master’s Student

In Geological Sciences

Research Corner



Data points:

• Fossilized shells

Research Corner



Data points:


(fossil beds up and down Eastern Seaboard)

Research Corner



Data points:


• Dated (by fossil bed)

Research Corner



Data points:



• Biologically categorized

Research Corner



Data points:




(family – genus – species, etc.)

Research Corner



Data points:




Goal: study extinctions over long periods

Research Corner



Data points:




Goal: study extinctions over long periods

(via last time saw each)

Research Corner



Oversmoothed:

nothing interesting

Research Corner



Undersmoothed:

many bumps appear

Research Corner



Undersmoothed:

many bumps appear

but not statistically significant

Research Corner



Intermediate Smoothing

two bumps appear

Research Corner



Intermediate Smoothing

two bumps appear

SiZer result: not statistically significant

Research Corner



Matthew’s Comment:

Whoah, those are times

of mass extinctions

Research Corner



Matthew’s Comment:

Whoah, those are times

of mass extinctions

Any way to show these are “really there”?

Research Corner



Any way to show these

are “really there”?

Research Corner





Standard Answer:

Get more data

Research Corner





Challenge:

Took 100s of year to get these!

Research Corner





Alternate Approach:

Refined from Genus level to Species

Research Corner



Species level result

Research Corner



Species level result:

Now both bumps are

significant

Research Corner



Species level result:

Now both bumps are

significant

Consistent with Global Climactic Events

Variable Relationships

Chapter 2 in Text


Chapter 2 in Text

Idea: Look beyond single quantities


Chapter 2 in Text

Idea: Look beyond single quantities, to how quantities relate to each other.


Chapter 2 in Text


E.g. How do HW scores “relate”

to Exam scores?


Chapter 2 in Text


E.g. How do HW scores “relate”

to Exam scores?

Section 2.1: Useful graphical device:

Scatterplot

Plotting Bivariate Data

Toy Example: Ordered pairs

(1,2)

(3,1)

(-1,0)

(2,-1)



Captures relationship between X & Y

(1,2) as (X,Y)

(3,1)

(-1,0)

(2,-1)




(1,2) as (X,Y)

(3,1) e.g. (height, weight)

(-1,0)

(2,-1)




(1,2) as (X,Y)

(3,1) e.g. (height, weight)

(-1,0) e.g. (MT Score, Final Exam Score)

(2,-1)


Toy Example:

(1,2) Think in terms of:

(3,1)

(-1,0) X coordinates

(2,-1)


Toy Example:


(3,1)


(2,-1) Y coordinates


Toy Example:


(3,1)


(2,-1) Y coordinates

And plot in x,y plane, to see relationship


Toy Example:

(1,2)

(3,1)

(-1,0)

(2,-1)

Toy Scatterplot, Separate Points

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

-2 -1 0 1 2 3 4

x

y


Toy Example:

(1,2)

(3,1)

(-1,0)

(2,-1)


-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

-2 -1 0 1 2 3 4

x

y


Toy Example:

(1,2)

(3,1)

(-1,0)

(2,-1)


-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

-2 -1 0 1 2 3 4

x

y


Toy Example:

(1,2)

(3,1)

(-1,0)

(2,-1)


-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

-2 -1 0 1 2 3 4

x

y


Sometimes:

Can see more

insightful patterns

by connecting

points

Toy Scatterplot, Connected points

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

-2 -1 0 1 2 3 4

x

y


Sometimes:

Useful to switch off

points, and only

look at lines/curves

Toy Scatterplot, Lines Only

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

-2 -1 0 1 2 3 4

x

y


Common Name: “Scatterplot”



A look under the hood in Excel



A look under the hood in Excel:

Insert Tab




Insert Tab

Charts




Insert Tab

Charts

Scatter Button




Insert Tab

Charts

Scatter Button

Choose Dots




Insert Tab

Charts

Scatter Button

Choose Dots

(but note other options)




Insert Tab

Charts

Scatter Button

Choose Dots

Manipulate plot as done before for bar plots




Insert Tab

Charts

Scatter Button

Choose Dots

Manipulate plot as done before for bar plots

(e.g. titles, labels, colors, styles, …)

Scatterplot E.g.Class Example 16:


Data from related Intro. Statistics Class





(actual scores)





(actual scores)

A. How does HW score predict Final Exam?





(actual scores)


xi = HW, yi = Final Exam





(actual scores)



(Study Relationship

using scatterplot)



How does HW score predict Final Exam?





(Scatterplot View)




i. In top half of HW scores




i. In top half of HW scores:Better HW Better Final





ii. For lower HW





ii. For lower HW:Final is more “random”

Scatterplots

Common Terminology:

When thinking about “X causes Y”,

Scatterplots

Common Terminology:


Call X the “Explanatory Var.”

Scatterplots

Common Terminology:


Call X the “Explanatory Var.” or “Indep. Var.”

Scatterplots

Common Terminology:



Call Y the “Response Var.”

Scatterplots

Common Terminology:



Call Y the “Response Var.” or “Dep. Var.”

Scatterplots

Common Terminology:




(think of “Y as function of X”)

Scatterplots

Common Terminology:




(think of “Y as function of X”)

(although not always sensible)

Scatterplots

Note: Sometimes think about causation

Scatterplots

Note: Sometimes think about causation,

Other times: “Explore Relationship”

Scatterplots

Note: Sometimes think about causation,

Other times: “Explore Relationship”

HW: 2.9

Class Scores Scatterplotshttp://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg16.xls

B. How does HW predict Midterm 1?

xi = HW, yi = MT1




xi = HW, yi = MT1

(Replace Final above

with 1st Midterm)




xi = HW, yi = MT1




xi = HW, yi = MT1

i. Better HW better Exam

(general upwards

tendency still

the same)




xi = HW, yi = MT1


ii. Wider range MT1 scores

(for each range

of HW scores)

(relative to final scores)




xi = HW, yi = MT1



iii. HW doesn’t predict MT1

(as well as HW

predicted the final)




xi = HW, yi = MT1




iv. “Outliers” in scatterplot




xi = HW, yi = MT1




iv. “Outliers” in scatterplot

e.g. HW = 72, MT1 = 94




xi = HW, yi = MT1




iv. “Outliers” in scatterplot may not be outliers in either individual variable

e.g. HW = 72, MT1 = 94

(bad HW, but good MT1?, fluke???)



C. How does MT1 predict MT2?

xi = MT1, yi = MT2




xi = MT1, yi = MT2

(Different choice of x and y, since

studying different relationship)




xi = MT1, yi = MT2

(Study Relationship

using tool of

scatterplot)




xi = MT1, yi = MT2

i. Idea: less “causation”, more “exploration”

(don’t expect better MT1

to lead to better MT2)




xi = MT1, yi = MT2


ii. High MT1 High MT2

(again clear overall

upwards trend)




xi = MT1, yi = MT2



iii. Wider range of MT2

(for each range of MT1)




xi = MT1, yi = MT2




i.e. “not good predictor”

(MT1) (of MT2)




xi = MT1, yi = MT2





iv. Interesting Outliers




xi = MT1, yi = MT2





iv. Interesting Outliers:MT1 = 100, MT2 = 56 (oops!)




xi = MT1, yi = MT2





iv. Interesting Outliers:MT1 = 100, MT2 = 56

MT1 = 23, MT2 = 74 (woke up!)




xi = MT1, yi = MT2





iv. Interesting Outliers:MT1 = 100, MT2 = 56

MT1 = 23, MT2 = 74

MT1 70s, MT2 90s (moved up!)


And now for something completely different

A thought provoking movie clip:

http://www.aclu.org/pizza/

http://www.aclu.org/pizza/

Important Aspects of Relations

I. Form of Relationship

(Linear or not?)



II. Direction of Relationship

(trending up or down?)




III. Strength of Relationship

(how much of data “explained”?)

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”





But saw others with

“rough linear trend”





But saw others with






But saw others with


Interesting question:

Measure strength of

linear trend





• Nonlinear: Data follows different pattern

(non-linear)





• Nonlinear: Data follows different pattern

Nice Example: Bralower’s Fossil Data




Bralower’s Fossil Datahttp://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg17.xls

From T. Bralower, formerly of Geological Sci.


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





Studies Global Climate



Studies Global Climate, millions of years ago



Studies Global Climate, millions of years ago:

• Small shells from ocean floor cores





• Ratios of Isotopes of Strontium






• Reflects Ice Ages, via Sea Level







(50 meter difference!)








• As function of time








• As function of time

• Clearly nonlinear relationship


• Positive Association



X bigger Y bigger



X bigger Y bigger

E.g. Class Scores Data above



X bigger Y bigger

• Negative Association



X bigger Y bigger


X bigger Y smaller



X bigger Y bigger


X bigger Y smaller

E.g. X = alcohol consumption, Y = Driving Ability

Clear negative association



X bigger Y bigger


X bigger Y smaller

Note: Concept doesn’t always apply:



X bigger Y bigger


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




• Final Exam is “closely related to HW”





• Midterm 1 less closely related to HW






• Midterm 2 even less related to Midterm 1






• Midterm 2 even less related to Midterm 1

Interesting Issue:

Measure this strength

Linear Relationship HW

HW:

2.11, 2.13, 2.15, 2.17

Documents

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