Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute...

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

So can write:

Replace by , then

has a distribution named:

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

nNX /,~

1,0~ N

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:

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

t - Distribution

Recall:

margin of error

from NORMINV

t - Distribution

Recall:

margin of error

from NORMINV

or CONFIDENCE

t - Distribution

Recall:

margin of error

from NORMINV

or CONFIDENCE

Using TINV?

t - Distribution

Recall:

margin of error

from NORMINV

or CONFIDENCE

Using TINV? Careful need to standardize

t - DistributionUsing TINV? Careful need to standardize

mXmXbyveredcoP ,95.0

# spaces on number line

Need to work into use TINV

mXP ns

Need to work into use TINV

t - Distribution

distribution

t - Distribution

distribution

t - Distribution

distribution

So want:

nTINV )1,05.0( nsm

t - Distribution

distribution

So want:

i.e. want:

nTINV )1,05.0(

nTINVm )1,05.0(

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

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

answers (a)

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])

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

(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

(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

3.1:0 H

3.1: AH

(from before)

3.1:0 H

3.1: AH

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

3.1|75.1 XP

3.175.1 nsns

3.1|75.1 XP

3.175.1 nsns

tP3.175.1

Now use TDIST

tP3.175.1

Now use TDIST

tP3.175.1

Now use TDIST

tP3.175.1

Now use TDIST

tP3.175.1

Now use TDIST

Degrees of Freedom = n – 1 = 4 - 1

tP3.175.1

Now use TDIST

tP3.175.1

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:

P-value =

[wrong way for TDIST(…,1)]

P-value =

Then use symmetry

P-value =

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

• 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

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

(-1,0)

(2,-1)

Captures relationship between X & Y

(1,2) as (X,Y)

(-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:

(-1,0) X coordinates

(2,-1)

Toy Example:

(2,-1) Y coordinates

Toy Example:

(2,-1) Y coordinates

And plot in x,y plane, to see relationship

Toy Example:

(-1,0)

(2,-1)

Toy Scatterplot, Separate Points

-2 -1 0 1 2 3 4

Toy Example:

(-1,0)

(2,-1)

-2 -1 0 1 2 3 4

Toy Example:

(-1,0)

(2,-1)

-2 -1 0 1 2 3 4

Toy Example:

(-1,0)

(2,-1)

-2 -1 0 1 2 3 4

Sometimes:

Can see more

insightful patterns

by connecting

points

Toy Scatterplot, Connected points

-2 -1 0 1 2 3 4

Sometimes:

Useful to switch off

points, and only

look at lines/curves

Toy Scatterplot, Lines Only

-2 -1 0 1 2 3 4

Common Name: “Scatterplot”

A look under the hood in Excel

A look under the hood in Excel:

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:

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

Data from related Intro. Statistics Class

(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

(Replace Final above

with 1st Midterm)

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

(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/

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

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.

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

• Clearly nonlinear relationship

• Positive Association

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

Interesting Issue:

Measure this strength

Linear Relationship HW

2.11, 2.13, 2.15, 2.17

Last Time Interpretation of Confidence Intervals Handling unknown μ and σ T Distribution Compute...

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